Author(affiliation)

Soohwan Kim(Chongshin University) Seonghun Kim (Korea Educational Broadcasting System) Seulki Kim(Seonbu Elementary School) Title Designing an Assessment Model of Computational Thinking in Elementary and Secondary Education.

Aims & Objectives

This proposal research is to create the assessment tools to evaluate students' learning of computational thinking. This research involves the following:

․ attempts to reshape the definition and elements of computational thinking within the current elementary and middle school of computing curriculum in Korea.

․ provides a computational thinking assessment model for teachers to assess what students know and how learning progresses ․ Verification and evaluation of the assessment model by applicability of the assessment contents according to the elementary and middle school level.

In order to complete this research, literature review of computational thinking is essential. we will argue trends of latest related research and limits of existing researches.

In use of Design and development, we will propose various methodologies applied with computational thinking assessment models; these include all forms of multimedia such as Evidence-Centered Design (ECD). (Mislevy, Behrens, Dicerbo, & Levy, 2012, p.11) In a verification part, the evaluation model was applied to specific elementary and junior high school teachers and then the teacher survey was conducted.

Background

Computational thinking

The viewpoint of computing thinking can be summarized in two ways. Papert (1981) focuses on specifying and expressing ideas, while Wing (2006) defines computing thinking as a process of problem solving. In other words, the way computers scientists think to solve problems is defined as Computational Thinking (CT). In addition, CSTA (2011), an association of computer science teachers in the United States, is a core principle of computer science focusing on abstraction, automation, and analysis that permeates all K-12 computer science standards, this is a problem-solving methodology that provides a clear means of analyzing and developing solutions to problems that can be solved by computers that can relate to one another. This is in line with Wing's (2006) claim that it is the ability to solve problems by integrating human thoughts and digital technologies by abstracting the key elements of the problem and automating the solution through computerized devices. Cuny, Snyder & Wing (2010) defines the problem process as a thinking process related to formalizing the problem and its solution, so that the problem solution is expressed in a form that can be efficiently executed by the information processing agent. In analyzing domestic and foreign literature, sub-elements of computing power can be summarized as in analyzing domestic and foreign literature, sub-elements of computing power can be summarized as in <Table 1> 부,2015b) The question "what should we assess in computational education?" directly links to the ultimate goal in computing education, which is developing student computational thinking. Particularly, the flow of educational evaluation nowadays is a Construct-centered approach which can be evaluated in the subject-oriented content evaluation, including the understanding of various complex thinking processes and content, and the ability to use the knowledge, [START_REF] Messick | The interplay of evidence and consequences in the validation of performance assessments[END_REF] suggest that

the constructivist approach may be valuable in society, such as critical thinking and creativity.

However, if it is difficult to assess students' abilities with existing assessment methods, claiming to evaluate constructs. If we need to evaluate constructs that are complex and difficult to assess, such as our thinking on computing, the existing content-based assessment method is not appropriate and the component-based assessment method is appropriate. The Evidence-centered Design

Assessment is a method that can be used to develop a structure evaluation system based on the evaluation logic. Evidence-based evaluation design is a comprehensive evaluation system that explains all elements of mathematical computation and reasoning methods used in the concept of education evaluation, task development, and evaluation " (Mislevy, Behrens, Dicerbo, & Levy, 2012, p.11) A theoretical and practical approach to the development and design of evaluation tasks in our research.

Proposed methodology

The characteristics of the computational thinking assessment model can be summarized as follows.

1. To derive the components of computing thinking through literature related to domestic and international computing thinking. The element of computing thinking appropriate from the perspective of elementary and middle school education is set through expert consultation 2. To develop the evaluation criteria for evaluating the components of computing thinking using ECD technique. The cognitive function of computing thinking is assessed by Dr. Scratch's modifying rules, and the attitude is evaluated by analyzing the student portfolio.

3. To develop the computational thinking assessment model satisfied with all such conditions, We will make web-application modules and code analysis DB system modules using web-programming languages such as Javascript, JAVA, php to improve information exchange as well as to evaluate alternative plans effectively. In addition, I will use MS-SQL to save data and information created and reorganized by participants.

4. To validate Computational thinking model, after applying the evaluation model to 20 elementary and junior high school teachers' classes, we will conduct teacher survey and get a conformity to the field application of the computational thinking ability model and system was confirmed through FGI (Focus Group Interview) targeting elementary and middle school teachers.

Table 1

 1 Table 1) are primarily knowledge and function elements, In recent years, besides the Approaches area in the CAS in the United Kingdom, the AP course in the United States offers examples of how to learn CT by presenting Big Idea and Computational Thinking

	Sub-elements of computational thinking
	KERIS(2015)	Wing	CSTA & ISTE (2011)	CAS (2016)	Google (2014)
			(2006)
	Data Collection	Abstraction	Data Collection	Patterns	Pattern
	Data Analysis		Data Analysis		recognition
	Structuring		Data Representation	Logic
	Abstraction	Decomposition		Problem Decomposition	Abstraction	Abstraction
		Modeling			Decomposition	Decomposition
		Algorithms		Abstraction
	Automation	Coding	Automation	Algorithms & Procedures	Algorithms	Algorithms
				Automation
		Simulation
				Simulation
	Generalization			Evaluation
				Parallelization

for a future workforce with necessary computer skills. In the Korea, there has been a governmental level push in computer science education. Starting computer science education in K-5 contributes critically to this initiative since teachers have faced is the lack of evaluation standards and tasks. (교육


Corso Padova

Jeffrey Barton

Michael G Dunn

Steve Manwaring

Gamaliel Young

Maurice Adams

Michael Adams

Maurice Michael Adams

Adams

Development of an Experimental Capability to Produce Controlled Blade Tip/Shroud Rubs at Engine Speed

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

Introduction

Contemporary gas turbines rely on complex blade-casing rubin-systems to improve the tip clearance behavior during operation from initial service to maximum lifetime ͓1,2͔. Typically a rub-insystem in the compressor section of the engine may consist of a specific circumferential area of the metal alloy case shaped to accept coatings of materials selected for in-service wear and, when required, fire shielding. A momentary eccentricity of a few tenths of 1 mm between a blade and the engine case, which may occur because of an abrupt maneuver in flight or in a hard landing, may cause the tips of a few airfoils to incur the rub-in surface.

Blade-to-case rubs can degrade the operation of an engine through the induction of high amplitude shaft vibration and severe blade/seal wear, and may lead to catastrophic failure of the engine in the worst occurrence. Precisely for these reasons, engines are already routinely tested for containment and survival of blade out events. But to enable the formulation of a comprehensive design methodology for rub-in-systems, particularly one capable to predict and take advantage of the potential offered by new and emerging rotor/blade concepts, advances in several underlying technologies are needed. Among them are: rotor dynamics under severe disturbances ͓3-6͔, forced vibrations of bladed disks ͓7-9͔, impact dynamics ͓10-13͔, and abradability of materials ͓14͔.

While the studies reported earlier and in the companion literature have provided excellent conceptual frameworks for developing the desired design methodology, there is much need for controlled experimental data obtained in experiments conducted with real engine components. The experimental capability described in this paper is intended to fill such need.

In the development of the facility and experimental method reported, considerable emphasis was placed on engine manufac-turer input regarding the facets of the rub phenomena where results would yield the highest design-advance payoff. Characterizing blade impact dynamics at engine speed was highest on the industry list. Seminal work on blade contact forces during rubbing has been published by Ahrens et al. ͓12͔; however, the measurements were done at a rotational speed about one order of magnitude below those typical of aircraft engines. In addition, engine hardware was not used in this earlier research activity. By contrast, characterizing the high-frequency disturbances occurring at blade first contact and at detachment during repeated rub events is important in the current research program and justifies the preference for experimenting with engine hardware rotating at design speed.

Experimental Arrangement

The arrangement of an in-ground spin-pit facility ͑SPF͒ specifically designed to investigate aeromechanic phenomena for gas turbine engine hardware rotating at engine speed is described in this section. The underground containment tank that houses both the rotating component and a high-stiffness spindle is cylindrical ͑1.52 m in diameter by 0.65 m high͒, and has 0.90 m diameter openings at the top and at the bottom. Figure 1 is a photograph of the facility with the protective top cover in place.

Vertical axis spin testing for design validation of turbomachinery components dates back to World War II applications, which sought to solve problems with military aircraft turbochargers. However, the research projects envisioned from the inception of this facility design dictated the use of a high-stiffness spindle. This requirement forced a significant departure from the common way to conduct a spin test, whereby a rotor is suspended in the heavily armored vacuum tank from a flexible shaft. That shaft arrangement allows the rotor to find its own balance axis and eliminates the need for extreme precision in mounting and balancing. Unfortunately, it is unsuitable to reproduce blade-to-case rubs that can degrade the performance of the engine through introduction of high amplitude shaft vibration and severe blade/seal wear.

Thus the first major subsystem component required for the facility is a vertical and high-stiffness spindle capable of operating at 20,000 rpm for a reasonable period of time. The system utilized here incorporates hybrid bearings and an oil mist lubrication system. This component will be described in more detail later in the paper.

A second major subsystem required to investigate blade-to-case rubs is a fast-acting mechanism that allows insertion of the engine casing into the path of the bladed rotor. Such a mechanism was designed and developed to research asymmetric and full rotor clearance closure for engine stages. In single blade asymmetric clearance closure investigations, a rapid contact of the bladed rotor with its housing is simulated as would occur in an abrupt flight maneuver. In full rotor clearance closure, an extended contact of the bladed rotor with its housing is simulated as could be experienced in some flight phases due to temporary thermal imbalances in different parts of the engine. The known requirements for those types of investigations have defined the general layout of the electromechanical, pneumatic, and measurement subsystems of the facility. The fast-acting incursion mechanism is described in detail later in this paper.

