Cédric Bain

Peter Davies

Guilhem Bles

Yann Marco

Julien Barnet

Influence of bedding-in on the tensile performance of HMPE fiber ropes

Keywords: HMPE, rope, bedding-in, stretched, fiber, braided

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Fiber ropes have been used for thousands of years, thanks to their excellent tensile strength, environmental resistance and light weight. Synthetic materials such as polyester, polyamide, aramid, and polyethylene are newcomers in the history of fiber ropes, and have extended the range of possible applications. They offer the potential to replace many steel wire ropes and chains, especially for mooring lines. However, synthetic fiber rope behavior is more complex than that of steel wire ropes; this is due to both the complex behavior of polymer fibers and the hierarchical structure of rope components (Leech, 2002). The most common rope construction used in mooring lines is based on twisted strands, and these have been widely studied [START_REF] Del Vecchio | Light Weight Materials for Deep Water Moorings[END_REF][START_REF] Ghoreishi | Modelisation analytique et caracterisation experimentale du comportement de cables synthetiques[END_REF][START_REF] Raoof | Analysis of Multilayered Structural Strands[END_REF]. Braided ropes are the other main type of rope construction, and have also been studied in the past decades [START_REF] Davies | OTC 14246 Synthetic Mooring Lines for Depths to 3000 Meters[END_REF][START_REF] Vu | Finite element simulation of the mechanical behavior of synthetic braided ropes and validation on a tensile test[END_REF]. A bedding-in protocol, a sequence of tensile loading to settle the fibers and their geometry, is essential for synthetic fiber ropes. It allows stabilizing the mechanical properties of ropes by applying a known loading history to the rope before use. To understand the consequences of the bedding-in protocol on the rope deformation mechanisms, some researchers have focused on the influence of rope construction on the mechanical behavior [START_REF] Hoppe | Modelling the static loading behavior of Dyneema in wire-rope construction[END_REF][START_REF] Leech | Modelling Tension and Torque Properties of Fibre Ropes and splices[END_REF][START_REF] Raoof | Analysis of Multilayered Structural Strands[END_REF], and others on the role of loading history [START_REF] Casey | OTC 12177 Short-and Long-Term Property Behaviour of Polyester Rope[END_REF][START_REF] Davies | OTC 14246 Synthetic Mooring Lines for Depths to 3000 Meters[END_REF][START_REF] Del Vecchio | Light Weight Materials for Deep Water Moorings[END_REF]. These two points concerning the construction and the history effect need further investigation.

Many investigations have focused on the understanding of the geometry of the rope, the interactions between each scale and the behavior of smaller scales, in order to be able to model numerically or empirically the mechanical behavior of synthetic fiber ropes. For example, [START_REF] Ghoreishi | Analytical modeling of synthetic fiber ropes. Part I: A linear elastic model for 1 + 6 fibrous structures[END_REF] developed a nonlinear elastic continuum model based on various earlier models for steel and synthetic stranded ropes such as those of [START_REF] Hoppe | Modelling the static loading behavior of Dyneema in wire-rope construction[END_REF][START_REF] Leech | Modelling Tension and Torque Properties of Fibre Ropes and splices[END_REF][START_REF] Raoof | Analysis of Multilayered Structural Strands[END_REF].

These models are based on the geometries and the properties of each scale, and friction is also considered in some of them. [START_REF] Davies | OTC 17872 Tension / Torsion Behavior of Deepwater Synthetic Mooring Lines[END_REF] applied Leech's model and found a good correlation of tension/torsion properties for results on full size long rope samples. Braided ropes have received less 3 attention than stranded ropes, but some researchers have investigated behavior models, in particular [START_REF] Rawal | Geometrically controlled tensile response of braided sutures[END_REF][START_REF] Vu | Finite element simulation of the mechanical behavior of synthetic braided ropes and validation on a tensile test[END_REF]. The development of finite element simulations and other analytical models are helpful to understand the complex behavior of fiber ropes and the mechanisms of the bedding-in process. These must account for internal friction between fibers and the reorganization of fibers under the load.

The bedding-in process and loading history effects have been studied in some detail during the development of the use of polyester ropes for floating platform mooring lines in the offshore industry. It is generally agreed that a bedding-in process will change the mechanical properties of ropes significantly.

For [START_REF] Banfield | Evaluation of fibre rope properties for offshore mooring[END_REF] the bedding process has to be recorded to interpret rope properties correctly. Work by [START_REF] Casey | OTC 12177 Short-and Long-Term Property Behaviour of Polyester Rope[END_REF][START_REF] Davies | OTC 14246 Synthetic Mooring Lines for Depths to 3000 Meters[END_REF][START_REF] Del Vecchio | Light Weight Materials for Deep Water Moorings[END_REF][START_REF] Hooker | Recent Investigation into the Physsical Properties of Superline Polyester Ropes[END_REF][START_REF] Liu | An experimental investigation on nonlinear behaviors of synthetic fiber ropes for deepwater moorings under cyclic loading[END_REF][START_REF] Weller | Synthetic rope responses in the context of load history : Operational performance[END_REF], has led to the conclusion that the mean load and the strain amplitude are the two main parameters which influence the mechanical behavior of ropes under cyclic loads. As [START_REF] Casey | OTC 12177 Short-and Long-Term Property Behaviour of Polyester Rope[END_REF] concluded, the dynamic stiffness after a full bedding-in process remains constant.

However, if the permanent strain, due to molecular rearrangement in the fibers and geometric construction changes, is not removed by the bedding-in sequence it will result in lower stiffness values. By understanding the effects of the bedding-in, simplified behavior models for ropes can be proposed; [START_REF] Davies | OTC 14246 Synthetic Mooring Lines for Depths to 3000 Meters[END_REF] developed an empirical expression for evaluating the dynamic stiffness of HMPE ropes. [START_REF] Liu | An experimental investigation on nonlinear behaviors of synthetic fiber ropes for deepwater moorings under cyclic loading[END_REF] further developed these expressions in order to obtain the dynamic stiffness from small size ropes and extrapolate to larger ones, and also to take into account the mean load, strain amplitude and number of loading cycles. (Lian et al., 2018a) concluded that different bedding-in levels can cause different values of strain, initial creep and dynamic stiffness for the same test samples. For this reason the investigation of the bedding-in behavior of synthetic fiber ropes is essential for most applications. Even if there is agreement on the need for a bedding-in procedure to stabilize rope behavior before measuring stiffness values, there are different recommendations: (American Bureau of Shipping (ABS), 2011;Bureau Veritas, 2007;International Standards Organization (ISO), 2004). Bedding-in protocols are not the same according to each guideline, but they have the same purpose, which is to stabilize the rope length and to improve rope and termination efficiencies. [START_REF] Del Vecchio | Light Weight Materials for Deep Water Moorings[END_REF]) used a bedding-in loading sequence of 1000 cycles between 10% and 30% of the minimum breaking load (MBL). [START_REF] Casey | OTC 17068 Factors Affecting the Measurement of Axial Stiffness of Polyester Deepwater Mooring Rope Under Sinusoidal Loading[END_REF]) concluded that at least 10000 cycles are required to get nearer to a fully bedded-in state, but they recommend that at least 500 cycles be applied. [START_REF] Lechat | Comportement mécanique de fibres et d'assemblages de fibres en polyester pour cables d'amarrage de plates-formes offshore[END_REF] followed a recommendation of the Oil Companies International Marine Forum (OCIMF) for bedding-in before testing which includes five steps of 5min at 50% of the MBL (minimum break load) each followed by a recovery step at 2% of the MBL. [START_REF] Humeau | Contribution to the study of coupling between moisture diffusion and mechanical stress, in high performance marine[END_REF] applied 5 cycles between 2% and 50% of the MBL. These different studies show how numerous bedding-in sequences are used and that there is not a unified procedure. That is why (Lian et al., 2018a) presented a quantitative method for determining the beddingin level of synthetic fiber ropes based on the tension-strain curve for static tensile loading. The permanent elongation [START_REF] Francois | Fibre rope deep water mooring: a practical model for the analysis of polyester mooring systems[END_REF][START_REF] Humeau | Contribution to the study of coupling between moisture diffusion and mechanical stress, in high performance marine[END_REF] of synthetic fiber ropes is the result of two types of elongation: the rope construction component and the material contribution. The permanent elongation related to the construction can be explained by compaction and the realignment of fibers in the loading direction. [START_REF] Chailleux | A Non-Linear Viscoelastic Viscoplastic Model for the Behaviour of Polyester Fibres[END_REF][START_REF] Humeau | Contribution to the study of coupling between moisture diffusion and mechanical stress, in high performance marine[END_REF][START_REF] Williams | Small-Scale Testing of Damaged Synthetic Fiber Mooring Ropes[END_REF] have included these contributions in elasto-viscoelastic-plastic models, which they applied to polyester and aramid yarns. [START_REF] Williams | Small-Scale Testing of Damaged Synthetic Fiber Mooring Ropes[END_REF] have observed that for a basic response for each sub-scale, the response of the rope is not the same due to the construction influence.

Most of the published work on bedding-in has been focused on polyester, polyamide and aramid fibers. Very few authors, with the exception of [START_REF] Costa | Modelling creep tests in HMPE fibres used in ultra-deep-sea mooring ropes[END_REF][START_REF] Davies | Mechanical behaviour of HMPE and aramid fibre ropes for deep sea handling operations[END_REF][START_REF] Humeau | Tension fatigue failure prediction for HMPE fi bre ropes[END_REF][START_REF] Lian | An investigation on internal damping behavior of fiber rope[END_REF]Lian et al., , 2018b[START_REF] Lian | A creep -rupture model of synthetic fiber ropes for deepwater moorings based on thermodynamics[END_REF] have considered HMPE. To the authors' knowledge no previous studies have presented results for ropes based on the Dyneema™ SK99 HMPE grade. This grade was specially developed for the nautical industry and shows high tensile stiffness and strength. Table 1 shows nominal properties of different grades of Dyneema™ [START_REF] Vlasblom | 18 -The manufacture, properties, and applications of high-strength, high-modulus polyethylene fibers[END_REF]. There are also few published test results from tests at different scales of a rope construction, and these are essential in order to establish the influence of load history on each one. In the present study different input parameters have been varied, such as maximum load during bedding-in; this has been done previously, but here the loading history path was also varied in order to understand its contribution.

In summary, bedding-in is primordial for synthetic fiber ropes, and a thorough understanding of the mechanisms involved is very important for many marine applications. For example, neither tensioning of sailboat rigging nor the installation of offshore mooring lines is possible without a good estimation of elongation during first loading. The aim of the present study is to separate the construction and material contributions of the bedding-in process to the mechanical behavior of HMPE braided ropes. Different bedding-in sequences will be applied to investigate the influence of the loading history on the mechanical behavior. First, the material and experimental devices will be presented, followed by the definition of the bedding-in protocols. The stress-strain curves from static tensile tests for each scale of synthetic fibers ropes will then be presented. Based on the test results, the discussion will provide conclusions about the optimal bedding-in process and the influence of rope construction.

HMPE

a. Material

The synthetic fiber ropes tested in the present study were made of Dyneema TM SK99 fibers. The rope was braided with a 16-strand construction and a nominal diameter of 4mm. A coating was applied by the manufacturer. The samples were provided by the Lancelin Corderie (Ernée, France). The results from these small diameter ropes can be used as a first approach towards understanding the behavior of bigger ropes, such as the ones used in offshore platform mooring lines.

As many researchers have shown previously, the geometrical structure of the rope is one of the major factors in determining the mechanical elongations that occur during bedding-in process. In order to rope levels, Figure 1. Pin Rope [START_REF] Leech | Modelling Tension and Torque Properties of Fibre Ropes and splices[END_REF][START_REF] Milne | An assessment of the strength of knots and splices used as eye terminations in a sailing environment[END_REF] have studied the best way to terminate a braided rope, and eye splices provide the maximal strength. Rope samples were loaded with steel pins with a diameter of 20mm.

Figure 3. Rope sample geometry.

A local strain measurement system was used in order to avoid measuring the slip that will occur in the end termination during loading. For the case of the rope, a RETIGA 6000 camera and NI Vision Assistance software allowed the movement of two markers, fixed to the rope, to be followed.

Each rope sample was weighed and its length measured before making the splices, in order to determine its linear density.

The initial dimensions of the rope samples were 10-20 cm of usable length in the central section, , each splice length measured 30-35cm with an eye length around 3.5cm, giving a total length of 87 cm, Figure 3.

Break loads were also determined: Eight monotonous loading tests until failure were performed to obtain a mean breaking load. The mean loading rate was 1%/min. The mean rope breaking load obtained was 14.7kN, with a standard deviation of 786N.

