Sarah Khellaf
email: khellaf@idmp.uni-hannover.de

Jana Peters
email: peters@idmp.uni-hannover.de

Design and analysis of an unusual curve sketching exercise for first year teacher students

Keywords: ATD, REM, Herbartian schema, Teachers' and students' practices at university level, Transition to, across and from university mathematics

This paper intends to show how certain theoretical elements and tools of the Anthropological Theory of the Didactic (ATD) can be used in the development of exercises that address specific mathematical difficulties of students at the transition from school to university and in a subsequent analysis of student solutions.

INTRODUCTION

The switch from school mathematics to a more reflected use of mathematical practices at university is still considered a challenge for students at the transition from school to university (Liebendörfer, 2018, chap. 2.3). Teacher students, specifically, experience what is known in German mathematics education as "double discontinuity" [START_REF] Winsløw | The ATD and other approaches to a classical problem posed by[END_REF]. To address such transition challenges in an introductory course of mathematics education for first-year teacher students1 , we employ exercises and teaching materials which aim to examine specific aspects of the relationship between school mathematics and university mathematics. The materials are created on the basis of design principles which address professionalisation aims of teacher education [START_REF] Ruge | Die Entwicklung reflektierter Handlungsfähigkeit aus subjektwissenschaftlicher Perspektive[END_REF]Ruge et al., 2021, p. 250) and were elaborated for the creation of mathematical exercises using the notion of praxeology of the ATD [START_REF] Khellaf | Aufgaben an der Schnittstelle von Schulmathematik, Hochschulmathematik und Mathematikdidaktik. Theoretische Überlegungen und exemplarische Befunde aus einer einführenden Fachdidaktikveranstaltung[END_REF].

This contribution focuses on one particular exercise that deals with the topic of curve sketching. Curve sketching is a standard topic in calculus courses in German schools and typically serves as application field for differentiation techniques. The exercise discussed in this contribution intends, on the one hand, to help students overcome certain mathematical difficulties connected with the topic of curve sketching (for an overview of typical difficulties see Roos, 2020, chap. 3). On the other hand, the exercise intends to bring to our students' attention specific limitations of typical school mathematical approaches to the topic. It is designed in such way that the application of standard solution strategies leads to a dead end. A solution can be found by directly applying the definitions of extremum and inflection point to the given graphs (Fig. 1). Our students typically experience difficulties in switching from solution strategies they learned in school to practices such as checking prerequisites of theorems and consulting definitions, which are more closely associated with university mathematics. When the usual criteria do not work, some students engage in a variety of 'unproductive' and 'unusual' practices and many have unexpected difficulty perceiving definitions as a resource. As most students attend several mathematics lectures before encountering our exercise (e. g. Analysis I), this persistence of school approaches surprised us and incited us to investigate this didactical phenomenon.

In this paper, we intend to do two things:

1. We employ the Herbartian schema of the ATD to describe in detail the rationale behind the exercise's design and successive modifications. This adds to previous work on exercise's design that drew mainly on Critical Psychology and the ATD's notion of praxeology (cf. [START_REF] Khellaf | Aufgaben an der Schnittstelle von Schulmathematik, Hochschulmathematik und Mathematikdidaktik. Theoretische Überlegungen und exemplarische Befunde aus einer einführenden Fachdidaktikveranstaltung[END_REF][START_REF] Ruge | Die Entwicklung reflektierter Handlungsfähigkeit aus subjektwissenschaftlicher Perspektive[END_REF]. 2. We present a reference epistemological model (REM) that was created to represent a range of (according to the reference institution) 'legitimate', 'plausible' and 'expected' solutions to the exercise. This REM shall in the future serve as analytical tool for the analysis of student solutions to the exercise.

We finish by discussing potentially generalizable aspects of our investigation and observations that might be of interest to for example DBR or SoTL communities (inside and outside the field of mathematics education).

THEORETICAL BACKGROUND: ELEMENTS OF THE ANTHROPOLOGICAL THEORY OF THE DIDACTIC

The Anthropological Theory of the Didactic (ATD) [START_REF] Chevallard | Fundamental concepts in didactics: Perspectives provided by an anthropological approach[END_REF][START_REF] Chevallard | Introducing the anthropological theory of the didactic: An attempt at a principled approach[END_REF]) is a research programme for the study of human practices from an institutional perspective2 . Institution in the sense of ATD means any form of legitimised social group. Any form of knowledge, and thus also actions in relation to this knowledge, is located in institutions and subject to institutional conditions and legitimisations. Institutional conditions do not merely represent external societal conditions but are constitutive for knowledge and the actions associated with it. In the ATD, knowledge is understood as human activitylegitimised, justified and explained within the institutionthat includes not only practical aspects of "know-how" (i.e. praxis, task and technique) but also knowledge in the sense of "know-why" (i.e. logos, technology and theory). This is subsumed under the term praxeology.