In addition to the two major subsystems noted earlier, a few other facility design features are key in making the facility a versatile tool for performing research on many areas of aeromechanics, including rubs, blade excitation, and blade dampers. A modern high-efficiency air turbine is used to spin-up rotors with large moment of inertia in a few minutes. Moderate vacuum is maintained in the containment tank during operation to eliminate high air friction losses and to reduce the risk of explosion of metal dust or oil fog during an experiment. A high-speed slip ring and hollow spindle is used to conduct strain gauge signals from an instrumented airfoil to the laboratory data collection instrumentation.

High Stiffness Spindle.

The vertical spindle of the SPF shown in Fig. 2 is capable of operating at 20,000 rpm. It consists of a hollow shaft supported on high speed, ceramic hybrid, angular contact ball bearings. The shaft is driven from the top by a reversible air motor. The single stage engine rotor is bolted beneath the spindle assembly and is surrounded by the containment test chamber. The outer spindle housing is made from an aluminum forging. A steel sleeve is inserted into the aluminum housing and serves as the bearing housing. Figure 3 show a photo of the spindle assembly with the slip ring and the drive cover removed. The connector assembly and drive are exposed at the top of the upper housing.

The first critical speed of the spindle is higher than the maximum operating speed. The analysis of the high-stiffness bearing arrangement indicates a shaft critical speed of 23,500 rpm for the design manufacturing tolerances and assumed operating temperature effects.

Incursion Mechanism System.

The function of the incur-sion mechanism is to move the engine-case segment into the blade path at an exact time for a precise distance and duration, which then determines the number of blade strikes. After a few strikes, the mechanism retracts the casing out of the blade path.

The experimental configuration allows insertion of a segment of engine casing into the path of single or multiple bladed disks while the disk is operating at engine speed. In the current configuration, a 90 deg sector of a representative engine casing is forced to rub the tip of a single bladed compressor disk for a limited number of times with predetermined blade incursion into the cas- ing. For the facility development measurements described herein, the speed at the time of rub was on the order of 17,000 rpm. For this rotational speed, the instrumented airfoil returns to the same circumferential position in 3.8 ms, thus several contacts between blade and shroud are experienced before the eccentricity disturbance is removed. The compressor disk and blade rotate on a vertical rigid spindle that is operated by an air turbine. The casing sector is mounted on a unit based on three piezoelectric load cells and attached to a support ring that swivels in the horizontal plane. The single throw by a fast acting gas piston attached to a cam follower is coupled to the support ring and brings the casing in and out of the path of the rotating blade for a few rubs over a time interval of about 20 ms. The incursion of the rubs is prescribed by setting the initial position of the mechanism in increments of 5 m.

The incursion mechanism can be broken down into three major subsystems. The structural subsystem consists of a support ring that carries the casing segment along during its motion, a linear cam device that controls the motion, and high-stiffness components anchoring all elements of the system to the containment tank. The actuation subsystem moves the cam block on a linear slide via a pneumatically operated piston and its associated controls. The third subsystem comprises the motion instrumentation and triggering electronics.

The function of the support ring is to carry the casing segment that is mounted to the load-measuring unit ͑LMU͒ through its travel cycle into and back out of the blade path. The ring surrounds the disk and pivots on a near zero clearance cylindrical bearing. The stationary pivot point of the bearing is a hollow steel shaft surrounded by a bronze oil-impregnated sleeve. Support for this pivot is provided by a gusseted weldment that attaches to the bottom surface of the containment tank. A cam-follower pin is built into the support ring at the location 180 deg from the pivot. The pin follows a sliding cam profile to effect ring motion. Vertical ring movement is held to less than 125 m by a set of oil impregnated bronze bearing pads held in a "C" clamp like assembly. The pads are in contact with the ring reinforcement webs. There is a second bolt-on reinforcing web that spans the lower diameter of the ring to help maintain dimensional stability under load. Figure 4 shows a schematic of the structural subsystem and identifies its main elements. Figure 5 shows a picture of the subsystem.

A slot in the cam block guides the cam-follower pin on the support ring. The block is mounted on a linear slide and both are moved thru their travel with a pneumatic piston. The slide and block travel path is along the centerline that extends from the ring follower pin to the ring pivot. A shock absorber is mounted near the end of the slide base to prevent over travel. This assembly is mounted to an adjustable and lockable mechanism that travels perpendicularly to the slide movement to provide initial static adjustment. Support for these components is a table mounted to the bottom surface of the vessel with substantial welded gussets to the vessel sides for stability. Figure 6 shows a schematic of the subsystem and identifies the main components.

The slide is of a square gib design with a 23 cm saddle ͑moving component͒ and a 51 cm base for a 28 cm total travel. Anti-friction surfaces are Roulon® and are set for zero clearance by adjusting a line of setscrews. Rated static loads are 170 kN in the vertical plane and 18.8 kN in the horizontal plane. The intended design of this slide is for high speed, high load, and high accuracy machinery. The high load rating and close tolerances allow for accuracy and minimal overshoot in ring travel.

The cam block is machined from C1018 steel ͑ASTM-A108͒ and bolted and pinned to the slide saddle. This particular steel was chosen for the finish quality obtainable and desired for a smooth surface at the follower pin contact. The profile of the cam is shown in Fig. 7. This profile, along with the travel speed characteristics provide for incursion timing ͑the casing must be fully extended into the blade path in less than one revolution͒, and length of time that the casing is in the blade path ͑referred to as dwell time͒. Maximum incursion distance is also controlled. Then, the casing must be retracted out of the blade path and the cam block/slide saddle slowed safely to a stop. The piston actuator is double acting with a 65 mm bore capable of 4.37 kN force.

The diagram of Fig. 8 shows the pneumatic and electrical components, and pressure instrumentation for controlling and monitoring the pressurization of the actuator piston. The reservoir is positioned close to the extend Valcor® valve in order to ensure adequate volume to the piston and is pressurized with gaseous helium. The Valcor® is an axial flow fast-acting full flow valve. Actuation times are on the order of 12 ms and very repeatable. The interval timer accepts either a local or remote trigger and has user programmable times for delay time and on time. The Valcor® valve requires a high initial voltage and so has its own supply in a metal enclosure separate from the control room. Pressure readouts are used as displays for initial setup, and paralleled to oscilloscopes and the data acquisition system for performance diagnostics. The retract side of the piston is connected to a three way valve using vacuum at prefire setup, shop air for post-test retraction, and a vent to ambient to relieve retract pressure. The operation of the subsystem is described later in Sec. 3.

Load Measuring Unit.

The segment of engine casing against which the blade rubs is carried by a metric support that includes three load cells. Each piezoelectric load cell measures three force components. Two end cells are placed below the load line, which is located in the midplane of the engine case. The center cell is located above. The arrangement is designed to minimize the moment loads on each cell and to obtain moment information around three mutually perpendicular axes from the entire unit.

Five accelerometers are also mounted on the LMU, which is shown in Fig. 9. Three radial directions are measured at locations very close to each load cell. A triaxial block, which is mounted near the center load cell, is equipped with the remaining two accelerometers that measure one axial and one circumferential direction in the middle of the LMU.

As in the actual engine, the radius of the casing cylinder is greater than the radius of the circular path followed by the blade tips under normal operating conditions. A tight but finite clearance is maintained at the rub-in-system surface. This condition is applied to ensure that no impact of the blade will occur first at the

Operation

During a typical experiment the rotor is spun-up to the target speed in less than 3 min. As the rotor speed approaches the target value selected for a rub to occur, the angular acceleration is gradually decreased so that the airfoil impact with the shoe occurs effectively at a constant value of rpm. A few rubs are obtained as follows. First, the casing movement toward the spinning blade tip is initiated by the triggering system described later in this section. This results in the motion shown in Fig. 10. Also shown in the figure are the markers that indicate the passing of the blade over the LMU for duration of about 1 ms. Second, during the casing motion, the tip sweep line is crossed as depicted in the figure. After this point in time, the LMU is in the path of the spinning Third, during the time that the casing is in the maximum incursion position, up to six rubs can take place for the values of rotational speed at which experiments were conducted. In fact, due to the settings chosen in the triggering circuit, five rubs were obtained in most cases. All zero-lead incursions occur with the same nominal interference between the path of the blade tip and the arc of the casing. The deviation of the actual interference from the nominal value was monitored in repeated experiments and found to be less than ±3.8 m.

Fourth, the continuing motion of the cam forces the casing to begin to retreat. In this phase, the rubs continue because the casing is still inside the incursion region. As before, the more severe the incursion, the more rubs occur before the casing clears the exit boundary of the incursion region.

Finally, the casing crosses the tip sweep line and the rubs end. In the experiments performed to this date, the maximum incursion imposed was for 400 m and resulted in a total of 15 rubs.

It is important to note that the measurements obtained during each experiment should be examined according to a number of different time scales. Five scales are illustrated in Fig. 11. In order of decreasing duration they are: the time of damping of blade excitation after a set of rubs, the time spanned by an entire set of rubs, the time for one revolution, the time interval a blade spends in free oscillation after a rub and before returning for a subsequent rub, the time interval during which the blade sweeps over the LMU, and the duration of an individual rub.

It is precisely to identify during an experiment the different scales of the rub phenomena that two types of markers are recorded while operating the SPF. First contact is detected by a sensor and is used to begin data acquisition in one type of experiments ͑disk/blade growth experiments͒ as described later. In the second type of experiments ͑ramp-incursion experiments͒ that are discussed in greater detail in this paper, the shoe motion toward the spinning blade is manually initiated and is followed, after a suitable delay, by the signal to begin data acquisition. In either case, the data acquisition system is already running in sampleand-hold mode at the precise instant when the command to acquire is received, and a prerub hold time is programmed into it. Thus any of the measurement histories shows typically a prerub value for about 200 ms, the onset of the rub event, the dynamic response resulting from the impact of the airfoil with the shoe, the release of the airfoil immediately following detachment from the shoe, the vibrational response of the airfoil as it spins around once more after having been excited by the impact, the effect of repeated impacts, and the decay in dynamic response after the conditions causing the incursion have been removed. Typical histories last 829 ms and consist of a few to several rubs depending on the type of experiment.