The linear density of the rope was measured and a value of 8.685 g/m was obtained.

Despite the fact that the strains are measured directly in the gauge region, the vicinity of the splices may have an effect on the strain measured. To investigate the influence of the distance from the splice on the strain measured, additional tests have been performed on a Dyneema SK99 rope with a diameter of 4mm, as shown in Figure 4. In order to observe the influence of the splices on the measured strain, different stress-strain curves are plotted.

Figure 6. Stress-Strain results for different markers Figure 6. shows the influence of the marker position on the measured strain. When measurements are made directly from the end of the splice a stiffening effect is noted (curve 1-6), whereas measurements away from the splice end (the other 4 curves in Figure 6) provide very similar stress-strain plots, within experimental scatter. Thus the strain field is certainly affected by the splices but provided markers are placed even a small distance away (as was the case for the values shown in this paper) the influence on measured strain is small. These are results from only one test but they do indicate that strain field is uniform over most of the gauge length.

ii. Yarn

The test machine used for yarns was an INSTRON model 5566, with a 500N load cell, Figure 7. The experimental setup is similar to the one used for the rope. Each marker was followed by a camera to determine the local deformation. Special pneumatic clamps were used to grip the samples, as shown in In the same way as for the ropes, the mean breaking load of the yarns was measured on 10 samples to be 277N (with a standard deviation of 10.1N). The mean loading rate was 0.169 %/min. Linear density was measured to be 9.73.10 -2 g/m, the gauge length of samples was 250mm.

iii. Filament

The same INSTRON 5566 test machine was used for filaments, but with a 50N load cell. This load cell was calibrated in 2018. Its accuracy is of +/-0.0021N for a 1N load. Therefore for a MBL of 0.402, the relative uncertainty is of +/-0.5%.

Figure 8. Attachment method for filament samples.

The filaments were extracted from the yarns. After extraction of the filament, each end of the latter was adhesively bonded between two pieces of cardboard. Pneumatic clamps were used to grip the cardboard (cf. Figure 8).

In order to determine the linear density of the filament, the section of a yarn was first observed by digital microscopy (KEYENCE TM ) to count the number of filaments. The linear density of the filament was then determined as follows:

= !" #$% &' '( ) $!*+ (! ) ,)%!
The difficulty of filament extraction and the very small filament diameter make fixing markers to the specimen difficult. Several preliminary tests were performed with local markers and these showed that there was no slip in the clamps. For subsequent tests, the clamp movement was therefore used to measure the filament strain.

In the same way as for the ropes, the mean breaking load of the filaments was measured to be 0.402N (12 samples with a standard deviation of 0.059). The mean loading rate was 5%/min. Linear density was 1.24.10 -4 g/m, the gauge length of filament samples was around 95mm. c. Definitions and Protocols i. Parameters

Figure 9 shows the parameters studied here.

Figure 9. Extracted data for the case of load/unload bedding-in (BI) protocol.

ii. Stress and Strain

As noted previously, synthetic fiber ropes have complex structures. The Cauchy stress based on the sample cross-section will be difficult to use in this study. A specific stress Σ . [START_REF] Costa | Modelling creep tests in HMPE fibres used in ultra-deep-sea mooring ropes[END_REF][START_REF] Fer | Thermodynamique macroscopique[END_REF], is preferred, which is based on the linear density of the rope:

Σ . = 1 / .
With: • / . : Cauchy stress tensor in Pa

In the 1D case (rope case), this specific stress leads to:

Σ = 0 ̿
With:

• ̿ : linear density in kg/m

• 0: Tensile Force in N

The S.I. specific stress unit is N.m/kg or J/kg. In 1D case, an equivalent unit is preferred, N/tex:

1 2 3 = 10 45 6 *$7 = 1 8) 9 3 5 : ;
Where 1 *$7 = 1 3/= .This specific stress will allow a comparison between the behaviors at each rope scale.

The nominal strain > will be used for this study:

> = ∆ @ = -@ @
With:

• @ : Initial length • : current length
For each scale, the detailed experimental procedure will be described.

iii. Bedding-in Protocols Three bedding-in protocols were studied: 1 cycle (load/unload), 10 cycles (load/unload) and relaxation. On figure 7, the loading path of each test sequence is plotted. After each of these tests, a 1-hour recovery step was applied, in order to remove the visco-elastic behavior resulting from the previous load.

The effects of bedding-in maximal load are investigated. The different bedding-in maximal loads were chosen as follows, based on the industrial use of this type of rope: 10%, 30% and 50% of the MBL (Mean Breaking Load) HMPE rope. Most guidance notes for synthetic fiber ropes recommend a maximal 13 bedding-in load of 50% of the MBL of the rope. However, for large diameters or very long ropes such as mooring lines, test machines may not be able to reach this level. For this reason investigations into the effects of lower maximal loads for a bedding-in protocol are important.

Figure 10. Bedding-in protocols studied

As shown in Figure 10, the different paths between point O and point A describe the different types of bedding-in chosen here. The time between these two points is the same for each type of bedding-in, in order to avoid any bias from visco-elastic effects. Also the relaxation time is the same as the time to perform 9 load/unload cycles.

These bedding-in protocols were applied to the different components of the HMPE rope: the filament the yarn, and the rope itself. Due to the difficulty in keeping the strand together after extracting it from the rope, that scale was not tested.

After the 1h recovery following the bedding-in protocol, a monotonous uniaxial loading up to failure was applied to the specimen to characterize its mechanical behavior. The extracted data were: the initial modulus A , the maximal tensile strength, the post bedding-in strain, and the maximal strains without and after bedding-in protocols (Figure 10).

Henceforth in this study, the bedding-in protocols will be called BI protocols, and simplified by the following acronyms:

• LU : one Load/Unload

• 10cLU : 10 cycles Load/Unload

• R : load, Relaxation and unload 14

The optimum bedding-in protocol has several criteria to satisfy:

• Be fast

• Minimize bedding-in loading level (particularly for large ropes)

• Stabilize the mechanical response by eliminating the construction effect

• Introduce no damage (No reduction of the maximal stress).

II. Results

a. Results without BI

Figure 11 shows the tensile test results without any bedding-in. The following figures present the mean data from the experimental bedding-in procedures described previously. These figures show the strong influence of bedding-in on the subsequent mechanical behavior of the braided rope. The BI load level also has an important role. The stress-strain curves after BI, at 30% and 50%MBL load levels, are very close and mostly linear. The stress-strain curves after BI at 10%MBL load level, are between the without-BI curve and the 30-50%MBL curves. The behavior after 10%MBL BI is not linear. This shows the gap between the samples with and without BI loading.

The results in Figure 12 show that the subsequent tensile behavior of the rope after BI sequences 10cLU, LU, and R, is almost the same; this means that the loading path is not important. The main parameter is the maximal load in Figure 9.d.

c. Comparison

In this section, the effects of each of the different bedding-in protocols will be compared for each scale (rope, yarn, and filament). Figure 13 presents the corresponding results. The use of a specific stress allows the different scales to be compared directly. The post bedding-in strain is the strain due to the bedding-in loading (cf. post-BI strain description at figure 6). The same relative classification of the post bedding-in strain for each BI was expected. It has been observed previously [START_REF] Davies | OTC 17872 Tension / Torsion Behavior of Deepwater Synthetic Mooring Lines[END_REF][START_REF] Humeau | Contribution to the study of coupling between moisture diffusion and mechanical stress, in high performance marine[END_REF][START_REF] Liu | An experimental investigation on nonlinear behaviors of synthetic fiber ropes for deepwater moorings under cyclic loading[END_REF] that the rope's construction influences the bedding-in behavior. The fact that filament post bedding-in strain is not zero shows that the material (fiber) history also influences the global behavior.

Scanning Electron Microscopy (SEM) observations were made on SK99 filaments (Figure 14). The coating is visible, and is not continuous along the filament. Fibrils can also be observed; these fibrils are a subscale of the filament microstructure. Different interpretations of the filament post bedding-in strain can be made. The first is to conclude that fibrils are being reoriented along the tension axis during the process.

After the reorientation of the fibrils, a permanent post bedding-in strain occurs. The second possibility is that we assume that the fibrils are perfectly oriented along their neutral fiber; the post BI strain is then due to perfect plasticity of the material. The third and the most likely hypothesis is a combination of these two possibilities, which several studies have assumed [START_REF] Davies | Mechanical behaviour of HMPE and aramid fibre ropes for deep sea handling operations[END_REF]. To verify this assumption, similar experimental tests would need to be performed at the fibril scale, but this would require very delicate measurements.

Whatever the scale, the post-BI strain increases with increasing BI load level. According to the third assumption, this plastic strain > measured at the end of the recovery step is the sum of a geometrical plastic strain > and a material plastic strain > :

> = > + >
The geometrical plastic strain > is due to the reorganization of fiber sub-scales for the case of rope, yarn and filament. And the material plastic strain > is due to an irreversible molecular rearrangement.

For the rope case, its fiber structure being more complex, the post-BI strain is increasing more than for the yarn and the filament. The material behavior and the rope's structure are both important for stabilizing the behavior of HMPE fiber ropes. The maximum strain is highest for the rope scale, whatever the BI protocol. The complex assembly of fibers increases the strain due to the rearrangement of the structure of fibers. Also the maximal strain is similar for the scale of filament and yarn. This is because the filaments within the yarn are nearly perfectly aligned in the loading direction. The BI protocol reduces the maximum strain significantly.

An interesting observation is that the maximal strain is higher for the filament. The fibrils orientation in the filament may change during the filament extraction. The fibril misalignment caused by the filament extraction will disappear by stretching in the load direction and thus increasing the modulus together with the maximal tensile strength and the maximal strain. The modulus values obtained here were lower than those given by the fiber producer, from 110N/tex for the filament to 136.8N/tex for the yarn compared with 159N/tex for DSM [START_REF] Vlasblom | 18 -The manufacture, properties, and applications of high-strength, high-modulus polyethylene fibers[END_REF]. There is a difference of 14% between the yarn experimental value and the manufacturer's modulus. DSM indicates a yarn linear density in the range 80-88 tex. The linear density measured for the yarn here is 97 tex, that is 9 to 18% higher than the manufacturer's values. This difference can be explained by the presence of the coating on the filaments for the present tests, that increases the linear density. If this difference is taken into account then the experimental initial modulus values obtained for the yarn are in the same range as those of the fibre supplier. The initial modulus values for filament and yarn are at least twice as high as that of the rope. This can be explained by the effect of the construction, the fibers in the rope are stretched in the direction of the tension, but they are not in uniaxial tension. Due to the braided construction the sub-elements will be initially off-axis before reorientation of the rope occurs during tension as shown in Figure 16. Figure 17 shows that the apparent diameter of the rope is decreasing significantly until 30% of the MBL and then stabilizes. The angle C between the strands was determined (cf. Figure 18). This shows the reorientation of the strands towards the rope axis. The evolution of the apparent diameter and the angle C were obtained during monotonous tensile tests on rope samples with a digital camera, which allowed the changes to the rope structure to be quantified on the recorded images using image analysis software. The reduction of the apparent diameter changes in a very similar way to the angle between strands and the neutral fiber of the rope, as shown in Figure 19. Also the plots become horizontal, characterizing a locking angle. This angle can be estimated from image analysis to be around 31 degrees, and corresponds to a reduction in the apparent diameter of 25%. The relationship between the lay angle of twisted ropes and their mechanical response is well known, but for braided constructions there is no simple expression (Vu et al, 2014). There appears to be a correlation here: Figure 16 shows both the maximal strain drop after different BI load levels and the evolution of the angle between strands and neutral fiber for the same loading. The similarity in the curves indicates that if the geometry is known then the mechanical behavior can be predicted. This suggests that a simple observation of the structural change of the rope during loading can indicate the optimum bedding-in loading level to reach in order to stabilize the mechanical properties of the rope. This is an interesting observation, and further investigation is needed to determine whether it is applicable for other types of rope (material, construction, diameter,…) in order to extend this conclusion.

Figure 19. Evolution of the maximal strain after BI at different load levels (triangle), the apparent diameter (square) and the evolution of the angle between strands with loading (circle).

As a first step, the evolution of the angle C during a monotonous loading has been measured on two other ropes which are:

• A Dyneema SK78 16-strand braided rope with a diameter of 4mm with a MBL of 11.2kN. The construction and coating are very similar to the main SK99 rope of this study.

• A Dyneema SK78 12-strand braided rope with a diameter of 22mm and a MBL of 274kN.

An exponential function was used to fit the experimental data as follows:

'D7E = )$ 4FG + C H
With:

• a and b constants

• C H constant which represents the locking angle The optimization algorithm used for this fitting was a polynomial least squares routine. Then with this method, the locking angle is 30.9° for the Dyneema SK99 rope with a 4mm diameter, 25.6° for the Dyneema SK78 rope with a 4mm diameter and 28.6° for the Dyneema SK78 rope with a 22mm diameter.

We assume that the optimum bedding-in loading level is reached when the realignment of the strands towards the rope axis stops. The criterion to reach this optimum level, was chosen to be:

I JC J0 K L MNO = 0.05 I JC J0 K LQ@
Where C is the strand-strand angle and 0 is the tension load applied on the rope.

Thus the optimum bedding-in loading level is 32.0% of the MBL for the Dyneema SK99 rope with a 4mm diameter, 27.4% for the Dyneema SK78 rope with a 4mm diameter and 22.0% for the Dyneema SK78 rope with a 22mm diameter.

As shown previously, the influence of the bedding-in loading path is small. One load/unload cycle at the optimum loading level is sufficient to stabilize the mechanical properties of the rope.

iv. Maximal tensile strength

According to the experimental data, the maximal tensile strength from the highest to the lowest can be ranked as follows: 1/Filament, 2/Yarn and 3/Rope. This ranking is not surprising; previous studies have already observed stress concentrations in rope construction resulting in lower strength than their constituent yarns (Del Vecchio, 1992) [START_REF] Lechat | Comportement mécanique de fibres et d'assemblages de fibres en polyester pour cables d'amarrage de plates-formes offshore[END_REF]. The filament path in a rope is more complex than in a yarn, and results in stress concentrations between fibers. This phenomenon is sometimes called the bundle effect, and can explain why rope construction has a lower tensile strength than a yarn and a filament.

The BI procedures studied here do not weaken the rope even with a 50% MBL bedding-in load level.

Previous studies have concluded that if the maximal bedding-in load does not exceed 60% of the MBL, fibers are not damaged [START_REF] Francois | Fibre rope deep water mooring: a practical model for the analysis of polyester mooring systems[END_REF]. In addition an increase of 10% of the maximal tensile strength can be observed on the rope scale after BI. As demonstrated previously, the fibers tend to realign in the tensile direction. The bedding-in protocol helps to organize the fibers along the neutral fiber of the rope and the bundle effect is reduced. The maximal tensile strength is slightly higher for the filament than for the yarn. This observation is in agreement with the bundle effect. 4 compares the evolution of post-BI strain, maximal strain and maximal stress depending on the BI load level and the rope scale in the case of the Load/Unload sequence BI protocol. The post-BI strain increases with the BI load level and with higher scale as already observed previously.

The evolution of the maximal tensile stress seems stable in a range of 10% for the filament and yarn scale.

The experimental noise is higher for the yarn and the filament than for the rope as shown by Figure 10.d.

Therefore the results for yarn and for filament can be considered stable. The experimental noise for the maximal tensile stress I the rope is a lot lower, and shows a 10% increase of the maximal tensile stress for the rope after BI. The reorganization of fibers during the BI of a rope decreases the bundle effect and increases the maximal tensile stress.

The maximum strain also seems to reach a maximum after a preload of 30% of the MBL. As observed earlier, the 'locking angle' between the strands is achieved with a 30% preload. This can explain the stabilization of the maximal strain for a preload level higher than 30% of the MBL.

Similar observations can be made for the 10 cycles Load/Unload and relaxation BI protocol.

Figure 21 shows how the rope stiffness changes for the different BI protocols. The change in the tangent modulus with nominal strain is nonlinear without the BI protocol. With a 10%

MBL BI protocol, the initial modulus is higher but still unstable. With 30% and 50 MBL BI protocols, the modulus is more stable than for lower BI load and the initial modulus is still around 30N/tex for both. Also the initial modulus is higher for the 30% and 50% preload levels than for 10%.

The modulus after a BI protocol seems to reach an optimum level. This optimum modulus is especially notable for the relaxation BI case, with a value around 60N/tex. But the tangent modulus does not seem to reach an optimum level for the case of rope without BI.

IV. Conclusion

This paper describes the influence of bedding-in loading on the mechanical performance of braided HMPE fiber rope developed for nautical applications. The results show that in order to improve rope properties significantly a preloading sequence should be applied, with loads at least to 30% of the break load. The loading path is of secondary importance. The improved rope behavior is the result of contributions both at the fiber level (polymer molecule reorganization and alignment), and from the rope construction (alignment and compaction). The observation of the apparent diameter of a rope and the change in the strand orientation during a monotonous loading allow an optimum bedding-in load level to be determined rapidly. This was applied successfully to a larger braid, but it would be very interesting to extend this study to bigger ropes and different rope constructions.

These results are important, because a rope which has not been fully bedded-in may de-tension in service. This may have significant consequences on marine structures for example, a sailing boat mast may be less securely held or a floating offshore platform may require mooring line re-tensioning to limit the platform offset. For the latter, permanent removal of bedding-in strain can also significantly reduce the amount of steel chain required during take-up on winches.

V.

Figure 1

 1 Figure 1. a) The studied braided rope and its components, b) section through new rope before loading, showing strands, yarns and filaments.

Figure 2

 2 Figure 2. a) Rope sample on test machine and b) attachment method by splice.

Figure 4 .

 4 Figure 4. Monotonous loading on rope with different markers

Figure 7 .

 7 Figure 7. Yarn sample on test machine (a) and attachment method (b).

Figure 7

 7 Figure 7.b).

Figure 11 .

 11 Figure 11. Monotonous loading without BI for each scale of the rope

Figure 12

 12 Figure 12. a. Rope results for a maximal BI load of 10% of the MBL.

Figure 12

 12 Figure 12. b. Rope results for a maximal BI load of 30% of the MBL.

Figure 12

 12 Figure 12. c. Rope results for a maximal BI load of 50% of the MBL.

Figure 12

 12 Figure 12. d. Rope results: comparison between a maximal BI load of 10%, 30% and 50% for Relaxation BI.

Figure 13

 13 Figure 13.a. Tension after 10%MBL BI Figure 13.b. Tension after 30%MBL BI

Figure

  Figure 14. Two SK99 filaments

Figure 15 .

 15 Figure 15. SEM image of extracted Dyneema filaments

Figure 16 .

 16 Figure 16. Images of the evolution of the rope's width during tension loading for Dyneema SK99 rope with a diameter of 4mm

Figure 18 .

 18 Figure 18. Change in the angle C between strands during tension loading for Dyneema SK99 rope with a diameter of 4mm. The error bars correspond to the minimum and the maximum angle measurements.

Figure 20 .

 20 Figure 20. Evolution of C for the different rope a) b)

Figure 21

 21 Figure 21.a. Change in tangent modulus for LU BI protocol

Table 2 .

 2 Post-BI strain for each BI and each scale

		% of MBL	% for LU	% for Relax	% for 10cLU
	Filament	10%	0,17	0,45	0,52
		30%	0,4	0,81	0,9
		50%	0,7	1,42	1,05
	Yarn	10%	1,01	0,97	1,06
		30%	1,25	1,13	1,32
		50%	1,41	1,4	1,46
	Rope	10%	1,84	1,93	2,11
		30%	2,56	2,84	2,95
		50%	2,97	3	3,71

Tension after 10 load/unload cycles BI Scales BI Loading Level in Post-BI Strain in Post-BI Strain in Post-BI Strain in

Table 3 .

 3 Table2shows these results for all bedding-in and all scales. Maximal Specific Stress for the three scales In Table3, the linear densities and corresponding maximum specific stress are presented.

	Scales	Linear density in tex	Maximal Specific Stress Without BI, N/tex

Table 4 .

 4 b. Influence of the Load/Unload sequence BI protocol Influence of LU bedding-in at different maximal BI loads for each rope scale Table

	Scale	Filament		Yarn			Rope
	Level of BI	10	30	50	10	30	50	10	30	50
	(% of the MBL)
	Post-BI Strain Without BI	/			/			/
	With BI	0.17	0.4	0.7	1.01	1.25	1.41	1.84	2.56	2.97
	%	/	/	/	/	/	/	/	/	/
	Maximal Strain Without BI	4.7			3.53			4.74
	With BI	4.57	3.57	3.43	2.7	2.72	2.82	3.91	3.1	3.13
	%	-3%	-24% -27% -23% -23% -20% -17% -35% -34%
	Maximal Stress Without BI	3.16			2.86			1.67
	With BI	3.24	3.52	3.14	2.56	2.72	2.85	1.82	1.82	1.83
	%	3%	11%	-1%	-10% -5%	0%	9%	9%	10%


Fabien Lespagnol

Gautier Dakin

High order accurate schemes for Euler and Navier-Stokes equations on staggered Cartesian grids

Keywords: Staggered scheme, High order accurate scheme, Euler equations, Cauchy-Kowalevskaya procedure, Navier-Stokes equations, Cartesian grids

We present here a new class of staggered schemes for solving the compressible 1D Euler equations in internal energy formulation on uniform grids. The schemes are applicable to arbitrary equation of states and can be extended to high order of accuracy in both time and space on smooth flows. High order accurracy in time is reached thanks to Cauchy-Kowalevskaya procedure. Modifications on the initial schemes are performed to give sufficient conditions for stability on 1D wave equations. Results obtained for wave equations are extended to 1D Euler equations and then to 2D compressible Navier-Stokes equations using directional splitting methods. Results on the conservation of total energy are given, proper shock capturing is observed experimentally. Numerical results are provided up to 4th-order accuracy in 1D and 2D.

Introduction

Staggered grids schemes are usually used for CFD application in industrial context because they require less degrees of freedom than colocated ones for acoustic propagation [START_REF] Arakawa | Computational design of the basic dynamical processes of the ucla general circulation model[END_REF], they are also known to be more robust and more stable [START_REF] Ferziger | Computational methods for fluid dynamics[END_REF]. It is acknowledged that staggered schemes for compressible hydrodynamics originate in the late 1940's with the work of Richtmyer and VonNeumann (3; 4). They proposed the first shock capturing hydrodynamic scheme. This scheme is a 1D Lagrange explicit scheme, formulated in internal energy and 2nd order accuracy in time and space. Truglio and Trigger [START_REF] Trulio | Numerical solution of the one-dimensional lagrangian hydrodynamic equations[END_REF] modified the original scheme to make it conservative in total energy retaining spatial staggering of the variables. From the 1970's, pionnered by DeBar, several multifluid hydrodynamics code for compressible non viscous flow (6; 7) relied on Trulio-Trigger implicit Lagrangian scheme using a Lagrange-remap approach with dimensional splitting. Lagrange-remap schemes are known to be less dissipative and capture better both contact discontinuities and shocks. A strictly explicit version of Trulio's code was also reported in [START_REF] Woodward | The numerical simulation of two-dimensional fluid flow with strong shocks[END_REF]. In 2018, Abgrall et al. [START_REF] Abgrall | Staggered grid residual distribution scheme for lagrangian hydrodynamics[END_REF] used staggered grid residual distribution scheme to solve Euler equations ensuring conservation of total energy. On the other hand, high order CFD solver are now required to improve the accuracy of complex simulation scenarios. For instance, Dakin and al. (10;[START_REF] Dakin | Couplage fluide-structure d'ordre (très) élevé pour des schémas volumes finis 2d lagrange-projection[END_REF][START_REF] Dakin | High-order staggered schemes for compressible hydrodynamics. weak consistency and numerical validation[END_REF] gave an extension path of schemes described before to high order accuracy using Runge-Kutta sequences. In this article, we also search to obtain high order accuracy in time but thanks to Cauchy-Kowalevskaya procedure. The first one-step scheme using Cauchy-Kowalevskaya procedure was proposed in 1960 by Lax and Wendroff [START_REF] Lax | Systems of conservation laws[END_REF]. The Lax-Wendroff scheme is 2nd order accuracy in time and space; its variables are centered in both time and space. It is commonly used for systems of conservation laws for correct shock capturing. Daru and Tenaud (14) proposed later an high order scheme relying on Cauchy-Kovalevskaya procedure to solve the 1D advection equation. Del Pino and Jourdren gave [START_REF] Del Pino | Arbitrary high-order schemes for the linear advection and wave equations: application to hydrodynamics and aeroacoustics[END_REF] an extension to any order of accurary for these schemes. See also (16; 17) for related work on collocated grids for Euler equations formulated in total energy and for magnetohydrodynamics system. In the context of high performance computing, Cauchy-Kowalevskaya procedure offers the advantage of requiring fewer communications, compared with Runge-Kutta sequences, which are expensive in CPU time. The other advantages are a smaller number of calls to the equation of state and a smaller memory footprint. The aim of this article is to propose a class of Cauchy Kovalevskaya scheme based on staggered grids and formulated in internal energy.