It is important to relate the above sequence of events to rub occurrences in flight. It is clear that in the experiments just like in flight, for a given incursion depth or severity of a rub event, the rate of incursion in each strike will be positive, null or negative during different phases of the set of rubs that makes up an event. By comparing measurements obtained at different experimental conditions, the effects of incursion depth and rate of incursion can be separated. In the middle rub of Fig. 11, one can see the blade detaching and reattaching during a single strike interval.

Furthermore, the effects of rub repetition can be identified. For the example shown in Fig. 11, the first rub is clearly brief and is followed by rubs of equal duration ͑250 s͒ later. Although not a full engagement, the 1st rub is important because it imparts to the airfoil an initial vibratory motion. In addition, in zero-lead rubs, the deviation of the actual interference from the nominal value was monitored in repeated experiments and found to be less than ±3.8 m ͑±1.5 mil͒. Rub Nos. 2 and 3 are equal in duration and incursion depth. However, as the blade comes around after each rub it is excited and one would intuitively expect that rub Nos. 2 and 3 will be different in blade vibratory response and in tip loads during contact to the casing. The extent to which this was found to be true is discussed in Sec. 4 later.

The actuation of the incursion mechanism system is obtained as follows ͑reference should be made to Fig. 8 earlier͒. Initially, the air piston is retracted with shop air, vented, and the retract side brought to a vacuum. Both extend side Valcor® valves are closed and the reservoir is charged with helium. The interval times are set and the timer is armed and ready for either a manual or remote trigger. When a trigger is received the timer waits for the delay time then fires the Valcor® valves through its power supply for a set-on time. The piston then pushes the cam block through its travel.

Ring and attached LMU, casing segment initial position, and casing actuated motion are measured with a capacitance probe referred to as the casing stand off ͑CSO͒ sensor. Lion Precision manufactures the probe and associated electronics. This system has a certified resolution of 0.36 m, a range ±1 mm, and a bandwidth of 16,800 Hz. The output is in volts. Cam block motion is measured with a LVDT manufactured by UniMeasure. It has a 25.4 cm stroke and has an in-house calibration of 0.3%. Several combinations of cam profile, helium pressure, and valve timing were tried during a total of 51 trial runs. The last of these runs is plotted in Fig. 12.

In disk/blade growth experiments, rubs of progressively in- creasing severity were obtained by using the expansion of the rotating component under increasing rotational speed as the mechanism driving the interference relative to a stationary casing. The total number of rubs obtained in these experiments was greater than the number of rubs in the ramp-incursion case by at least one order of magnitude. The focus of this class of experiments is on the vibratory response of the blade when undergoing a multitude of excitation cycles.

The trigger to start both the data acquisition and incursion systems must be precisely controlled. Two different triggering schemes were used. One for the disk/blade growth experiments where the casing segment is stationary and the blade grazes it as a result of tip growth. The other for the ramp-incursion experiments where the casing is suddenly brought into the blade path. The high-speed data acquisition has a selectable window in which to collect data and record the rub events, therefore it must trigger at the appropriate moment. The incursion mechanism is stationary for the disk/blade growth experiments, requiring no trigger. For the ramp-incursion experiments it must be precisely timed.

For the disk/blade growth experiments, a voltage signal needed for the trigger to start the acquisition is produced when the blade tip first touches the casing segment. This is accomplished by setting the rotor to 5 V and the casing to ground. The blade tip touching the shoe completes a circuit and triggers the data system. Figure 13 shows a simplified diagram.

The triggering circuitry for the ramp-incursion experiments is fairly involved. The object is to get the casing sector to move out and set in place during the time interval when the blade cannot interfere with the casing surface. This time interval spans several blade revolutions, as can be seen from Fig. 10. Using a selectable time delay from a trigger that is related to instantaneous blade position, the object is achieved by delay increments equal to a fraction of one revolution. An optical sensor placed to sense blade passage at the leading edge of the shoe supplies the trigger signal. The incursion system timer can resolve to a millisecond, which is too coarse for this application, so a delay generator capable of resolving 0.1 ms is placed in front. A manual trigger-enable switch disables the sensor trigger until the rotor set speed is reached. The data acquisition system is triggered with the incursion system timer so that casing movement can be monitored during the experiment and correct position relative to blade location can be verified. Figure 14 shows a simplified diagram.

The facility instrumentation is conveniently grouped into two categories: rotating sensors and stationary sensors.

The rotating sensors consist of miniature dynamic strain gauges arranged on the suction and pressure side of the blade. Excitation power and signal response are carried by electrical wires routed through the hollow shaft of the spindle to the rotating terminal of a slip ring unit. The stationary terminal of the unit is directly patched into the high-speed data acquisition system. In the present experiments all strain gauges were operated in constant current mode.

The primary stationary sensors consist of the three piezoelectric load cells that are part of the LMU and of the five piezoelectric accelerometers, also mounted on the LMU.

Results and Discussion

The measurements obtained simultaneously from load cells, accelerometers, and strain gauges during nearly a second of data acquisition time, when combined, offer a detailed description of the rub phenomena. The purpose of this paper is to describe the facility design and operation and the measurements later are used merely to illustrate the capabilities of the facility. Extensive results from specific applications and research projects will be the subject of subsequent papers. Load cell and strain gauge measurements are discussed briefly later. The measurements of acceleration are qualitatively very similar to those obtained from the load cells.

Typical measurement traces are shown in Fig. 15 and 16. In these figures, the initial prerub level of the sensor, the response during about 10 ms of multiple rubs, and the postrub sensor response are easily identified.

The three load cell measurements from top to bottom of Fig. 15 are for the first cell, center cell, and last cell passed by the blade during each revolution. Note how quickly after the rub interval the measured loads return to the prerub level. The two typical strain gauge measurements from top to bottom of Fig. 16 are for a gauge located at the tip and for a gauge located at the root of the blade, respectively. Note that the tip gauge indicates localized disturbances during the rubs, a temperature-induced displacement of the baseline, and no postrub oscillations. In contrast, the root gauge shows oscillations building up from rub to rub followed by an exponential decay after the last rub. As the blade rubs on the engine-case segment attached to the LMU, blade tip loads are distributed to the three LMU load cells via the rigid transfer bar. Thus the LMU acts as a mechanical filter on the tip loads. A transfer frequency response function ͑FRF͒ algorithm is needed in relating engine-casing loads to load cell measurements. In brief, the theory for the FRF algorithm is based on characterizing the LMU dynamics by impulse vibration testing, whereby the output response at location x 0 is given by u͑x 0 ,t͒ = H oi ͑͒Fie jt where H oi ͑͒ is the output FRF at location o per unit force applied at location i.

In order to implement the FRF algorithm we proceeded as follows. Prior to conducting rub experiments, input loads from an impulse hammer were applied at three locations, in three direc-tions for each location. At the same time, nine response measurements were recorded from the load cells. The elements of the matrix H oi ͑͒ were computed as ratios of measured outputs to known inputs in the frequency domain. Then the inverse of the output FRF matrix was computed and stored for later use in conjunction with rub measurements.

During a rub experiment, nine response measurements are recorded from an unknown force at the rub surface. It is precisely this resultant force that is computed from the known instantaneous blade position and from the inverse of the output FRF operating on the load cell measurements transformed to the frequency domain. The computation earlier is then repeated for a discrete set of blade positions during the rub.

At present, issues associated with the detailed requirements of impulse vibration testing have restricted the application of the FRF algorithm to two components of the blade tip load, the radial and the circumferential component. However, finite element structural modeling is in progress to remove such restriction. In the Much of the paper to this point has been devoted to describing the facility design and operation, but Figs. 10 and 11 given earlier and Fig. 17 given later illustrate an example of measurement of transient dynamics for the blade and casing vibroimpact response that is important to the industry. The reader attention is to be focused here on the shoe-sweep time scale to compare one of the zero-lead rubs, No. 3, to the first rub ͑see Fig. 11͒. Figure 17 shows the measurements again for the root gauge and the tip gauge. It is noticeable that the traces show distinct spikes at the beginning and end of the most perturbed region. The time interval between these two spikes correlates well in all experiments with the indications of electrical contact shown earlier as the bladecontact marker, so the first spike is taken as indicating an attachment stress wave and the last spike a detachment stress wave. In this case the blade contact lasted about 200 s.

Looking first at the root gauge, the first rub is not a full engagement rub. The centrifugal load has been subtracted out and we see that relative stress is zero up to first attachment. Then there is some high frequency content probably related to a changing area of contact as the airfoil flexes and twists under load, and after a low frequency oscillation gradually develops. It is a first flex frequency that persists also at later times. The detachment spike is visible. Notice that it is a bit early relative to the companion event in the third rub. Again, this first rub is not a full engagement strike. Now let us follow the third rub of the root gauge. The blade comes around and is excited in the first flex mode shape. It attaches first and it results in a history more suggestive of an increasing-decreasing contact area. Then it detaches and resumes the first flex motion.

Looking now at the tip gauge, the following occurs: the start of the first rub contact is clearly marked by a first positive spike followed by smaller disturbances because of varying contact area and local blade detachment. A distinct negative spike marks final blade detachment. In rub No. 3 the spike marking the initial blade impact is stronger and is followed by a more uniform load during contact consistent with a deeper incursion. The detachment spike again ends the strike at about the same level encountered during rub No. 1.