This article proposes a new class of high order one-step staggered schemes for solving the 1D Euler equations in internal energy formulation on uniform grids. The integration in time in the Lagrange step is based on a Cauchy-Kovalevskaya procedure leading to high-order accuracy in time.

Section 2 is devoted to the construction of the finite volume schemes for wave equations with results on stability for orders until 4. Section 3 de-tails the extension of the schemes to Euler equations and 2D Navier-Stokes equations and gives conservation results. Section 4 presents numerical results illustrating the accuracy, the convergence and the robustness of the schemes. Eventually a comparison of performance with the Runge-Kutta based schemes is done.

1D wave equation with constant coefficients

We consider the 1D acoustic wave equation with constant coefficients ρ 0 and c 0

ρ 0 ∂ t u + ∂ x p = 0, ∂ t p + c 2 0 ∂ x u = 0, (2.1)
where ρ 0 ∈ R * + designs the initial mass density and c 0 ∈ R * + is the sound velocity. From now, we set ρ 0 = 1 to make formal computations easier. However, it does not affect the linear stability study.

High-order finite volume schemes on staggered grid

We define two uniform Cartesian grids: a primal grid {x j-1

2

} and a dual grid {x j }. p is cell-centered (values on each elements center of the primal grid) when u is staggered (values on each elements center of the dual grid). In the following, φ and φ will respectively denote space averaged value of φ and its pointwise value

           φ n j = 1 ∆X x j+ 1 2 x j-1 2 φ(x, t n )dx, and φ n j = φ(x j , t n ), φ n j-1 2 = 1 ∆X x j x j-1 φ(x, t n )dx, and φ n j-1 2 = φ(x j-1 2 , t n ).
We also define

∆ j φ = φ j+ 1 2 -φ j-1 2 and ∆ j-1 2 φ = φ j -φ j-1 such that ∆X = ∆ j x = ∆ j-1 2
x. To get the semi-discrete formulation of the scheme, system depicted in equation (2.1) is integrated in time between t n and t n+1 over a cell [x j-1 2 , x j+ 1 2 ] for p and over a cell [x j-1 , x j ] for u. It yields

         ∆X(u n+1 j-1 2 -u n j-1 2 ) = - t n+1 t n p j (θ) -p j-1 (θ)dθ, ∆X(p n+1 j -p n j ) = -c 2 0 t n+1 t n u j+ 1 2 (θ) -u j-1 2 (θ)dθ.
The high-order approximation in time is obtained thanks to the Cauchy-Kovalevskaya procedure. In [START_REF] Del Pino | Arbitrary high-order schemes for the linear advection and wave equations: application to hydrodynamics and aeroacoustics[END_REF], Cauchy-Kovalevskaya procedure is applied to Riemann invariants associated to (u, p). Here, this procedure is directly applied to (2.1). For any positive integer k, the procedure gives for u

∂ 2k+1 t u = -c 2k 0 ∂ 2k+1 x p, ∂ 2k t u = c 2k 0 ∂ 2k x u, (2.2)
and for p

∂ 2k+1 t p = -c 2(k+1) 0 ∂ 2k+1 x u, ∂ 2k t p = c 2k 0 ∂ 2k x p.
(2.3)

Then Taylor expansion of u and p at respectively point (x j-1 2 , t n ) and (x j , t n ) leads to

         t n+1 t n u j-1 2 (θ)dθ = N -1 k=0 (∆t) k+1 (k+1)! ∂ k t u n j-1 2 + O(∆t N +1 ), t n+1 t n p n j (θ)dθ = N -1 k=0 (∆t) k+1 (k+1)! ∂ k t p n j + O(∆t N +1 ).
For 1 ≤ k ≤ N -1, we replace ∂ k t u and ∂ k t p by their expression given in (2.2) and (2.3) so only spatial derivatives appear on the right member of the equations. Building an order N scheme requires to discretize ∂ k

x at least at order N -k for k ∈ {1, ..., N }. We set δ k a discretisation of ∂ k

x at the expected order and ν = ∆t/∆X. We define u * j-1 2 and p * j with

         u * j-1 2 = N -1 2 k=0 ν 2k (2k+1)! c 2k 0 δ 2k u n j-1 2 - N 2 -1 k=0 ν 2k+1 2(k+1)! c 2k 0 δ 2k+1 p n j-1 2 , p * j = N -1 2 k=0 ν 2k (2k+1)! c 2k 0 δ 2k p n j - N 2 -1 k=0 ν 2k+1 2(k+1)! c 2(k+1) 0 δ 2k+1 u n j .
(2.4)

Then we have

         t n+1 t n u j-1 2 (θ)dθ = ∆tu * j-1 2 + O(∆t N +1 ) + O(∆X N +1 ), t n+1 t n p j (θ)dθ = ∆tp * j + O(∆t N +1 ) + O(∆X N +1 ).
And the schemes at order N writes

u n+1 j-1 2 -u n j-1 2 = -ν∆ j-1 2 p * , p n+1 j -p n j = -νc 2 0 ∆ j u * .
Building an order N scheme also requires to compute the point-wise (resp average) values from the average (resp. pointwise) at least at order N. Although other reconstructions are possible, centered and symmetric ones are chosen like in [START_REF] Dakin | High-order accurate lagrange-remap hydrodynamic schemes on staggered cartesian grids[END_REF] because they are sufficient for uniform Cartesian grids.

                     φ ξ(j) = k C k φ ξ(j)+k , φ ξ(j) = k Ĉk φ ξ(j)+k , φ ξ(j) = (ρ 0 φ) ξ(j) (ρ 0 ) ξ(j) , δ 2m φ ξ(j) = k≥0 d 2m k (φ ξ(j)+k+ 1 2 -φ ξ(j)-k-1 2
),

δ 2m+1 φ ξ(j) = d 2m+1 0 φ ξ(0) + k≥1 d 2m+1 k (φ ξ(j)+k + φ ξ(j)-k ),
with ξ(j) = j on primal grid and ξ(j) = j + 1 2 on dual grid. The values of C k , Ĉk and d 0 k are given in Table 1 and Table 2. They are given by solving a linear system based on Taylor Lagrange expansion. The order is the order of the scheme and not the order of the derivative. The choice for the order of the derivative is explained below.

Order

C 0 C ±1 C 0 C ±1 1 and 2 1 0 1 0 3

Stability's study

However, the discretisation of ∂ k x at order N -k turns out to be not sufficient for the stability of the schemes. To gain stability, it is necessary to add extra terms in the initial scheme or to discretize the space partial derivatives at orders higher than the initial discretisation. The stability under study is the Von Neumann's stability [START_REF] Allaire | Numerical analysis and optimization: an introduction to mathematical modelling and numerical simulation[END_REF]. Von Neumann's stability is based We define the sequences φ,k (x, t) = e α φ t e ikπx and n φ,k,j = φ,k (j∆x, n∆t). If we define λ φ = e α φ ∆t we obtain the following properties on n φ,k,j :

n+1 φ,k,j = λ φ n k,j , n φ,k,j+m = e imθ n φ,k,j .
The amplification factor is introduced as a function of θ = kπ∆x, ∆x and ∆t as :

G(θ, ∆x, ∆t) = n+1 φ,k,j n φ,k,j = e α φ t .
The stability condition yields

|G(θ, ∆x, ∆t)| ≤ 1.

Order 1 et 2

At order 1, the scheme writes

u n+1 j-1 2 -u n j-1 2 = -ν[(p n j -p n j-1 )], p n+1 j -p n j = -νc 2 0 [(u n j+ 1 2 -u n j-1 2 )]. (2.5)
Considering that at order 1, p j = p j and u j-

1 2 = u j-1 2
the system (2.5) rewrites

u n+1 j-1 2 -u n j-1 2 = -ν[(p n j -p n j-1 )], (2.6)
p n+1 j -p n j = -νc 2 0 [(u n j+ 1 2 -u n j-1 2 )]. (2.7)
To study the stability of the scheme, we need to get separate equations on u and p. To obtain the equation on u, we write (2.6) at time n + 2,

u n+2 j-1 2 -u n+1 j-1 2 = -ν[(p n+1 j -p n+1 j-1 )]. (2.8)
Then we replace p n+1 j and p n+1 j-1 in (2.8) thanks to (2.7),

u n+2 j-1 2 -u n+1 j-1 2 = -ν[p n j -p n j-1 -νc 2 0 (u n j+ 1 2 + u n j-3 2 -2u n j-1 2 ].
Eventually, we replace (p n j -p n j-1 ) using (2.6),

u n+2 j-1 2 -u n+1 j-1 2 = u n+1 j-1 2 -u n j-1 2 + ν 2 c 2 0 (u n j+ 1 2 + u n j-3 2 -2u n j- 1 2
).

(2.9)

We replace u n j+ 1 2 by u,k,j+ 1 2 in (2.9) and mutliply by e i θ 2 ,

λ 2 u -2λ u + 1 + 4ν 2 c 2 0 sin 2 ( θ 2 ) = 0. Then λ ± u = 1 ± 2ic 0 ν sin( θ 2 ) and λ ± u 2 = 1 + 4ν 2 c 2 0 sin 2 ( θ 2 ) > 1.
We can check by symetry that λ p = λ u . So the initial scheme at order 1 is not stable. Using the same procedure, we obtain at order 2,

λ ± = 1-2c 2 0 ν 2 sin 2 ( θ 2 )±2iνc 0 sin( θ 2 ) and λ ± 2 = 1 + 4ν 4 c 2 0 sin 4 ( θ 2 ) > 1.
So the initial scheme at order 2 is not stable either.

Order 3 and 4

If we discretize the scheme at order 3 or 4 accordingly to the description above, the scheme is not stable either. However, by discretizing all the partial derivatives ∂ k

x and the average values at chosen order, we can obtain CFL conditions for stability (see Fig. 1). Indeed, if we discretize δ and δ 2 at order 3 according to the table 2, the 3rd order scheme is stable for c 0 ν ≤ 0.726. For the 4th order, we choose to discretize δ and δ 3 at order 4 and δ 2 at order 6. The scheme at order 4 is then stable for c 0 ν ≤ 1.073. Another solution to obtain stability conditions is to discretize the operator δ at order N and to iterate to obtain the operators δ n at order N where

δ n = δ • δ n-1 .
The main drawback of this method is the use of larger stencils. However, it gives straight forwardly the implementation for the 1D Euler equations in the non linear case. We obtain a 3rd order stable scheme for c 0 ν ≤ 0.742 and a 4th order stable scheme for c 0 ν ≤ 1.138. In the following, this solution is retained for the non linear case.

Remark : The high-order schemes presented here may be extended to order higher than 4.

Stabilization of order 1 and 2

We are motivated in getting stable schemes for order 1 and 2. For that purpose, we modify the schemes by adding the term of superior order but multiplied by a coefficient ζ. Our goals are to conserve the initial order of the scheme and to obtain stable schemes with a standard CFL condition, depending on ζ. The stability condition is determined by the sign of a seconddegree polynomial. At order 1, the scheme writes

u n+1 j-1 2 -u n j-1 2 = -ν[∆ j-1 2 p n -ζc 2 0 ν∆ j-1 2 (δu) n ], p n+1 j -p n j = -νc 2 0 [∆ j u n -ζ ν ρ 0 ∆ j (δp) n ].
We obtain the following sufficient and necessary condition of stability :

4ζ 2 c 2 0 ν 2 -2ζ + 1 ≤ 0. Let ζ = 1- √ 1-4α 2 4α 2
solution of 4ζ 2 α 2 -2ζ + 1 = 0, then the scheme is stable for c 0 ν ≤ α. Therefore maximising the CFL condition, one gets α = 1 2 and ζ = 1. At order 2, the scheme writes

u n+1 j-1 2 -u n j-1 2 = -ν[∆ j-1 2 p n -ν 2 c 2 0 ∆ j-1 2 (δu) n + ζν 2 ∆ j-1 2 (δ 2 p) n ], p n+1 j -p n j = -νc 2 0 [∆ j u n -ν 2 ∆ j (δp) n + ζν 2 c 2 0 ∆ j (δ 2 u) n ].
We obtain the following sufficient and necessary condition of stability :

16ζ 2 c 2 0 ν 2 -8ζ + 1 ≤ 0. Let ζ = 1- √ 1-α 2 4α 2
solution of 16ζ 2 α 2 -8ζ + 1 = 0, then the scheme is stable for c 0 ν ≤ α. Therefore maximising the CFL condition, one gets α = 1 and ζ = 1 4 .