It must be stressed that these are the durations involved in a real engine rub. It is necessary to understand the physics under these conditions, not at some longer duration that is easier to obtain experimentally. The detailed measurements provided in this paper have illustrated that the excitation state of the airfoil on the second and subsequent encounters with the shroud is very different from the initial encounter. This finding has resulted in re-evaluation of the modeling commonly used in design tools used for predicting the blade/shroud incursion interactions.

Conclusions

An in-ground spin-pit facility specifically designed to investigate aeromechanic phenomena for gas turbine engine hardware rotating at engine speed has been utilized successfully to obtain specific information related to prediction and modeling of bladecasing interactions as reported herein.

The results presented in this paper demonstrate that sudden incursions of varying severity can be controlled to increments of 5 m. Rubs can be repeated for detailed investigation of tip loads and vibratory stresses. In addition, the amplification of blade oscillations under successive rubs can be quantified. In the future, refinements in tip load measurements will be obtained from finite elements modeling and the effect on rub forces from varying incursion rate will be investigated.

The purpose of this paper was to describe the development of the experimental capability along with results that demonstrate the operability of the facility and the importance of these results to the engine designer. It is interesting to note that a blade is capable of sustaining incursions that are more significant than might have been initially thought. The dynamics of high-frequency disturbances occurring at blade first contact and at detachment can be characterized relative to low-frequency oscillations in repeated rubs. An indication of tip loads can also be deduced from the measurements.

In combination with refined design methods and refined modeling of frictional forces, the greater understanding of the phenomena that affect blade-casing interactions is expected to result in savings in gas turbine maintenance costs applicable not only in propulsion engines but for stationary advanced power systems as well. Incursion measurements obtained using actual engine hardware at engine rotational speed have demonstrated the need for improved modeling of the event.

Fig. 1 Fig

 1 Fig. 1 General view of facility "with protective top cover…

Fig. 4

 4 Fig. 4 Incursion mechanism support ring

Fig. 5

 5 Fig. 5 Incursion mechanism photo

Fig. 8

 8 Fig. 8 Pneumatic control system

Fig

  Fig. 11 Time Scales of blade rubs

Fig. 13

 13 Fig. 13 Blade contact trigger circuit

Fig. 14

 14 Fig. 14 Optical sensor trigger circuit

Fig. 17

 17 Fig. 17 Strain gauge measurements for shoe-sweep time scale

Acknowledgment

The work described in this paper was performed as part of the General Electric University Strategic Alliance program at The Ohio State University. The assistance of GE personnel in all stages of the project is greatly appreciated.


Léo Glangetas

Ibrahim Jrad

NUMERICAL COMPUTATION FOR THE NON-CUTOFF RADIALLY SYMMETRIC HOMOGENEOUS BOLTZMANN EQUATION

Keywords: July 7, 2017. 2010 Mathematics Subject Classification. 34K08, 35Q20, 35-04, 35P05, 35P30, 80M22 Boltzmann equation, kinetic equations, spectral decomposition, symbolic computation, numerical computation

For the non cutoff radially symmetric homogeneous Boltzmann equation with Maxwellian molecules, we give the numerical solutions using symbolic manipulations and spectral decomposition of Hermit functions. The initial data can belong to some measure space.

1. Introduction 1.1. The Boltzmann equation. The Boltzmann equation, derived by Boltzmann in 1872 (and Maxwell 1866), models the behavior of a dilute gas (see [START_REF] Boltzmann | Weitere studien über das wärmegleichgewicht unter gas-molekülen[END_REF]). As we know, Boltzmann has created a theory which described the movement of gases as balls which could bump and rebound against each other [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF][START_REF] Dimarco | Numerical methods for kinetic equations[END_REF]. This model can be considered by one of many cases which represent the so-called kinetic equation. Presently, the diversity of sciences and applications contains these models such as rarefied gas dynamics, semiconductor modeling, radiative transfer, and biological and social sciences. This type of equations is made by including a combination of a linear transport term and several interaction terms which provide the time evolution of the distribution of particles in the phase space. The equation that bears his name is the following

∂ t f + v.∇ x f = Q(f, f )
where f = f (t, x, v) ≥ 0 is the probability density to find a particle at the time t, on the position x and with velocity v where the physical and the velocity space are located in three dimensions. The term v.∇ x f describes the free action of particles and Q(f, f ) is a bilinear operator which describes the binary collision process. It is called the Boltzmann collision operator and given by

Q(g, f )(v) = R 3 S 2 B(v -v * , σ)(g(v ′ * )f (v ′ ) -g(v * )f (v))dv * dσ
where for σ ∈ S 2 , the symbols v ′ * and v ′ are abbreviations for the expressions,

v ′ = v + v * 2 + |v -v * | 2 σ, v ′ * = v + v * 2 - |v -v * | 2 σ,
which are obtained in such a way that collision preserves momentum and kinetic energy, namely

v ′ * + v ′ = v + v * , |v ′ * | 2 + |v ′ | 2 = |v| 2 + |v * | 2
where |•| is the Euclidean norm on R 3 . Note that v, v ′ are the velocities before collision and v * , v ′ * the velocities after collision. The non-negative cross section B(z, σ) depends only on |z| and the scalar product z |z| • σ = cos θ where θ is the deviation angle. Without loss of generality, we may assume that this cross section is supported on the set cos θ ≥ 0. See for instance [START_REF] Lerner | Spectral and phase space analysis of the linearized non-cutoff Kac collision operator[END_REF] for more details on the cross section and [START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF] for a general collision kernel. For physical models, it usually takes the form

B(v -v * , σ) = Φ(|v -v * |)b(cos θ), cos θ = v -v * |v -v * | • σ, 0 ≤ θ ≤ π 2 ,
where Φ(|vv * |) = |vv * | γ is a kinetic factor and γ > -3.

In this work, we consider the spatially homogeneous case, that means the density distribution f = f (t, v) depends on the variables t ≥ 0, v ∈ R 3 and is uniform with respect to x. So that the Boltzmann equation reads as

∂ t f = Q(f, f ), f (0, v) = F (v) (1.1)
where the initial data F is depends only on v. For the collision kernel, we study only the Maxwellian molecules and non-cutoff cases (see [START_REF] Desvillettes | Regularization for the non Cutoff 2D Radially Symmetric Boltzmann Equation with a Velocity Dependant Cross Section[END_REF][START_REF] Desvillettes | Regularization Properties of the 2-Dimensional Non Radially Symmetric Non Cutoff Spatially Homogeneous Boltzmann Equation for Maxwellian Molecules[END_REF][START_REF] Desvillettes | On the Smoothing Properties of a Model Boltzmann Equation without Grad's Cutoff Assumption[END_REF][START_REF] Desvillettes | On a Model Boltzmann Equation without Angular Cutoff[END_REF][START_REF] Desvillettes | Smoothness of the Solution of the Spatially Homogeneous Boltzmann Equation without Cutoff[END_REF][START_REF] Gressman | Global Classical Solutions of the Boltzmann equation without angular cut-off[END_REF][START_REF] Lerner | Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff[END_REF]), that means the kinetic factor Φ ≡ 1 and

b(cos θ) ≈ 1 |θ| 2+2s , 0 < s < 1, θ ∈ 0, π 2
.

(1.2)

1.2. Results on the Boltzmann equation. With the previous assumption (the non-cutoff case) on the cross-section, there is existence of a weak solution for the Boltzmann equation (1.1) for a positive initial value F ∈ L 1 2+δ (R 3 ). See [START_REF] Villani | On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations[END_REF] and many others. Moreover, it is well-known that there is a regularization effect in Sobolev and Schwartz or analytic spaces for any time t > 0 (we refer the reader to [START_REF] Desvillettes | Regularization for the non Cutoff 2D Radially Symmetric Boltzmann Equation with a Velocity Dependant Cross Section[END_REF][START_REF] Desvillettes | Regularization Properties of the 2-Dimensional Non Radially Symmetric Non Cutoff Spatially Homogeneous Boltzmann Equation for Maxwellian Molecules[END_REF] and recently [START_REF] Alexandre | Regularizing Effect and Local Existence for the Non-Cutoff Boltzmann Equation[END_REF]) and that the solutions converge to the Gaussian when the time tends to infinity ( [START_REF] Gressman | Global Classical Solutions of the Boltzmann equation without angular cut-off[END_REF]).

An important point that our distribution lives in a multidimensional space: this reason make us think that we have a numerical problem because in this case the computational cost is more or less forbidden [START_REF] Dimarco | Numerical methods for kinetic equations[END_REF]. The study of the numerical part for kinetic equations is not obvious due to many difficulties come from the computational cost. To clarify more, we mention two of these difficulties: It is clear the appearing of multiple scales, and then to get out of the resolution of the stiff dynamics, one should build suitable numerical schemes [START_REF] Jin | Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms[END_REF][START_REF] Jin | Efficient asymptotic-preserving (ap) schemes for some multiscale kinetic equations[END_REF][START_REF] Bennoune | Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics[END_REF][START_REF] Dimarco | High order asymptotic-preserving schemes for the Boltzmann equation[END_REF][START_REF] Jin | Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations, a review[END_REF][START_REF] Dimarco | Asymptotic preserving implicit-explicit Runge-Kutta methods for nonlinear kinetic equations[END_REF]. The other one is that the collision operator is defined by multidimensional integrals and to compute one should solve it point by point as physical space [START_REF] Pareschi | Numerical solution of the Boltzmann equation I: Spectrally accurate approximation of the collision operator[END_REF][START_REF] Filbet | Solving the Boltzmann equation in N log2 N[END_REF]. To treat kinetic equations numerically, there is several ways which are used over the centuries until now: probabilistic numerical methods such as Direct Simulation Monte Carlo (DSMC) schemes [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF][START_REF] Bird | Molecular gas dynamics and the direct simulation of gas flows[END_REF], and, deterministic numerical methods such as finite volume, semi-Lagrangian and spectral schemes [START_REF] Dimarco | Numerical methods for kinetic equations[END_REF].

There are two important deterministic methods which are used in the past decades : the discrete velocity method (DVM) [START_REF] Goldstein | Investigations of the motion of discrete-velocity gases[END_REF][START_REF] Rogier | A direct method for solving the Boltzmann equation[END_REF][START_REF] Bobylev | On approximation of the Boltzmann equation by discrete velocity models[END_REF][START_REF] Buet | Regularized Boltzmann operators, Simulation methods in kinetic theory[END_REF][START_REF] Panferov | A new consistent discrete-velocity model for the Boltzmann equation[END_REF][START_REF] Cai | Entropy monotonic spectral method for Boltzmann equation[END_REF] and the Fourier spectral method (FSM) [START_REF] Bobylev | Difference scheme for the Boltzmann equation based on fast fourier transform[END_REF][START_REF] Pareschi | A Fourier spectral method for homogeneous Boltzmann equations[END_REF][START_REF] Pareschi | Numerical solution of the Boltzmann equation I: Spectrally accurate approximation of the collision operator[END_REF][START_REF] Gamba | A fast spectral method for the Boltzmann collision operator with general collision kernels[END_REF][START_REF] Cai | Entropy monotonic spectral method for Boltzmann equation[END_REF]. Due to its discrete nature, the DVM preserves positivity of the distribution function, the H-theorem and the exact conservation of mass, energy and momentum. Note that the Fourier spectral method is based on two main things : the truncation of the collision operator and the restriction of the distribution function to an appropriate cube, for more details see [START_REF] Pareschi | Numerical solution of the Boltzmann equation I: Spectrally accurate approximation of the collision operator[END_REF][START_REF] Mouhot | Fast algorithms for computing the Boltzmann collision operator[END_REF].

Our goal is to present an alternative method to solve formally and numerically the homogeneous Boltzmann equation in the non-cutoff case. In this work, we consider the radial symmetric case and we use a spectral method : we first compute the spectral coefficients of the solution with a formal computation software (Maple R 13; the codes can be provided). We then approximate these exact solutions and check the numerical results.

The used method helps us to motivate our work in several ways: It let us in the physical view understand more the behavior of the solutions and as we compute the first exact projections of the solutions on the spectral basis, that is in the numerical view, some other algorithms can be tested in the non-cutoff case (recall that the explicit 2D "BKW" solutions, obtained independently in [START_REF] Bobylev | Exact solutions of the Boltzmann equation[END_REF][START_REF] Krook | Exact solutions of the Boltzmann equation[END_REF] are used to test the accuracy of the numerical methods in the case of a regular collision kernel B ≡ 1, see for example [START_REF] Cai | Entropy monotonic spectral method for Boltzmann equation[END_REF]). Finally, we do hope that our work will give some clues to formulate new mathematical conjectures.

The paper is organized as follows. In section 2, we state the main theoretical results. The numerical details and algorithms are provided in section 3. Sections 4 and 5 present the numerical results of the Boltzmann equation with different initial data for the Cauchy problem: we discuss in section 4 the results for a small L 2 initial data (bi-Gaussian); in section 5, we consider the case of a measure initial data. After that, we give a conclusion for this work. The paper ends with an appendix where we set some technical results.

Theoretical results

In this section, we present some theoretical parts: we begin by linearizing the Boltzmann equation and giving the spectral decomposition of this equation.

2.1. Linearization of the Boltzmann equation. We remark that Q(µ, µ) = 0 where the Gaussian function is defined by

µ(v) = 1 (2π) 3/2 e -|v| 2 2
and it is a stationary solution of the Boltzmann equation. We consider now a perturbation g of the Gaussian. Then the solution f of (1.1) can be written as

f (t, v) = µ(v) + µ(v) g(t, v), F (v) = µ(v) + µ(v) G(v).
It is easy to show that g is a solution of the Cauchy problem

∂ t g + L(g) = Γ(g, g), g| t=0 = g(0, v) = G(v) (2.1)
where

L(g) = - 1 √ µ [Q( √ µg, µ) + Q(µ, √ µg)]
is a linear operator and

Γ(g, h) = 1 √ µ Q( √ µg, √ µh)
is a nonlinear operator. We decompose the solution of (2.1) into a linear and nonlinear part:

g(t, v) = e -tL G(v) linear part + e -tL h(t, v) nonlinear part
where e αL is the exponential of the linear operator defined by his spectral decomposition (see below) and the new function h(t, v) satisfies the following equation

∂ t h = e tL Γ(e -tL (G + h), e -tL (G + h)), h(0, v) = 0. (2.2)
The linearized operator L is a positive unbounded symmetric operator on L 2 (R 3 v ) (see [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF][START_REF] Lerner | Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators[END_REF][START_REF] Lerner | Spectral and phase space analysis of the linearized non-cutoff Kac collision operator[END_REF][START_REF] Lerner | Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff[END_REF]) with the kernel

N = span √ µ, √ µv 1 , √ µv 2 , √ µv 3 , √ µ|v| 2 .
From a rescaling argument (see Appendix 7.1), we can always assume that the initial condition G satisfies

G ∈ N ⊥ .
In [START_REF] Lerner | Spectral and phase space analysis of the linearized non-cutoff Kac collision operator[END_REF], For the radially symmetric case, the authors show that the linear Boltzmann operator behaves like the fractional harmonic oscillator H s (0 < s < 1) with

H = -∆ + |v| 2 4 .
We study in the next section the spectral properties of the operators L and Γ.

2.2.

The spectral problem. We introduce now an orthonormal basis of L 2 r (R 3 ) the radial symmetric functions of L 2 (R 3 ) involving the generalized Laguerre polynomials L [ℓ+ 1 2 ] n : for that, we set for any n ≥ 0

ϕ n (v) = n! √ 2Γ(n + 3/2) 1/2 e -|v| 2 4 L [ 1 2 ] n |v| 2 2 1 √ 4π (2.3)
where Γ( • ) is the standard gamma function, for all x > 0,

Γ(x) = +∞ 0 t x-1 e -x dx
and the Laguerre polynomial L (α) n of order α, degree n is

L (α) n (x) = n r=0 (-1) n-r Γ(α + n + 1) r!(n -r)!Γ(α + n -r + 1)
x n-r .

We have the spectral decomposition for the linear Boltzmann operator

L ϕ n = λ n ϕ n n ≥ 0, with φ 0 = √ µ, λ 0 = 0 and for n ≥ 1 λ n = 2 π 4 0 β(θ) 1 -(sin θ) 2n -(cos θ) 2n dθ (2.4)
where β(θ) is defined from the collision kernel (see (1.2))

β(θ) = sin θ b(cos θ) ≈ 1 |θ| 1+2s , 0 < s < 1, θ ∈ 0, π 2 .
(2.5)

The two families (ϕ n (v)) n≥0 and (λ n ) n≥0 represent the eigenvectors and the eigenvalues of L. Remark that this diagonalization of the linearized Boltzmann operator with Maxwellian molecules is also verified in the cutoff case (see [START_REF] Bobylev | The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules[END_REF][START_REF] Cercignani | The Boltzmann equation and its applications[END_REF][START_REF] Dolera | On the spectrum of the linearized Boltzmann collision operator for Maxwellian molecules[END_REF][START_REF] Lerner | Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators[END_REF][START_REF] Lerner | Spectral and phase space analysis of the linearized non-cutoff Kac collision operator[END_REF]). We consider the spectral expansion

g(t, v) = ∞ n=0 g n (t) ϕ n (v), G(v) = ∞ n=0 G n ϕ n (v) (2.6)
where

g n (t) = g(t, •), ϕ n (•) L 2 and G n = G, ϕ n L 2
. By definition, we have

e -tL G(v) = ∞ n=0 e -λnt G n ϕ n (v).
It is the solution of the equation

∂ t g ℓin + L g ℓin = 0, g ℓin (0, v) = G(v).
Then the operator Γ satisfies Γ(ϕ p , ϕ q ) = µ pq ϕ p+q where the non-linear eigenvalues are given by

µ pq = (2p + 2q + 1) (2p + 1)(2q + 1) C 2p 2p+2q 1 2 |θ|≤ π 4 β(θ) (sin θ) 2p (cos θ) 2q dθ
(2.7) for p ≥ 1, q ≥ 0 and

µ 0q = - |θ|≤ π 4 β(θ) (1 -(cos θ) 2q )dθ
for q ≥ 1. Following [START_REF] Lerner | Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff[END_REF], we therefore derive from (2.1) the following infinite system of ordinary differential equations :

           g ′ 0 (t) = 0, g ′ 1 (t) = 0, for all n ≥ 2, g ′ n (t) + λ n g n (t) = p+q=n 0≤p, q≤n µ pq g p (t) g q (t) (2.8)
with the initial conditions (see (2.6))

g n (0) = G n for n ≥ 0.
The goal is to study the behavior of each function t → g n (t).

In the rest, we will focus on the computation and properties of this intermediate solution.