Euler equations

We consider now the 1D Euler equations :

   ∂ t ρ + ∂ x (ρu) = 0, ∂ t (ρu) + ∂ x (ρu 2 + p) = 0, ∂ t (ρe) + ∂ x ((ρe + p)u) = 0, (3.1)
where ρ, u, p, e denote respectively the mass density, the velocity, the pressure and the total energy. Introducing the traditional Euler-Lagrange change of variables, (x, t) → (X, t) satisfying ρdx = ρ 0 dX, e kin = u 2 /2 the kinetic energy and the internal energy (e = + e kin ), one writes formally

       D t (ρ 0 τ ) -∂ X u = 0, D t (ρ 0 u) + ∂ X p = 0, D t (ρ 0 ) + p∂ X u = 0, D t (ρ 0 e kin ) + u∂ X p = 0. (3.2)
The system is closed by an equation of state linking the pressure, the internal energy and the specific volume p = EOS(τ, ). 1D Acoustic wave equation (2.1) can be interpreted as a linear approximation of 1D Euler equations with u = O( ) and p = p + O( ).

Lagrange step

The principle of Lagrange-remap approach selected here is to integrate the Lagrange system (3.2) and then to perform a conservative remap of the variables on the initial grid. The same uniform grids are used to integrate (3.2) and (2.1): ρ 0 τ and ρ 0 are cell-centered (primal grid) when ρ 0 u and ρ 0 e kin are staggered (dual grid). As for (2.1), system depicted in equation (3.2) is integrated in time between t n and t n+1 over a cell [x j-1 2 , x j+ 1 2 ] for the thermodynamics variables ρ 0 τ and ρ 0 and over a cell [x j-1 , x j ] for ρ 0 u and ρ 0 e kin :

                             ∆X(ρ 0 τ n+1 j -ρ 0 τ n j ) = t n+1 t n u j+ 1 2 (θ) -u j-1 2 (θ)dθ, ∆X(ρ 0 u n+1 j-1 2 -ρ 0 u n j-1 2 ) = t n+1 t n p j (θ) -p j-1 (θ)dθ, ∆X(ρ 0 n+1 j -ρ 0 n j ) = t n+1 t n x j+ 1 2 x j-1 2 p∂ X u(θ)dθ, ∆X(ρ 0 kin n+1 j-1 2 -ρ 0 kin n j-1 2 ) = t n+1 t n x j x j-1 u∂ X p(θ)dθ.
Using the fact that p = EOS(τ, ), we reach high-order approximation of the system (3.2) thanks to Cauchy-Kovalevskaya procedure applied on the system

D t (ρ 0 u) + ∂ x p = 0, D t (ρ 0 p) + (p∂ p -∂ τ p)∂ x u = 0.
In an isentropic case, the second equation is equivalent to

D t (ρ 0 p) + (ρc) 2 ∂ x u = 0.
In the acoustic regime (ρ 0 and c constant in time and space), using u * and p * defined in (2.4), the discrete scheme for Euler equations writes

         ρ 0 τ n+1 j -ρ 0 τ n j = ν∆ j u * , ρ 0 u n+1 j-1 2 -ρ 0 u n j-1 2 = -ν∆ j-1 2 p * , ρ 0 n+1 j -ρ 0 n j = -ν(pδu) * n j , ρ 0 kin n+1 j-1 2 -ρ 0 kin n j-1 2 = -ν(uδp) * n j-1 2 ,
where (pδu) * and (uδp) * write at order N

       (pδu) * = N -1 k=0 k l=0 σ l,k ∂ k t δu∂ k-l t p, (uδp) * = N -1 k=0 k l=0 σ l,k ∂ k t uδ∂ k-l t p, with σ l,k = ν k (k+1)! k! l!(k-l)! for k, l ∈ N and ∂ k t u, ∂ k t p are given by (2.
3) and (2.2). Remark : For orders 1 and 2, we can make the 2nd order accurate linear approximation :

ρ 0 n+1 j -ρ 0 n j = -νp * j ∆ j u * , ρ 0 kin n+1 j-1 2 -ρ 0 kin n j-1 2 = -νu * j-1 2 ∆ j-1 2 p * .
(3.3)

Non linear case

The linear approximation -ρ 0 and c locally constant for all x and t -is only first order accurate. Given that ρ 0 and c don't appear in 1rst order terms, this approximation is 2nd order accurate but is false for the 3rd and 4th order scheme. At order 3, the Cauchy-Kovalevskaya procedure rewrites for u

D t u = -1 ρ 0 ∂ x p, D tt u = -1 ρ 0 ∂ x (-(p∂ p-∂τ p) ρ 0 )∂ x u,
and for p

   D t p = -1 ρ 0 (p∂ p -∂ τ p)∂ x u, D tt p = -1 ρ 0 (p∂ p -∂ τ p)∂ x (-1 ρ 0 ∂ x p) + ( ∂xu ρ 0 ) 2 (∂ p(p∂ p -∂ τ p) +p∂ (p∂ p -∂ τ p) -∂ τ (p∂ p -∂ τ p))).
Therefore, it is more convenient to use the second discretization (where δ n is obtained by iterating δ n times) for orders higher than 3. Then u * and p * writes at order 3

         u * j-1 2 = u n j-1 2 -ν 2 1 ρ 0 (δp) n j-1 2 + ν 2 6 1 ρ 0 δ( p∂ p-∂τ p ρ 0 (δu)) n j-1 2 , p * j = p j -ν 2 (ρc) 2 ρ 0 (δu) n j + ν 2 6 (-1 ρ 0 (p∂ p -∂ τ p)δ(-1 ρ 0 δp) n j +( 1 ρ 0 ) 2 ((δu) n j ) 2 (∂ p(p∂ p -∂ τ p) + p∂ (p∂ p -∂ τ p) -∂ τ (p∂ p -∂ τ p)))).
We apply the same procedure for order 4.

Conservation of energy

Conservation of mass and momentum is immediate, we only prove the conservation of total energy. Here total energy at time t = t n is defined as

E n = i ρ 0 e n i = i ρ 0 n i + ρ 0 e kin n i+ 1 2
. The demonstration is inspired by the demonstration made in [START_REF] Dakin | High-order accurate lagrange-remap hydrodynamic schemes on staggered cartesian grids[END_REF] in the case of Runge-Kutta integration. To make it easier to read, we note Π := p the pressure and

∂ := ∂ t the time derivative. ∆E = E n+1 -E n = i ρ 0 e n+1 i -ρ 0 e n i = i (ρ 0 n+1 i -ρ 0 n i ) + i (ρ 0 e kin n+1 i+ 1 2 -ρ 0 e kin n i+ 1 2 ) = -ν i Πδu n i + uδΠ n i+ 1 2 = -ν i,k 0≤m≤l≤N -1 Ĉk σ m,l [(∂ m Πδ∂ l-m u) i+k + (∂ m uδ∂ l-m Π) i+k+ 1 2 )] = -ν i,k,k 0≤m≤l≤N -1 d k Ĉk σ m,l [∂ m Π i+k ∂ l-m u i+k+k + 1 2 + ∂ m Π i+k+k +1 ∂ l-m u i+k+ 1 2 -∂ m Π i+k ∂ l-m u i+k-k -1 2 -∂ m Π i+k-k ∂ l-m u i+k+ 1 2 ]
Making the change of index i ← i + k in the first term and i ← i + k + 1 in the second term, we get the result for wall or periodic conditions.

∆E = -ν i,k,k 0≤m≤l≤N -1 d k Ĉk σ m,l [∂ m Π i+k-k ∂ l-m u i+k+ 1 2 + ∂ m Π i+k ∂ l-m u i+k-k -1 2 -∂ m Π i+k ∂ l-m u i+k-k -1 2 -∂ m Π i+k-k ∂ l-m u i+k+ 1 2
] = 0 By introducing the internal energy corrector defined in [START_REF] Dakin | High-order staggered schemes for compressible hydrodynamics. weak consistency and numerical validation[END_REF] we can prove the conservation of the quantity

E n = i ρ 0 n i + ρ 0 u 2 n i+ 1 2
while not affecting the conservation of the quantity E n . This internal energy corrector is defined as an a posteriori corrector

ρ 0 n+1, i = ρ 0 n+1 i + ∆K n+1 i , ρ 0 e kin n+1, i+ 1 2 = ρ 0 e kin n+1 i+ 1 2 -∆K n+1 i+ 1 2 , where ∆K n+1 i+ 1 2 = ρ 0 e kin n+1 i+ 1 2 -1 2 ρ 0 u 2 n+1 i+ 1 2 and ∆K n+1 i = 1 2 (∆K n+1 i-1 2 + ∆K n+1 i+ 1 2
).

Remap step

The remap step consists in projecting the Lagrangian quantities on the initial uniform Cartesian grids so that ones get a Cartesian Euler scheme. The projection is equal to the one proposed in (16; 10). Variables ρ and ρ are remapped on the primal grid while ρu and ρe kin are remapped on the dual grid. The remap step is conservative in mass, momentum, internal and kinetic energies.

2D extensions with dimensional splitting

2D Euler equations

As presented in (16; 10), the extension to the multidimensional case is realised thanks to dimensional splitting. The Euler equations in 2D write

       ∂ t ρ + ∂ x (ρu) + ∂ y (ρv) = 0, ∂ t ρu + ∂ x (ρu 2 + p) + ∂ y (ρuv) = 0, ∂ t ρv + ∂ x (ρuv) + ∂ y (ρv 2 + p) = 0, ∂ t ρe + ∂ x (ρue + pu) + ∂ y (ρve + pv) = 0.
This system can be split according to x-direction

       ∂ t ρ + ∂ x (ρu) = 0, ∂ t ρu + ∂ x (ρu 2 + p) = 0, ∂ t ρv + ∂ x (ρuv) = 0, ∂ t ρe + ∂ x (ρue + pu) = 0, and y-direction        ∂ t ρ + ∂ y (ρv) = 0, ∂ t ρu + ∂ y (ρuv) = 0, ∂ t ρv + ∂ y (ρv 2 + p) = 0, ∂ t ρe + ∂ y (ρve + pv) = 0.
Splitting techniques relies on solving alternatively first and second system with weighted time-steps in order to reach high order of accuracy as described in [START_REF] Duboc | High-order dimensionally split lagrange-remap schemes for compressible hydrodynamics[END_REF]. Each of the subsystem is solved using the 1D schemes proposed in this paper. Variables are distributed according to an Arakawa C-type distribution (see Fig. 2).

Extension to the 2D compressible Navier-Stokes equations with gravity

The compressible Navier-Stokes equations are quite similar to the Euler equations with an additive viscous term denoted Υ and a constant gravity source term g. The system of equation in 2D writes in conservative form as

   ∂ t ρ + ∇ • ρu = 0, ∂ t ρu + ∇ • (ρu ⊗ u + pI -Υ) = g, ∂ t ρe + ∇ • (((ρe + p)I -Υ) • u) = gu,
where Υ = µ(∇u + (∇u) t ) + λ(∇ • u)I, µ and λ being two parameters which describe the viscous properties of the considered fluid. The 2D staggered hydrodynamic schemes are based on directional splitting. We take the same splitting as described in [START_REF] Dakin | Couplage fluide-structure d'ordre (très) élevé pour des schémas volumes finis 2d lagrange-projection[END_REF]. In the x-direction, the Lagrangian formulation is

       D t (ρ 0 τ ) -∂ x (ρu) = 0, D t (ρ 0 u) + ∂ x (p -Υ 1,1 ) = 0, D t (ρ 0 v) + ∂ x (-Υ 2,1 ) = 0, D t (ρ 0 e) + ∂ x ((p -Υ 1,1 )u -Υ 2,1 v) = 0, and in the y-direction        D t (ρ 0 τ ) -∂ y (ρv) = 0, D t (ρ 0 v) + ∂ y (p -Υ 2,2 ) = ρ 0 g, D t (ρ 0 u) + ∂ y (-Υ 1,2 ) = 0, D t (ρ 0 e) + ∂ y ((p -Υ 2,2 )v -Υ 1,2 u) = ρ 0 gv, where Υ 1,1 = (2µ + λ)∂ x u + ∂ y v, Υ 2,2 = (2µ + λ)∂ y v + ∂ x u and Υ 1,2 = Υ 2,1 = µ(∂ y u + ∂ x v).
Using the formulation in both kinetic and internal energies, it yields formally (in the y-direction)