Proposition 2.1. We assume that G ∈ N ⊥ . Then the intermediate solution h(t, v) defined by (2.2) satisfies h(t, v) = ∞ n=0 h n (t) ϕ n (v) (2.9)
where

h 0 ≡ h 1 ≡ h 2 ≡ h 3 ≡ 0 and for all n ≥ 4 h n (t) = p+q=n 2≤p, q≤n-2 t 0 µ pq e -(λp+λq-λn)s G p + h p (s) G q + h q (s) ds.
(2.10) Remark 2.2. As we have seen before, we divide the function g in two parts as follows:

g(t, v) = ∞ n=0 e -λn t G n ϕ n (v) g ℓin (t,v) + ∞ n=0 e -λn t h n (t) ϕ n (v) g nℓ (t,v) , (2.11)
therefore the formal solution f (t, v) can be written as

f (t, v) = µ(v) + µ(v) ∞ n=0 e -λn t G n + e -λn t h n (t) ϕ n (v). (2.12)
Proof of proposition 2.1. : As G ∈ N ⊥ , we get G 0 = G 1 = 0 and we can verify from (2.8) that

g 0 (t) = g 1 (t) = 0, g 2 (t) = G 2 e -λ 2 t , g 3 (t) = G 3 e -λ 2 t
and therefore h 0 ≡ h 1 ≡ h 2 ≡ h 3 ≡ 0. By (2.8), we may write

g n (t) = e -λn t G n + e -λn t h n (t) (2.13) and g ′ n (t) + λ n g n (t) = p+q=n 2≤p, q≤n-2 µ pq g p (t) g q (t).
(2.14)

We plug again the value of g n from (2.13) into the equation (2.14) and we get h ′ n (t) = e λn t p+q=n 2≤p, q≤n-2 µ pq g p (t) g q (t).

Note that h n (0) = g nℓ n (0) = 0. Finally, plugging the expression of g p and g q from (2.13) into the previous equation and integrating we prove (2.10). Concerning the exact expression of the eigenvalue λ n and µ pq , see [START_REF] Lerner | Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff[END_REF]. This concludes the proof.

We introduce now the following notations. For a k-uplet α ∈ N k ,

Λ α = λ α 1 + λ α 2 + • • • + λ α k , G α = G α 1 × G α 2 • • • × G α k .
Proposition 2.3. For each integer n ≥ 4, we define I n a set of admissible indices

I n = α ∈ N k k ∈ N * , α i ≥ 2, |α| = n, .
Then for each multi-index α, β, ∈ I n there exists some real coefficients c α β which depends only on λ 2 , . . . , λ n and µ pq for 2 ≤ p, q ≤ n -2, p + q ≤ n such that

h n (t) = α,β∈In c α β G α 1 -e -(Λ β -λn) t . (2.15)
Proof. We compute directly from (2.10)

h 4 (t) = c (2,2) (2,2) G 2 2 1 -e -(Λ (2,2) -λ 4 ) t
where c

(2,2) (2,2) = µ 22 (Λ (2,2) -λ 4 ) and h 5 (t) = c (2,3) (2,3) G 2 G 3 1 -e -(Λ (2,3) -λ 5 ) t + c (3,2) (3,2) G 3 G 2 1 -e -(Λ (3,2) -λ 5 ) t where c (2,3) (2,3) = µ 23 (Λ (2,3) -λ 5 )
and c

(3,2) (3,2) = µ 32 (Λ (3,2) -λ 5 )
.

We prove the result by induction. Then we can suppose that (2.15) is true for each h n ′ (4 ≤ n ′ ≤ n -1). We will use the integral expression (2.10) of h n . We consider two integers p, q such that 2 ≤ p , q ≤ n -2 and p + q = n. Then from (2.15)

h p (t) = α,β∈Ip c α β G α 1 -e -(Λ β -λp) t , h q (t) = α ′ ,β ′ ∈Iq c α ′ β ′ G α ′ 1 -e -(Λ β ′ -λq) t .
From the integral formula (2.10) we get

h n (t) = t 0 p+q=n 2≤p, q≤n-2 (A + B + C + D) ds with A = µ pq G p G q e -(λp+λq-λn) s , B = α ′ ,β ′ ∈Iq µ pq c α ′ β ′ G p G α ′ (e -(λp+λq-λn) s -e -(λp+Λ β ′ -λn) s ), C = α,β∈Ip µ pq c α β G α G q (e -(λp+λq-λn) s -e -(Λ β +λq-λn) s ), D = α,β∈Ip α ′ ,β ′ ∈Iq µ pq c α β c α ′ β ′ G α G α ′ × (e -(λp+λq-λn) s -e -(Λ β +λq-λn) s -e -(λp+Λ β ′ -λn) s + e -(Λ β +Λ β ′ -λn) s ).
Expanding each previous terms and integrating over [0, t], we get the result (2.15) since each number

λ p +λ q -λ n , Λ β +λ q -λ n , λ p +Λ β ′ -λ n , Λ β + Λ β ′ -λ n are positive from the next lemma and |α| = |β| = p, |α ′ | = |β ′ | = q and p + q = n.
Lemma 2.4. The linear eigenvalues λ n for the non-cutoff radially symmetric spatially homogeneous Boltzmann equation

λ n = |θ|≤ π 4 β(θ) 1 -(sin θ) 2n -(cos θ) 2n dθ, n ≥ 2,
verify the following property

λ α 1 +•••+α k < λ α 1 + • • • + λ α k (= Λ α ) for multi-index α ∈ (N \ {0, 1}) k .
Proof. By [START_REF] Lerner | Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff[END_REF], we may write

λ α 1 +α 2 < λ α 1 + λ α 2 ,
then by iteration, we have

λ (α 1 +•••+α k )+α k+1 < λ α 1 +•••+α k + λ α k+1 < (λ α 1 + • • • + λ α k ) + λ α k+1 .

Numerical computations

From now on, for sake of simplicity, we consider the specific case s = 1 2 and β(θ) = (sin θ) -2 .

For the general case s ∈]0, 1[ and other kernel β which satisfies (2.5), we can compute some numerical approximations of the eigenvalues. We think that the results do not change.

3.1. Computation of the eigenvalues. By the following assumption β(θ) ≈ 0 1 |θ| 2 , we obtain (see [START_REF] Lerner | Spectral and phase space analysis of the linearized non-cutoff Kac collision operator[END_REF])

λ n ≈ ∞ √ n (3.1)
where the linear eigenvalues λ n of L was defined in (2.4). We recall the value of λ n for n ≥ 2:

λ n = 2 π 4 0 β(θ) 1 -(sin θ) 2n -(cos θ) 2n dθ.
We compute the exact and approximate values of λ n by the following algorithm :

λ 0 ← 0 for n from 1 to N do expr ← algebraic simplification of 1-sin 2n θ-cos 2n θ sin 2 θ λ exact n ← symbolic computation of 2 π 4 0 expr dθ λ approx n ← numerical computation of λ exact n
The "algebraic simplification" of "expr" removes the singularity when θ → 0 coming from the collision kernel β(θ) = sin -2 θ (see (2.5)).

It consists in a factorization of trigonometric polynomials. The symbolic computation of λ exact n is reduced to compute the exact integral of a trigonometric polynomial. Then λ exact n is approached numerically with a number of significant digits (equal to 10 in 1). The approximation of eigenvalues can be controlled to be sufficiently precise for upcoming computations. For a general kernel β(θ), there is in general no more explicit values. But some classical numerical methods can be easily applied. Nevertheless, there is no more any algebraic simplification, and it is necessary to treat carefully the singularity.

3.2.

Computation of the nonlinear eigenvalues. We recall the coefficients µ pq from (2.7): for some p, q ≥ 1

µ pq = (2p + 2q + 1) (2p + 1)(2q + 1) C 2p 2p+2q |θ|≤ π 4 β(θ) (sin θ) 2p (cos θ) 2q dθ.
We compute the exact and the approximate value of µ pq (again with a relative error ≈ 10 -10 ) for 1 ≤ p + q ≤ N by the following algorithm :

for p from 1 to N do for q from 0 to N-p do expr ← symbolic computation of 2

π 4 0 sin 2p-2 θ cos 2q θ dθ µ exact pq ← (2p+2q+1) (2p+1)(2q+1) C 2p 2p+2q × expr µ approx pq ← numerical computation of µ exact pq
We present in the table 2 of results for p+q = n = 2, . . . , 5, 20. The sin- gularity coming from the collision kernel β(θ) = sin -2 θ is removed by a simple simplification (remark the exponent (2p -2) of the sinus term of µ pq ). Again for a general collision kernel, the values of these nonlinear eigenvalues can be approximated by classical numerical methods.

n = 2 n = 3 n = 4 n = 5 n = 20 µ 1,1 ≈ 2.

3.3.

Numerical solutions of the linear problem. We introduce from (2.11) the approximation of the linear solution

g ℓin N (t, v) = N n=0 e -λn t G n ϕ n (v) (3.2)
where the reals G n are the given initial spectral coefficients. In order to compute the value of the linear solution, we use the formula (2.3) of the eigenfunction ϕ n which involves the generalized Laguerre polynomials L

[ℓ+ 1 2 ] n . We get the following algorithm : for n from 0 to N do

ϕ n (v) ← n! √ 2Γ(n+3/2) 1/2 e -|v| 2 4 L [ 1 2 ] n |v| 2 2 1 √ 4π
Finally, we obtain the linear solution by the sum :

g ℓin N (t, v) ← 0 for n from 2 to N do g ℓin N (t, v) ← g ℓin N (t, v) + e -λnt G n ϕ n (v)
We estimate the L 2 theoretical error (g ℓing ℓin N ) for the different initial data G used for computation in the next sections. Proposition 3.1. We consider the solution of the following linear problem ∂ t g ℓin + L g ℓin = 0, g ℓin (0, v) = G(v).

(3.3)

We have the following estimates : 1) For initial data G ∈ L 2 ,

g ℓin (t, •) -g ℓin N (t, •) L 2 e -c √ N t G L 2 .
2) For the measure initial data G defined by (5.1) (see also proposition 7.

3), there exist some constants C > 0 and c > 0 such that for t > 0

g ℓin (t, •) -g ℓin N (t, •) L 2 1 t b e -γ √ N t .
Proof. The solution of (3.3) is

g ℓin (t, v) = ∞ n=0 e -λn t G n ϕ n (v).
The exact error in L 2 is

g ℓin (t, •) -g ℓin N (t, •) 2 L 2 = ∞ n=N +1 e -2λn t |G n | 2 . 1) If G ∈ L 2 (R 3 v )
, then as we have from (3.1)

g ℓin (t, •) -g ℓin N (t, •) 2 L 2 = ∞ n=N +1 e -2λn t |G n | 2 e -2 c √ N t G 2 L 2 .
We can deduce that the exact error tends to zero when N tends to infinity.

2) We suppose now that F is the measure initial data µ + δ. We can approximate the spectral coefficients G n of G by n 1 4 and by 3.1 we can then find some positive constants c and C such that

g ℓin (t, •) -g ℓin N (t, •) 2 L 2 ≤ C ∞ n=N +1
e -c √ n t n 2 .

We consider the function ρ t defined on R + by ρ t (x) = e -c √ x t x 2 . So that ρ t is positive, continuous and decreasing for x ≥ 16/(c t) 2 , therefore by using the Cauchy integral criterion, we can write the following inequality :

g ℓin (t, •) -g ℓin N (t, •) L 2 ≤ C t b e -γ √ N t -→ 0 N →∞
where b and γ are some positive constants.

3.4.

Numerical solutions of the non-linear part. Concerning the nonlinear part g nℓ = e -tL h of the solution, we consider the partial series

g nℓ N (t, v) = N n=0 e -λn t h n (t) ϕ n (v). (3.4)
We then use the decomposition of h in the spectral basis (2.9) and the integral formula (2.10) to compute h n (t). Therefore we solve explicitly the system (2.8) by the following algorithm: h 0 (t), h 1 (t), h 2 (t), h 3 (t) ← 0 for n from 4 to N do S ← 0 for p from 4 to n do

q ← n -p S ← S + µ pq (G p + h p (t)) (G q + h q (t)) e -(λp+λq-λn) t h n (t) ← symbolic computation of t 0 S
The exact computation of the integral t 0 S is straightforward since, from proposition 2.3, the symbolic expression S is an linear combination of exponential terms e αt . We get the exact following solutions of the system of integral formula (2.10):

h 0 = h 1 = h 2 = h 3 = 0, h 4 = µ 22 λ 2 + λ 2 -λ 4 G 2 2 1 -e -(λ 2 +λ 2 -λ 4 ) t , h 5 = µ 23 + µ 32 λ 2 + λ 3 -λ 5 G 2 G 3 1 -e -(λ 2 +λ 3 -λ 5 ) t , • • •
From the symbolic expression of h n we compute the numerical approximation :

h 0 = h 1 = h 2 = h 3 = 0, h 4 =2.51 G 2 2 1 -e -0.279 t , h 5 =1.62 G 2 G 3 1 -e -0.698 t , h 6 =0.322 G 3 2 1 -e -1.
g nℓ N (t, v) ← 0 for n from 2 to N do g nℓ N (t, v) ← g nℓ N (t, v) + e -λnt h n (t) ϕ n (v)
The symbolic and numerical computation of the nonlinear part of the solution plays the main difficulty of our method. We analyze the computation time and rounding off error in the next section.

f N = µ + √ µ (g ℓin N + g nℓ N ). (3.5)
The method using the software Maple R 13 and its internal function "int(f (x), x = a..b)" for symbolic computation of integrals seems limited to a number N around 20, since for N = 20, the number of terms of h 20 is around 5000 and the computation time is around 50 seconds. Moreover they are both exponentially increasing (see Figure 1). We now estimate the truncation and rounding error due to the software computations. For a regular L 2 initial data we have computed the solution f N for N = 20 and different number of digits (we can control the number of digits that Maple R 13 uses when making calculations with software floating-point numbers). We set P 1 and P 2 two numbers of digits and we compare the two numerical solutions f P 1 N and f P 2 N computed respectively using P 1 and P 2 . We define the rounding relative error as

error = f P 1 N -f P 2 N ∞ f P 2 N ∞
and we get the following results for different choices of (P 1 , P 2 ) : We check from the table 3 that the relative error is roughly 10 times the run on a computer having 8 Xeon processors 2.33 GHz with 8 GB of memory. The method using Maple R 13 on this computer seems limited to a number around N = 20. Surprisingly, the computation time is roughly the same (around 90 seconds) for a number of digits between 20 and 1000. The main part of this time is therefore used for algebraic manipulation.

We present in the two upcoming sections the results of the computation for different initial values.

Radial bi-Gaussian initial value

We set the initial data :

F (w) = 1 (2π) 3 2 exp - 1 2 (|w| + 1) 2 + exp - 1 2 (|w| -1) 2 .
We next rescale the initial data following lemma 7.1. We show in figure 3 the spectral approximation

F N (v) of the initial data F (v) such that F N (v) = µ(v) + µ(v) G N (v) where G N (v) = N n=0 G n ϕ n (v).
We then compute the solutions h n (t) from the proposition 2.

1 for n = -6 -4 -2 0 2 4 6 0 1 • 10 -2 2 • 10 -2 3 • 10 -2 v F F 20 F 10 F 5
Figure 3. Approximation of the initial data 4, 5, . . . , N with N = 20. For each integer n, the function t → h n (t) is monotone and tends to a finite limit when t tends to infinity (See Figure 4 (a)). We recall that h n (t) is a finite sum of decreasing exponential terms (see section 3.4). Since the initial data G is a regular function, the spectral coefficients G n are exponentially decreasing. The numerical computation of h n (t) shows also that h n ∞ is exponentially decreasing with respect to n (See Figure 4 (b)). In this special case, the linear part e -tL G and the nonlinear part e -tL h have roughly the same behavior. We present in figure 5 the graph of the linear part and nonlinear part and the ratio in

L 2 -norm R N (t) = g nℓ N (t, •) L 2 g ℓin N (t, •) L 2 = N n=4 |e -λnt h n (t)| 2 1 2 N n=2 |e -λnt G n | 2 1 2 . ( 4.1)
We observe that the nonlinear part is very small compared to the other. We remark that in this case the series (3.2) and (3.4) behave as

g ℓin N (t, v) ≈ e -λ 2 t G 2 ϕ 2 (v), g nℓ N (t, v) ≈ 2.51 e -λ 4 t G 2 2 (1 -e -(2 λ 2 -λ 4 ) t ) ϕ 4 (v)
, because the terms of h n (t) are composed of products of terms which are numerically converging to zero. The quotient (for G 2 = 0) of the two previous approximations behaves closely like the ratio (see (4.1))

R N (t) ≈ R(t) def = e -(λ 4 -λ 2 ) t |G 2 |(1 -e -(2 λ 2 -λ 4 ) t ). (4.2)
We finaly compute (see figure 6) the solution f = µ + √ µg using the spectral Hermite eigenfunctions ϕ n (v) and the expansion (2.12) of g in this basis. Since the function g(t, •) ∈ N ⊥ for all time t ≥ 0, the approximate solution f N (t, •) is naturally orthogonal to ϕ 0 and ϕ 1 . Therefore is a conservation of the mass and the energy. Finally, we check that the approximate solution f N converges to the Gaussian function when the time tends to infinity.

Numerical results for initial measure data

We consider the initial measure data F = gaussian + Dirac = µ + δ.

Following the lemma and rescaling the solution F (v) = 2 -5 2 F (2 -1 2 v), we get the normalized initial data

F (v) = 2 -5 2 µ(2 -1 2 v) + 2 -1 δ(v), G(v) = 2 -13 4 π -3 4 -µ(v) + 2 -1 4 π 3 4 δ(v). (5.1)
We verify that G, ϕ 0 = G, ϕ 1 = 0 and therefore G ∈ N ⊥ . We then compute the spectral coefficients for n ≥ 0 (see proposition 7.1):

G n = G, ϕ n = 1 + (-1) n 2 (2 n + 1)! 2 2n (n!) 2 1 2
.

Note that the coefficients G 2n+1 are equal to zero and we have the following approximation of G:

G(v) ≈ ∞ n=1 n 1 4 ϕ 2n (v).
We set F reg (v) = 2 -5 2 µ(2 -1 2 v) the regular part of the distribution F . We check in the left figure 7 that the approximate initial data behaves as a Dirac function. Remark that to capture the approximation of the regular part F reg , we have to rescale the cote y-coordinate. We observe the oscillations of F N which are expected since the functions F N approach the Dirac function when N tends to infinity (see the right figure 7). We now focus on the evolution problem. As the initial data is a distribution, we can check that the linear part of the solution is singular :

g ℓin (t, •) 2 L 2 = ∞ n=2 G 2 n e -2 λnt ≈ 1 t α , when t → 0 (5.2)
for some α > 0 (since G 2n ≈ n 1 4 and λ n ≈ n

2 ). We next compute the nonlinear part h n (t) of the solution (see the left figure 8). We observe some numerical evidences that these functions are increasing less than a power of n :

sup t≥0 |h n (t)| ≤ C n a with a close to 1. Since G n ≈ n 1 4
, the behavior of a term of the series (g ℓin N (t) + g nℓ N (t)) is dominated by the nonlinear part. We next calculate the linear part and nonlinear part of solution (see Figure 9). g nℓ N (t) is a regular function for all time and we verify also that g ℓin N (t) is singular as t → 0 as pointed in (5.2). For a large time, the L 2 norm of the linear part (∼ e -λ 2 t ) dominates the norm of the nonlinear part (∼ e -λ 4 t ), and the ration R N (t) has the same behavior as in the previous section (see (4.2)).

We then compute the numerical approximation f N of the solution f for N = 20 and we check that the solution behaves as a Dirac function as t → 0 and tends to the Gaussian as t → ∞ (see Figure 10).

Since λ n ≈ c √ n and if the behavior of sup t≥0 |h n (t)| is dominated by a power of n (which is numerically verified), then we have for some b, γ > 0 :

∀t > 0, f N (t, •) -f (t, •) L 2 1 t b e -γ √ N t → 0 as N → ∞.
We observe some other numerical evidences that the series converges in L 2 for t > 0 and the solution converges to a Gaussian as t → ∞.