               D t (ρ 0 τ ) -∂ y (ρv) = 0, D t (ρ 0 v) + ∂ y (p -Υ 1,1 ) = ρ 0 g, D t (ρ 0 u) + ∂ y (-Υ 1,2 ) = 0, D t (ρ 0 ) + (p -Υ 1,1 )∂ y v -Υ 1,2 ∂ y u = 0, D t (ρ 0 e kin,v ) + v∂ y (p -Υ 2,2 ) = ρ 0 gv, D t (ρ 0 e kin,u ) + u∂ y (-Υ 1,2 ) = 0. (4.1)
The Cauchy-Kowlalevski procedure can be performed in the x-direction thanks to the system

     D t u = -1 ρ 0 (∂ x p -(2µ + λ)∂ xx u + λ∂ xy v), D t p = -(ρc) 2 ρ 0 ∂ x u, D t v = µ(∂ xy u + ∂ xx v),
and in the y-direction

     D t v = -1 ρ 0 (∂ y p -(2µ + λ)∂ yy v + λ∂ xy u + ρ 0 g), D t p = -(ρc) 2 ρ 0 ∂ y v, D t u = µ(∂ xy v + ∂ yy u).
An Arakawa C-type (1; 11) like grid is chosen with a nodal distribution for the non-diagonal viscous terms. In addition, the values of the terms Υ 1,2 ∂ y u and Υ 2,1 ∂ x v on indices (i, j) (see equations 4.1) necessary to update i,j need to be computed from their values on the indices (i -1 2 , j -1 2 ) thanks to coefficients given in Table 1.

r   p Υ 1,1 Υ 2,2   i,j r u i-1 2 ,j r v i,j-1 2 r v i,j+ 1 2 r u i+ 1 2 ,j r Υ 1,2 Υ 2,1 i+ 1 2 ,j+ 1 2 r Υ 1,2 Υ 2,1 i-1 2 ,j+ 1 2 r Υ 1,2 Υ 2,1 i-1 2 ,j-1 2 r Υ 1,2 Υ 2,1 i+ 1 2 ,j-1 2

Numerical results

Cook-Cabot Breaking Wave test case (19)

In the Cook-Cabot Breaking Wave test case, initial conditions are sinusoidal for density, pressure and velocity. The variables are smooths until a given time T shock when a discontinuity occurs. The breaking wave initial data are set as follows

       ρ = ρ 0 (1 + α sin(2πx)), p = p 0 ( ρ ρ 0 ) γ , c = c 0 ( ρ ρ 0 ) (γ-1)/2 , u = 2 γ-1 (c 0 -c).
for -0.5 ≤ x ≤ 0.5

An analytical solution is known before the shock which makes it possible to evaluate the order of the schemes until the time T shock . Computations are performed until T = 0.9 * T shock with CFL = 0.7. In Table 3 3: l 1 -error in momentum and order of convergence of the Lagrange-remap staggered scheme on the Cook-Cabot breaking wave test problem. † : corresponds to the 3rd order in space with linear approximation in time (3.3).

2D Isentropic vortex advection (20)

This case involves advection of an isentropic vortex in an inviscid flow. The initial data are given by (with

r 2 = x 2 + y 2 )          ρ 0 (x, y) = (1 -(γ-1)β 2 8γπ 2 e 1-r 2 ) 1 γ-1 , u 0 (x, y) = (2, 1) t + β 2π e (1-r 2 )/2 .(-y, x) t , p 0 (x, y) = ρ 0 (x, y) γ , γ = 1.4.
As the precedent test case, an analytical solution is known which corresponds to the pure advection of the vortex. Computations are performed until T = 20 with a CFL = 0.7 and periodic boundary conditions are imposed. In Table 4, convergence results on density in l 1 norm are proposed.

Sod test case (21)

The Sod shock tube is a common one dimension Riemann problem. The resolution of this test case leads to three characteristics describing the propagation speed of the various regions of the system : the rarefaction wave, the contact discontinuity and the shock discontinuity. It is useful to determine if a scheme recovers properly discrete Rankine-Hugoniot relations on the shock. The initial data are

N x Order 1
Order 2 Order 3 † Order 3 Order 4 50 7.6e-1 2.4e-1 1.1e-1 9.6e-2 5.7e-2 100 4.3e-1 0.802 7.1e-2 1.745 2.1e-2 2.372 1.7e-2 2.459 5.5e-3 3.361 200 2.4 e-1 0.885 1.9e-2 1.915 3.5e-3 2.577 2.4e-3 2.862 3.4e-4 4.021 400 1.2e-1 0.936 4.8e-3 1.975 6.4e-4 2.465 3.1e-4 2.969 2.1e-5 4.013 800 6.3e-2 0.966 1.2e-3 1.991 1.32e-4 2.282 3.8e-5 2.992 1.3e-6 4.000 Table 4: l 1 -error in density and order of convergence for the Lagrange-remap scheme taken on the isentropic vortex advection test problem. † : designs the 3rd order in space with linear approximation in time (3.3).

       ρ 0 (x) = 1.0χ {x<0.5} + 0.125χ {x>0.5} , p 0 (x) = 1.0χ {x<0.5} + 0.1χ {x>0.5} , u 0 (x) = 0, γ = 1.4.
Wall boundaries are imposed. Computations are performed until T=0.2 with CFL = 0.7. In Fig. 3, profiles of density and internal energy are depicted with the analytical solution in line for two meshes containing 100 cells and 200 cells. It can be seen that the three characteristics of the shock are correctly rendered. We observe a undershoot in the density and a peak in the internal energy that can be caused by the absence of artificial viscosity.

123 problem

The 123 problem is a Riemann problem resulting in two strong rarefaction waves and a stationnary contact discontinuity. It is useful to assess the performance of numerical methods on low-density flows . The initial data are

       ρ 0 (x) = 1, p 0 (x) = 0.4, u 0 (x) = -2χ {x<0.5} + 2χ {x>0.5} , γ = 1.4.
Wall conditions are imposed. Computations are performed until T = 0.15 with CFL = 0.7. In Fig. 4, profiles of density, pressure, internal energy and velocity are depicted with the analytical solution in line for a mesh containing 1000 cells.

Shu-Osher test case (22)

The Shu-Osher test case is Mach 3 shock wave interacting with a sinusoidal density field. This test assesses the ability of numerical schemes to capture both small-scale smooth flow and shocks. The initial data are

       ρ 0 (x) = 27 7 χ {x<-4} + (1 + sin(5x) 5 )χ {x>-4} , p 0 (x) = 31 3 χ {x<-4} + 1χ {x>-4} , u 0 (x) = 4 √ 35 9 χ {x<-4} , γ = 1.4.
In Fig. 5, profiles of density are depicted with a reference solution in line for two meshes containing 200 and 400 cells. On the right, a 6th order remap is performed. It is possible to see the significant improvement in performance with the 6th order remap at a resolution of 200 cells, all of the short-wavelenght extrema are captured. However this also leads to a slight overevaluation of the extrema at a resolution of 400 cells. There is no analytical solution for this test case, so we compare the results with a reference solution computed thanks to the Godunov Anti-Diffusé (GAD) schemes [START_REF] Heuzé | Dissipative issue of high-order shock capturing schemes with non-convex equations of state[END_REF] with a mesh containing 10 5 cells

Woodward test case (8)

The Collela-Woodward blastwave test case is the interaction of two Riemann problems. It is a three states shock tube. This test case highlights the robustness of the scheme. We first assess the numerical scheme on the left half of the blast wave problem with the initial data

       ρ 0 = 1, u 0 = 0, γ = 1.4, p 0 (x) = 1000χ {x<0.5} + 0.01χ {x>0.5} .
Computations are performed until T = 0.012 with CFL = 0.7. In Fig. 6, profiles of density, pressure, internal energy and velocity are depicted with the analytical solution in line for a mesh containing 1000 cells. The initial data of the full problem are In Fig. 7, profiles of density are depicted with a reference solution in line for a mesh containing 300 cells. On the right, a 6th order remap is performed. There is no analytical solution for this test case, so we compare the results with a reference solution computed thanks to the Godunov Anti-Diffusé (GAD) schemes ( 23) with a mesh containing 10

       ρ 0 = 1, u 0 = 0, γ = 1.

Sedov Test case (24)

With the Sedov test case, we assess the robustness of the staggered schemes as well as their ability to restitute correct cylindrical symmetry. Let where sedov = 0.851072. In Fig. 8, profile of density obtained with the 3rd order remap is depicted with a reference solution in blue line for a mesh containing 100 cells. A 6th order Lagrangian projection is performed during the remap procedure. In Fig. 8, the discontinuity is captured correctly, however, the maximum is slightly underevaluated.

r sedov = 1 √ 2 √ ∆X 2 + ∆Y 2 . Initial data are        ρ 0 (x, y) = 1, u 0 (x, y) = 0, p 0 (x, y) = (γ-

Taylor-Green Vortex (25)

The Taylor-Green vortex is used to assess the accuracy of the proposed schemes. It has an exact closed form of the incompressible Navier-Stokes equations in Cartesian coordinates. Here, enforcing a very high sound speed, the compressible Navier-Stokes equations are in near incompressible regime

       ρ 0 (x, y) = 1, u 0 (x, y) = cos(x) sin(y), v 0 (x, y) = -sin(x) cos(y), p 0 (x, y) = p 0 +
       ρ 0 (x, y) = 2χ {y>0} + 1χ {y<0} , u 0 (x, y) = 0, v 0 (x, y) = 0.25a(1 + cos(4πx))(1 + cos(3πy))χ {-1 6 <y< 1 6 } , p 0 (x, y) = K 0 + ρ 0 (x, y)gy,
where g = -0.1, K 0 = 2.5 and a = 10 -2 . The viscous parameters are chosen very small µ = 10 -4 and λ = -2 3 µ. Computations are performed until T = 9.5 (left) and T = 12.75 (right) with 200 cells in the x-direction and 600 in the y-direction see Fig. 9. In the case of Euler equations, turbulent modes appear due to the numerical noise and increase with the order of the scheme. This modes are smoothed by viscosity in Navier-Stokes equations.

Performance

Most of the reconstruction procedures we use in this note are similar to the ones described in [START_REF] Dakin | High-order accurate lagrange-remap hydrodynamic schemes on staggered cartesian grids[END_REF]. The difference lays in the time integration, in [START_REF] Dakin | High-order accurate lagrange-remap hydrodynamic schemes on staggered cartesian grids[END_REF], the time integration is based on a Runge-Kutta integration including several integration steps at each time when the schemes described in this note are based on a one-step time integration procedure. Error rates and accuracy on standard test cases are quite similar in both situations. The performance tests are performed on the Sod test case described above. Only the integration time of the Lagrange step is taken into account here. The calculations are performed on a haswell partition. Each node is divided into four sockets of eight cores. We can see that results on one process are quite similar. Indeed, Runge-Kutta time integrations is based on an iterative procedure with a low computational cost, it requires less intel on the EOS and a narrower stencil, when Cauchy-Kowalevskaya procedure requires more calculations. However, Cauchy-Kovaleskaya procedure requires less communications and [START_REF] Dakin | High-order accurate lagrange-remap hydrodynamic schemes on staggered cartesian grids[END_REF] and integration in time based on Cauchy-Kowalevskaya procedure

Conclusion

In this article, we present one-step Lagrange-remap schemes with integration in time based on Cauchy-Kovalevskaya procedure for the resolution of the 1D Euler equations. We find sufficient conditions for stability at order 1 and 2 by adding higher order terms. At order 3 and 4, we adapt the stencil of spatial derivatives to stabilize the schemes. We also give a multidimensional extension using directional splitting as well as an extension to compressible Navier-Stokes equations. The proposed schemes were tested successfully in 1D and 2D both in linear and non linear formulation. We currently work on an implicit version of the schemes to withdraw the parabolic CFL condition for viscous flows.

7 Figure 1 :

 71 Figure 1: Value of the squared module of the amplificator factor λ in function of θ with two different values of ν for the 3rd order scheme. On the left (ν = 0.8) |λ| exceeds 1 when on the right (ν = 0.7) it remains under 1.

Figure 2 :

 2 Figure 2: Arakawa C-type like grid for the compressible Navier-Stokes equation with a special distribution for the non-diagonal viscous terms[START_REF] Dakin | Couplage fluide-structure d'ordre (très) élevé pour des schémas volumes finis 2d lagrange-projection[END_REF]

Figure 3 :Figure 4 :

 34 Figure 3: Density profiles on [0 : 1] for the Sod test case problem (21) at time T=0.2, CFL=0.7, 100 cells (top) and 200 cells (down) for the 3rd order staggered scheme with Cauchy-Kovalevskaya procedure.

Figure 5 :

 5 Figure 5: Density profiles on [-5 : 5] for the ShuOsher test case problem (22) at time T=1.8, CFL=0.7, 200 cells (top) and 400 cells (down) for the 3rd staggered scheme with Cauchy-Kovalevskaya procedure. On the right, a 6th order Lagrangian projection is performed during the remap step.

Figure 6 :

 6 Figure 6: Density, internal energy, velocity and pressure profiles on [0 : 1] for the left half of the Collela-Woodward blastwave test case problem at time T=0.012, CFL=0.7, 1000 cells for the 3rd order staggered scheme with Cauchy-Kovalevskaya procedure.

22

 22

Figure 7 :

 7 Figure 7: Density and pressure profiles on [0 : 1] for the Woodward test case problem (8) at time T=0.038, CFL=0.7, 300 cells, for the 3rd staggered scheme with Cauchy-Kovalevskaya procedure. On the right, a 6th order Lagrangian projection is performed during the remap step.

Figure 9 :

 9 Figure 9: Density profiles on the Rayleigh-Taylor mono-mode instability for the Euler equations (top) and the compressible Navier-Stokes (bottom) using 2nd and 3rd order scheme at time T = 9.5 (left) and T=12.75 (right) with 200 cells in the x-direction and 600 in the y-direction (∆X = ∆Y green

Table 1 :

 1 Coefficients for the finite volume computation of point-wise values from cellaverage ones and vice versa.

	13	-1	11	1
	12	24	12	24

Table 2 :

 2 Coefficients for the δ m coefficient

	on the Fourier decomposition of the numerical error. It is often used for
	Cauchy problem with linear partial differential equations. For φ ∈ L 2 (R ×
	R + ), under boundary conditions, it yields
	φ(x, t) = e α φ t	ψ φ,k e ikπx with ψ φ ∈ l 2 (Z)
	k∈Z

Table

  , convergence results on momentum in l 1 norm are proposed.

	N x	Order 1	Order 2	Order 3 †	Order 3	Order 4
	50 1.3e-1	3.2e-3	5.9e-4	1.1e-4	6.2e-6
	100 6.7e-2 0.968 7.9e-4 2.013 1.6e-4 1.904 1.4e-5 2.907 4.0e-7 3.947
	200 3.4e-2 0.966 2.0e-4 1.966 4.1e-5 1.921 1.8e-6 2.942 2.7e-8 3.891
	400 1.7e-2 0.983 5.1e-5 1.982 1.1e-5 1.961 2.4e-7 2.973 1.8e-9 3.942
	800 8.7e-3 0.991 1.3e-5 1.991 2.7e-5 1.981 3.0e-8 2.986 1.1e-10 3.969
	1600 4.4e-3 0.995 3.2e-6 1.995 6.7e-7 1.990 3.7e-9 2.993 1.1e-11 3.386

  1) sedov πr 2 sedov χ {x 2 +y 2 <r 2 sedov } + 10 -14 χ {x 2 +y 2 >r sedov } , γ = 1.4,

Table 5 :

 5 l 1 -error in momentum and order of convergence for the Lagrange-remap scheme taken on the Taylor-Green vortex test problem[START_REF] Taylor | The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. i[END_REF]

Table 6 :

 6 Comparative table of the integration time of the Lagrangian phase with Runge-Kutta time integration

The analytical solution for incompressible flows writes

, v(x, y, t) = -sin(x) cos(y)e -2µt , p(x, y, t) = p 0 + 1 4 (sin(2x) + sin(2y))e -4µt , with p 0 = 10. The pressure is set such that the regime is nearly incompressible, using a stiffened gas EOS which writes p = (γ -1)ρ -γp * , γ = 1.4, p * = 10 8 .

In Table 5, convergence results on momentum in l 1 norm are proposed. Errors are taken as absolute errors. The convergence saturation observed is due to the fact that incompressibility hypothesis is not fully verified and that machine precision is reached rapidly.

Rayleigh Taylor test case

The Rayleigh Taylor instability is used to determine the ability of a scheme to handle instabilities, and if those instabilites are accentuated by the high-order accuracy. The initial data are


N Filippov

L Troitsyna
email: larisa.troitsyna@univ-fcomte.fr

R Asfin

N Gennadiev

J Buldyreva

Vibrational shifts of absorption bands of linear molecules diluted in high-density rare gases

Keywords: Infrared absorption spectra, Band-origin shift, Spectral moments, Vibrational dependence, DID model, Carbon dioxide, Carbonyl sulfide, Rare gases

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Though molecular spectra contain information on the nature of intermolecular interactions and features of molecular structure, their interpretation is associated with numerous difficulties [START_REF] Gordon | Molecular Motion and the Moment Analysis of Molecular Spectra. III. Infrared Spectra[END_REF] still to be resolved. For instance, spectral band shapes change significantly with increasing pressure and a set of characteristics is required for description of the band under various experimental conditions. In particular, the spectral moments M n account for numerous band properties and thus can be used for practical purposes, though even a complete set of spectral moments does not characterize a unique band. For examining the interaction mechanisms that result in specific-band properties, particularly suitable candidates are benchmark systems composed of linear molecules, such as CO 2 and OCS, perturbed by rare gases.

These two molecules are present in planetary atmospheres (94.9% and 96.5% of CO 2 for Mars [START_REF] Franz | Initial SAM calibration gas experiments on Mars: Quadrupole mass spectrometer results and implications[END_REF] and Venus [START_REF] Basilevsky | The Surfaces of Venus[END_REF], respectively, and trace-gas level [START_REF] Horn | The composition of the Martian atmosphere: Minor constituents[END_REF][START_REF] Arney | Spatially resolved measurements of H 2 O, HCl, CO, OCS, SO 2 , cloud opacity, and acid concentration in the Venus near-infrared spectral windows[END_REF] of OCS), comets [START_REF] Russo | Carbonyl Sulfide in Comets C/1996 B2 (Hyakutake) and C/1995 O1 (Hale-Bopp): Evidence for an Extended Source in Hale-Bopp[END_REF][START_REF] Moretto | Abundances and spatial distributions of H 2 O and CO 2 at comet 9P/Tempel 1 during a natural outburst[END_REF], and interstellar space [START_REF] Matthews | Observations of OCS and a search for OC 3 S in the interstellar medium[END_REF][START_REF] Minissale | CO 2 formation on interstellar dust grains: A detailed study of the barrier of the CO+O channel[END_REF]. Precise knowledge of their spectral band shapes and band origins is therefore essential for reliable radiative transfer simulations and remote sensing of atmospheric environments. Line positions, intensities, widths and shifts have been closely considered in numerous works. For OCS-Ar e.g., line widths in the ν 1 [START_REF] Bouanich | Diode-laser measurements of Ar-and CO 2 -broadened linewidths in the ν 1 band of OCS[END_REF] and hot [START_REF] Blanquet | Line-mixing effects in Ar-broadened doublets of a hot band of OCS[END_REF] bands, line widths and shifts in the 2ν 3 band [START_REF] Domenech | Pressure Lineshift and Broadening Coefficients in the 2v3 Band of 16 O 12 C 32[END_REF] have been measured. For CO 2 -Ar, line shifts in the 00 0 3-00 0 0 band [START_REF] Thibault | Pressure induced shifts of CO 2 lines: Measurements in the 0003-0000 band and theoretical analysis[END_REF], line shifts, speed-dependence parameters and line mixing coefficients in the 30013-00001 and 30012-00001 bands [START_REF] Benner | Constrained multispectrum analysis of CO 2 -Ar broadening at 6227 and 6348 cm -1[END_REF] have been reported. In contrast, studies of band-origin shifts remain very scarce up to now due to two major reasons.

From the experimental standpoint, determination of the band-origin shift (defined by M 1 -see Sec. 3.2) requires accurate intensity measurements up to very far spectral wings, which is a tedious and time-consuming task because of the numerous neighboring bands to be removed. That is why, traditionally, the band-origin shift is rather estimated from the position of the intensity maximum, as it was done in the work of Ozanne et al. [START_REF] Ozanne | Line-mixing, finite duration of collision, vibrational shift, and non-linear density effects in the ν 3 and 3ν 3 bands of CO 2 perturbed by Ar up to 1000 bar[END_REF] who investigated the ν 3 and 3ν 3 bands of CO 2 -Ar at pressures up to 1000 bar. The band shifts were determined as a deviation of the measured intensity maximum from its position computed by the Energy-Corrected Sudden model. The shifting coefficients were found to be γ(ν 3 ) = (-5.4±1.0)×10 -3 cm -1 amagat -1 and γ(3ν 3 ) = (-17.0±1.0)×10 -3 cm -1 amagat -1 .

From the theoretical viewpoint, accurate band-origin shift calculations require vibrationally dependent potential energy surfaces (PESs) for both initial (ground) and final (excited) states, that are currently available only for a limited amount of systems. If the vibrational dependence of the final-state PES is unknown, mechanisms responsible for inducing the band-origin shift must be examined (e.g., Ozanne and coauthors [START_REF] Ozanne | Line-mixing, finite duration of collision, vibrational shift, and non-linear density effects in the ν 3 and 3ν 3 bands of CO 2 perturbed by Ar up to 1000 bar[END_REF] suggested that the polarizability matrix elements of a linear molecule are vibrationally dependent).

In the present work we perform accurate high-pressure measurements of CO 2 -Rg (up to 100 atm) and OCS-Rg (up to 300 atm) band shapes and rigorously compute the band-origin positions from the first moment definition (see Sec. 4.1). Moreover, we suggest that, in the absence of vibrationally-dependent PES, the interaction between the dipole moment induced on the atom and the dipole of the molecule (DID interaction) can also result in a band-origin shift and test this hypothesis by comparing the results of the DID model and those obtained with vibrationally-dependent PES for both initial and final states. In Section 2 we briefly describe the experimental conditions. Theoretical background is summarized in Section 3. Details of modeling for CO 2 and OCS are given in Sec. 3.2. The bulk of our experimental and theoretical results is discussed and compared with that of Ref. [START_REF] Ozanne | Line-mixing, finite duration of collision, vibrational shift, and non-linear density effects in the ν 3 and 3ν 3 bands of CO 2 perturbed by Ar up to 1000 bar[END_REF] in the final section.

Measurements

IR absorption spectra of CO 2 -Ar, CO 2 -Kr, CO 2 -Xe, OCS-Ar and OCS-Kr gas mixtures at 298 K were registered with Bruker IFS-28 and Bruker IFS 125/HR spectrometers in the region 400-4500 cm -1 with a spectral resolution 0.8 cm -1 . Active-gas concentrations were at trace-gas level, and the buffer-gas pressures varied from 15 to 100 atm for CO 2 -Rg and from 25 to 300 atm for OCS-Rg. In function of the studied spectral range and the mixture pressure, two cells were used: from 400 to 1000 cm -1 and pressures up to 100 atm a 13.5 cm cell with thallium bromoiodide (KRS-5) windows and from 1000 to 4500 cm -1 and pressures up to 300 atm a 7.4 cm cell with fluorite (CaF 2 ) windows. The mixtures were composed by pumping the compounds into a cell. The purity was controlled spectroscopically (no inpurity-gas bands were detected above the noise level).

Figure 1 shows examples of OCS band-shape transformation with increasing buffer-gas pressure. For both active molecules, the band origin was tracked by the first moment definition (numerical integration of Eq. (2) of Sec. 3.2 with the subsequent normalization). Examples of experimental band-origin shifts versus gas density and their linear fits can be seen in Fig. 2. The experimental band-origin shift coefficients are given in Tables 1 and2.

Table 1: Spectral ν 3 band-origin shift coefficients γ (in 10 -3 cm -1 amagat -1 ) measured and calculated by the direct (Eq. ( 5)) and DID (Eq. ( 6)) methods for CO 2 perturbed by various rare gases. The references indicate the PES used for computations. expt direct DID CO 2 -Ar -9.6(2) -2.06 [START_REF] Cui | A new potential energy surface and predicted infrared spectra of the Ar-CO 2 van der Waals complex[END_REF] -6.28 [START_REF] Cui | A new potential energy surface and predicted infrared spectra of the Ar-CO 2 van der Waals complex[END_REF], -6.57 [START_REF] Parker | Intermolecular potential surfaces from electron gas methods. I. Angle and distance dependence of the He-CO 2 and Ar-CO 2 interactions[END_REF], -6.27 [18] CO 2 -Kr -18.2(2) -9.51 [START_REF] Chen | Intermolecular potential energy surface, microwave and infrared spectra of the Kr-CO 2 complex from ab initio calculations[END_REF] -8.96 [19] CO 2 -Xe -32.0(9) -10.20 [START_REF] Chen | Potential energy surface, microwave and infrared spectra of the Xe-CO 2 complex from ab initio calculations[END_REF] -13.29 [START_REF] Chen | Potential energy surface, microwave and infrared spectra of the Xe-CO 2 complex from ab initio calculations[END_REF], -2.83 [START_REF] De Dios | The 129 Xe nuclear shielding surfaces for Xe interacting with linear molecules CO 2 , N 2 , and CO[END_REF], -3.36 [START_REF] Buck | Observation of multiple-collision rotational rainbows in Xe-CO 2 : Comparison between TOF measurements and scattering calculations[END_REF] Table 2: Experimental and DID-computed spectral band-origin shift coefficients γ (in 10 -3 cm -1 amagat -1 ) for various bands of OCS perturbed by Ar (PES [START_REF] Zhu | Ab initio potential energy surface and predicted microwave spectra for Ar -OCS dimer and structures of Arn -OCS (n = 2-14) clusters[END_REF]) and Kr (PES [START_REF] Sun | A three-dimensional potential energy surface and infrared spectra for the Kr-OCS van der Waals complex[END_REF]).