Conclusion

We have considered the perturbation g of the solution f of the Boltzmann equation defined by

f = µ + √ µg where g(t, v) = n g n (t)ϕ n (v)
and we have studied the behavior of the spectral coefficients We have then computed formally the spectral coefficients h n (t) for n = 0, 1, . . . , N with N = 20. We have checked also the results for small L 2 initial data and distribution type initial data µ + δ.

g n (t) = e -λnt (G n + h n (t)), g n (0) = G n .
• For small L 2 initial data, our method was tested with several L 2 initial conditions : F is a sum of two Gaussian, G n = 0.1 n , G n = 1 n . The results show that there are some numerical evidences that the spectral series n e -λnt (G n + h n (t))ϕ n is convergent in L 2 for any time t ≥ 0 and the solution converges to a Gaussian. Moreover, for large times, the linear part G n is preponderant with respect to the non-linear part h n (t).

• For the distribution type initial data µ + δ, the simulations show some numerical evidences that the spectral series converges in L 2 and there is a regularization of the solution for t > 0.

We have computed the formal solutions of the spectral coefficients h n of the solution of the Boltzmann equation. If there exists a regular solution for t > 0, then the solutions h n are the exact projections of the solution on the spectral basis. These calculations were made in the case of a non-cutoff kernel. The numerical results are coherent for small L 2 initial data or for the distribution case µ + δ. There is conservation of the mass, momentum and energy of the approximated solution (since g N (t, •) is orthogonal to the kernel N for all time). Moreover the approximated solution f N (t, •) (defined in (3.5)) converges to a Gaussian when t tends to infinity. Then we have the following limit in the sense of distribution as ε → 0:

F ε → F = 2 -5 2 µ(2 -1 2 •) + (2 1
2 ) 3 δ ,

G ε → G = - √ µ + 2 -13 4 π -3 4 + 2 -1 4 π 3 4 δ.
The coordinates of G ε in the spectral basis (ϕ n ) n≥0 are given by:

G ε,0 = G ε,1 = 0, G ε,n = 1 + (-1) n 2 (1 -ε 2 ) n (1 + ε 2 ) n (2 n + 1)! 2 2n (n!) 2 1 2
, ∀n ≥ 2.

Moreover we have G ε,n → G n as ε tends to 0 as ε → 0 where G n is given in (7.1).

Remark. There is continuity of the spectral coefficients : G ε,n → G n as ε tends to 0.

Proof. From Lemma 7.1, we set

F ε (v) = α ε Fε (β ε v)
where

α ε = 1 3 R 3 w 2
Fε (w) dw

3. 5 .

 5 Discussions on the symbolic computation. From the computation of the linear (3.2) and nonlinear (3.4) part, we calculate the approximated solution of the Boltzmann equation (1.1)

Figure 1 .Table 3 .

 13 Figure 1. Number of terms for h n and computation time in seconds for h 1 , . . . , h n .

Figure 2 .

 2 Figure 2. Computation time for different number of digits.

Figure 4 .Figure 5 .

 45 Figure 4. Behavior of the nonlinear part h n .

Figure 6 .

 6 Figure 6. Graph of (t, v) → f N (t, v) for N = 5, 20 and µ(v).

- 1 • 10 - 2 0 1 • 10 - 2 vFigure 7 .

 110211027 Figure 7. Approximation of the initial data.

n

  → h n (t) for n = 4, 5, . . . , 20. sup t |hn(t)| |Gn| for n = 4, 6, . . . , 20.

Figure 8 .

 8 Figure 8. Behavior of the nonlinear part h n .

Figure 9 .

 9 Figure 9. Comparaison of the non linear part with respect to the linear part (see (3.2), (3.4), (4.1), (4.2)).

Figure 10 .

 10 Figure 10. Graph of (t, v) → f N (t, v) for N = 5, 10, 20 and µ(v).

Remark 7 . 2 . 3 F 3 µ 3 v 2 F 3 v 2 √ µ L 2 = G, |v| 2 √ µ L 2 = 0 .Recalling that ϕ p , ϕ q L 2 = δ pq , ϕ 0 = √ µ and ϕ 1 = 6 -1 2 ( 3 - 3 F

 72333232220216233 Therefore f (t, v) is a solution of the Boltzmann equation (1.1) with initial data F . Moreover, if we setF = µ + √ µ G, we then have G ∈ N ⊥ . If F is such that R (v) dv = R (v) dv = 1, R (v) dv = R µ(v) dv = 3, then the function G defined by G = 1 √ µ (Fµ) belongs to N ⊥ . Proof. It is easy to check that f (t, v) is a solution of the Boltzmann equation. Since G is a radial function, it is enough to check that G, |v| 2 ) √ µ, it is equivalent to prove (F/ √ µ, ϕ 0 ) L 2 = 1 and (F/ √ µ, ϕ 1 ) L 2 = 0,which gives the equationsR (v) dv = R 3 µ(v) dv = 1, R 3 |v| 2 F (v) dv = R 3 |v| 2 µ(v) dv = 3.Following lemma 7.1, the rescaled initial data of Fε isF ε = µ + √ µ G ε where G ε ∈ N ⊥ and G ε (v) =µ(v) + 2 -

3 2 R 3 Fε 1 3 R 3 w 2 Fε (w) dw 1 2 R 3 Fε 2 . 2 3 4 ( 1 + ε 2 ) 3 / 2 16π 3 / 4 e -ε 2 v 2 4 + 1 ε 3 e -v 2 4 ε 2 orG 2 1 2 ( 1 + ε 2 ) 2 1 2 ( 1 + ε 2 ) 3 / 2 8 (I 2 +

 32312232241232344132222122212322 Then F ε = µ + √ µ G ε where G ε (v) =µ(v) + ε (v) =µ(v) + (v/ε) ,then G ε ∈ N ⊥ . We compute from lemma 7.5G ε,n = (G ε , ϕ n ) L 2 = I 1 + I 3 )

Table 1 .

 1 The approximation is easily controlled by the estimate of the relative error |λ exact Symbolic and numerical computation of λ n .

	n numerical table 1. -λ appr. n n |/λ exact	. Using the software Maple R 13, we finally get the
		Exact value	Approximate value Relative error
	λ 1	0		0	-
	λ 2 λ 3 λ 4 λ 5 λ 10 λ 15 λ 20	1 + 1 2 π 3 2 + 3 4 π 23 12 + 15 16 π 55 24 + 35 32 π 61717 16128 + 109395 65536 π 41349267 8200192 + 35102025 16777216 π 60225247403 9906683904 + 83945001525 34359738368 π	2.570796327 3.856194490 4.861909780 5.727783632 9.070756042 11.61545300 13.75454524	8.0 × 10 -11 5.0 × 10 -11 1.2 × 10 -10 8.2 × 10 -11 9.0 × 10 -11 3.2 × 10 -10 2.5 × 10 -11

Table 2 .

 2 Numerical computation of µ pq .

	µ 4,1 ≈ 0.084 µ 4,16 ≈ 0.46 . . .
	µ 19,1 ≈ 10 -5

35 µ 1,2 ≈ 2.88 µ 1,3 ≈ 3.29 µ 1,4 ≈ 3.62 µ 1,19 ≈ 6.68 µ 2,1 ≈ 0.519 µ 2,2 ≈ 0.702 µ 2,3 ≈ 0.84 µ 2,18 ≈ 1.55 µ 3,1 ≈ 0.196 µ 3,2 ≈ 0.30 µ 3,17 ≈ 0.75

  20 t + 1.17 1e -0.928 t G 2 G 4 + -2.95 e -0.928 t + 0.677 + 2.26 e -1.20 t G 2 3 , h 7 =0.501 G 2 G 5 1e -1.09 t + 0.220 1e -1.51 t G 3 G 4 + 0.201 + 0.478 e -1.79 t -0.274 e -1.51 t -0.407 e -1.09 t G 2 2 G 3 , ... We finally get from (3.4) the approximation g nℓ N of the nonlinear part of the solution g nℓ (t, v) by the following algorithm :

Acknowledgments

The authors wish to thank Chao-Jiang Xu for interesting discussions. 7. Appendix 7.1. Rescaling of the solution. We consider a radial solution f (s, w) of the Boltzmann equation :

The second author is supported by a grant from Lebanon.

Using the change of variable w = β v, we can check that if we set the values of α and β given in the lemma, the previous equations are fulfilled.

7.2. Measure initial data. We define the following distribution initial data : F = µ + δ.

Following the rescaling of lemma 7.1, we compute

and then α = 2 -5 2 and β = 2 -1 2 . Using the change of variable w = βv, we get the new rescaled distribution initial data

2 ) 3 δ .

Proposition 7.3. We consider the initial data

2 ) 3 δ and we set G such that F = µ + √ µG. Then we have

We consider the coordinates G n = G, ϕ n of the distribution G in the spectral basis (ϕ n ) n . We can check that

and for all integer n ≥ 2,

Proof. The expression of G follows from the definition of the Gaussian µ. We then compute

ϕ n (0) and the conclusion results directly from lemma 7.5.

We consider now a special Gaussian approximation F ε ∈ L 2 of the distribution initial data F = µ+δ and we obtain some spectral stability result in this case. Proposition 7.4. We consider the initial data for ε > 0

where

Finally we get :

.

7.3. Some results on the spherical harmonics. We recall that

where the Laguerre polynomial L (α) n of order α, degree n is

x n-r .

Lemma 7.5. For a > 0 and n ≥ 0 we have

(

Proof. These equalities come from classical properties of the Hermite funtions (we have checked them using Maple R 13 for integers n ≤ 20).