Absorption coefficient and correlation function

An incident electromagnetic wave with angular frequency ω and intensity I 0 , having propagated over a path length l in a media containing n a absorbers per unit volume, looses its intensity I according to the Beer-Lambert law:

I(ω) = I 0 (ω)e -naK(ω)l .
The absorption coefficient K(ω) is expressed [START_REF] Hartmann | Collisional effects on molecular spectra[END_REF] as

K(ω) = 4π 2 ω 3 c 1 -e -ω k B T F (ω),
where , c and k B stand for the Plank constant, speed of the electromagnetic wave in vacuum and Boltzmann constant, respectively, T is the temperature in Kelvin and F (ω) is the so-called spectral function. F (ω) may be regarded as the Fourier transform of a time correlation function C(t) [START_REF] Gordon | Correlation Functions for Molecular Motion[END_REF]:

F (ω) = 1 2π ∞ -∞ dte -iωt C(t),
which is related to the dipole-moment absorption as

C(t) = d † (0)d(t) . (1)
Here, the dagger denotes the Hermitian conjugation and

d = |v f v f |D|v i v i | with v f |D|v i standing for the vibrational f ← i transition moment of the active-molecule dipole D.
The function C(t), when known, determines completely F (ω) and, consequently, the full absorption band. To characterize the band-origin position we employ the spectral moments approach.

Spectral moments

The spectral moment of order n is defined by

M n = ∞ -∞ F (ω)ω n dω. (2)
Due to the fact that M 0 represents simply the integral intensity of the spectral band, normalized spectral moments µ n = M n /M 0 are frequently used. From the Fourier-transform relationship

C(t) = ∞ -∞
dωe iωt F (ω) one easily gets

d n C(t) dt n t=0 = (i) n ∞ -∞ F (ω)ω n dω.
Substitution of this equation in Eq. ( 2) gives

M n = (-i) n d n C(t) dt n t=0 .
For our purposes the key interest is in the first spectral moment giving the band origin:

M 1 = -i dC(t) dt t=0 .
With the definition (1) of C(t) the time derivative can be re-written as

dC(t) dt t=0 = d † (0) d dt d(t) t=0 , where d dt d(t) = i [H, d(t)]
is determined by the commutator containing the full Hamiltonian H of the interacting molecule. The first spectral moment becomes

M 1 = d † (0) [H, d(t)] / . (3)
Moreover, H can be decomposed as a sum of the free-active-molecule Hamiltonian H a and the absorber-perturber interaction term (PES) W :

H = H a (Ω, q) + W (R, Ω, q).
Here R is the intermolecular distance vector (between the centers of mass of the absorber and perturber), Ω specifies the active-molecule axis orientation and q is a set of vibrational coordinates. Thus, the rhs side of Eq. ( 3) reads

d † (0) [H, d(t)] / = d † (0) [H a , d(t)] / + d † (0) [W, d(t)] / .
The intermolecular potential W , in its turn, consists of the isotropic part W iso (R) and two compounds related to rotations and vibrations:

W (R, Ω, q) = W iso (R) + W rot (R, Ω) + W vib (R, Ω, q).
The rotational perturbations W rot and W iso commute with the dipole moment d of a vibrational transition, thus the presence of solely rotational perturbation doesn't influence the value of the first spectral moment. For the vibrational transition f ← i one gets

M 1 = M 0 µ 0 1 + M 0 W vib f f -W vib ii / (4)
with the normalized first spectral moment in the absence of vibrational perturbation

µ 0 1 = v f |H a |v f -v i |H a |v i
and the double indices ii(f f ) denoting the vibrational diagonal matrix elements of W vib (R, Ω, q). Dividing Eq. ( 4) by M 0 one obtains

µ 1 = µ 0 1 + γn Rg ,
where n Rg is the number density of the perturbing gas and the band-origin shift coefficient

γ = W vib f f -W vib ii /(n Rg ) is introduced.

Band-origin shift computations

Direct computation of band shifts

Since the binary collision regime is confirmed by the linear dependence of measurements on n Rg (see Fig. 2), the average required for the shifting coefficient calculation can be performed with the pair distribution function [START_REF] Frommhold | Collision Induced Absorption in Gases[END_REF]:

W vib f f -W vib ii = n Rg dR dΩe - W ii k B T (W vib f f -W vib ii ), (5)
where the first integral is a volume integral and the second one is the integral over the active molecule orientation; the diagonal matrix element W ii corresponds to the full potential. Practical computations with Eq. ( 5) are determined by the availability of PES with vibrational dependence in the initial (typically, ground) and final (excited) states. Owing to relative simplicity of "linear molecule + atom" systems, their ab initio potential energy surfaces have been intensively studied in the literature. In the pioneering works, the absorbing molecule was treated as a rigid rotor [START_REF] Parker | Intermolecular potential surfaces from electron gas methods. I. Angle and distance dependence of the He-CO 2 and Ar-CO 2 interactions[END_REF][START_REF] Preston | Classical trajectory studies of rotational transitions in Ar-CO 2 collisions[END_REF][START_REF] De Dios | The 129 Xe nuclear shielding surfaces for Xe interacting with linear molecules CO 2 , N 2 , and CO[END_REF][START_REF] Buck | Observation of multiple-collision rotational rainbows in Xe-CO 2 : Comparison between TOF measurements and scattering calculations[END_REF][START_REF] Zhu | Ab initio potential energy surface and predicted microwave spectra for Ar -OCS dimer and structures of Arn -OCS (n = 2-14) clusters[END_REF][START_REF] Sun | A three-dimensional potential energy surface and infrared spectra for the Kr-OCS van der Waals complex[END_REF][START_REF] Harris | Intermolecular potential between an atom and a linear molecule: The structure of ArOCS[END_REF], thus the vibrational dependence was totally neglected. However, accounting for antisymmetric stretching vibrations is essential for calculating band shifts [START_REF] Cha Lasiński | State of the Art and Challenges of the ab Initio Theory of Intermolecular Interactions[END_REF] for both CO 2 and OCS. For direct computations we employed therefore only the vibrationally dependent potentials [START_REF] Cui | A new potential energy surface and predicted infrared spectra of the Ar-CO 2 van der Waals complex[END_REF][START_REF] Chen | Intermolecular potential energy surface, microwave and infrared spectra of the Kr-CO 2 complex from ab initio calculations[END_REF][START_REF] Chen | Potential energy surface, microwave and infrared spectra of the Xe-CO 2 complex from ab initio calculations[END_REF] constructed with noniterative inclusion of connected triples (CCSD(T)) and basis functions sets depending on the considered system.

Computation via dipole-induced dipole interaction

In the cases where vibrationally dependent PES are unavailable and Eq. ( 5) can not be used, evaluation of γ can be made with a simple model suggested bellow. Collisions with a polar active molecule may alter the spherical symmetry of the atomic perturber and induce on it an instantaneous dipole moment which will interact with the permanent (or transient) dipole of the molecule. This interaction can be a cause of the band origin shift. Indeed, the dipole moment D ind induced on an atom placed in the electric field E is proportional to the atom polarizability α Rg :

D ind = α Rg E,
and its potential energy reads

W ind = - 1 2 α Rg R 6 (3 cos 2 θ + 1)D 2
with the angle between D and R denoted by θ. For a transition from the ground state i to a state f Eq. ( 5) becomes

γ = πα Rg ( v f |D 2 |v f -v i |D 2 |v i ) ∞ 0 dR R 4 π/2 -π/2 e -W ii kT (3 cos 2 θ + 1) sin θdθ. (6)
In the harmonic approximation (D is proportional to q) one gets 1|D

2 |1 -0|D 2 |0 = 2|D|1 2 = 2 1|D|0 2 and 2|D 2 |2 -0|D 2 |0 = 4 1|D|0 2 .
For our calculations, the transition dipole moments d = 1|D|0 were taken as 0.374 D and 0.0473 D for the ν 3 and ν 2 OCS bands, [START_REF] Tanaka | Dipole moment function of carbonyl sulfide from analysis of precise dipole moments and infrared intensities[END_REF] respectively, whereas d = 0.3212 D was used for the ν 3 band of CO 2 [START_REF] Johns | Absolute intensities in CO 2 : The 4.3-and 2.7-µm regions revisited[END_REF].

The polarizabilities α Rg were 1.6404 [START_REF] Hohm | Interferometric measurements of the dipole polarizability a of molecules between 300K and 1100K II. A new method for measuring the dispersion of the polarizability and its application to Ar, H 2 , and O 2[END_REF], 2.4844 [START_REF] Lide | CRC Handbook of Chemistry and Physics[END_REF], 4.044 [START_REF] Teachout | The static dipole polarizabilities of all the neutral atoms in their ground states[END_REF] Å3 for Ar, Kr and Xe, respectively. The results of computations are presented in Tables 1 and2.

Discussion

The bulk of our experimental and theoretical results is collected in Tables 1 (for CO 2 ) and 2 (for OCS). From the experimental standpoint, the band-origin shift coefficients appear to grow with increasing polarizability of the perturbing gas. Moreover, for the OCS case, the order of magnitude for the band shift for the ν 2 band is significantly smaller in comparison with that for the ν 3 band, which confirms the proportionality of the band shift to the dipole matrix element associated with the transition. Our experimental γ-value for CO 2 -Ar ((-8.9 ± 0.2) × 10 -3 cm -1 amagat -1 ) has the same order of magnitude as the datum published by Ozanne et al. [START_REF] Ozanne | Line-mixing, finite duration of collision, vibrational shift, and non-linear density effects in the ν 3 and 3ν 3 bands of CO 2 perturbed by Ar up to 1000 bar[END_REF] (γ(ν 3 ) = (-5.4 ± 1.0) × 10 -3 cm -1 amagat -1 ) and, likely their results for the ν 3 and 3ν 3 bands of CO 2 -Ar, we find roughly the factor two between our measurements for ν 3 and 2ν 3 bands of OCS-Ar and OCS-Kr.

Due to the absence of vibrationally dependent potentials for OCS-Rg, direct calculations with Eq. ( 5) were feasible solely for CO 2 -Rg cases, and comparison with the DID model (Eq. ( 5)) can be made for this absorbing molecule only (Table 1). In the case where the same PES is used, for CO 2 -Kr both methods give close results, but for CO 2 -Ar and CO 2 -Xe direct computation leads to a smaller absolute value of the band shift than the DID model. It can be noted also that the DID results for CO 2 -Ar are very similar for various PES tested, while for CO 2 -Xe choice of the potential plays a significant role.

To summarize, both computational methods may be reliably applied to CO 2 /OCS mixtures with Ar, but with increase of the perturber polarisability the experiment gives significantly larger values. The nature of this discrepancy currently remains unclear and deserves further detailed consideration.

Figure 1 :

 1 Figure 1: 2ν 3 OCS-Ar (a) and OCS-Kr (b) band shapes at buffer-gas pressures of 25 (1), 100 (2), 200 (3) and 300 (4) atm (active gas present at trace level).

Figure 2 :

 2 Figure 2: Deviations of the first moment ∆µ 1 = µ 1 -µ 0 1 with increasing gas density and corresponding linear fits for the ν 3 CO 2 (a) and 2ν 3 OCS (b) bands: perturbation by Ar (1), Kr (2) and Xe (3).

  2) -7.25 -23(2) -14.50 -2(2) -0.12 OCS-Kr -19(2) -11.32 -34(1) -22.63 -5(2) -0.183. Theory

Acknowledgments

The authors are grateful for the financial support of the Russian Foundation of Fundamental Research (grant No. 19-03-00830). The spectra were recorded with core facilities of the "Geomodel" research center of St. Petersburg State University.