The study of the dissemination of knowledge through institutions and among persons is at the heart of the ATD. One important tool to capture study processes is the Herbartian schema (cf. [START_REF] Bosch | Study and Research Paths: A Model for Inquiry[END_REF]

: [S(X; Y; Q ♥ ) → M={A ◊ i , ..., Q i , …, W i , …, D i , …}] => A ♥ .
It consists of the Didactic System S(X; Y; Q ♥ ) 3 around a question Q ♥ that is studied by X (one or more students) with the help of Y (one or more teachers) to find an answer A ♥ . The study process takes place in a didactic milieu M consisting of different types of objects: established answers A ◊ i to questions Q i that come up in the study process and have to be deconstructed and reconstructed in order to arrive at A ♥this process is known as question-answer-dialectic; works W i and data D i , which help with or provide feedback on the study process. Another category for researching study processes is media. Media are all types of systems that issue a message or statements. This can be textbooks, other works, but also fellow students or teachers. The usefulness of a statement to arrive at A ♥ is evaluated against the milieu and fed back to X and Y. This is called the media-milieu-dialectic. Media-milieu-and question-answerdialectics are analytical tools to describe the dynamics of study processes.

The institutional perspective of the ATD on study processes means that X and Y are institutional positions, Q ♥ and A ♥ as well as the milieu are part of institutionally legitimised knowledge. In each institution, e.g. school, there is a predominant way of describing and presenting the knowledge, i.e. the set of relevant questions, what answers, works and data are regarded as legitimate or adequate, in what way the knowledge is used etc. This is called the dominant epistemological model (DEM). To study didactic phenomena of an institution, the ATD proposes to build a reference epistemological model (REM) (cf. [START_REF] Lucas | The phenomenotechnical potential of reference epistemological models: The case of elementary differential calculus[END_REF]. The REM can be seen as a phénoménotechnique (in the sense of Bachelard) with which didactic phenomena linked to the DEM can be produced and thus studied in the research process.

As mentioned in the introduction, we suspect that institutional differences between school mathematics and university mathematics might have played a role in the genesis of the phenomenon to be studied. Differences between the knowledge of different educational institutions and their implications for possible study processes are central concerns of the ATD. Therefore, ATD is particularly useful for us, as it makes possible a detailed description of practices typical for the involved institutions.

Give correct mathematical answers to the following questions and provide an acceptable mathematical justification. 1) How many inflection points does graph 1 have?

2) How many extrema does graph 2 have? Fig. 1 shows the curve sketching exercise given to students (cf. Ruge et al., 2021, section 14.2.2). Students are additionally given a chapter on curve sketching from a German mathematics textbook for upper secondary school (Freudigmann et al., 2012, pp. 38-67) as reference material. The chapter contains definitions, theorems, examples and exercises. Figures 2 and3 show the definitions of extremum and inflection point.

Definition: A function has a local maximum 𝒇(𝒙 𝟎 ), local minimum 𝒇(𝒙 𝟎 ),
at 𝑥 0 , if there exists an interval 𝐼 with 𝑥 0 ∈ 𝐼, so that for all 𝑥 ∈ 𝐼:

𝒇(𝒙) ≤ 𝒇(𝒙 𝟎 ) 𝒇(𝒙) ≥ 𝒇(𝒙 𝟎 ).
In this case, the point (𝑥|𝑓(𝑥 0 )) [sic] is called "high point" of the graph "low point" of the graph. Among the theorems are four which specify criteria for the existence of extremums and inflection points on certain differentiable functions; Fig. 4 shows one such theorem.

Theorem: Let the function 𝑓 be arbitrarily often differentiable on an interval 𝐼 and let 𝑥 0 be an inner point of the interval.

1. If 𝑓 ′′ (𝑥 0 ) = 0 and 𝑓 ′′ has a sign change in the vicinity of 𝑥 0 , then 𝑓 has an inflection point at 𝑥 0 .

2. If 𝑓 ′′ (𝑥 0 ) = 0 and 𝑓 ′′′ (𝑥 0 ) ≠ 0, then 𝑓 has an inflection point at 𝑥 0 . Questions 1) and 2) can be answered with the help of the definitions of extremum and inflection point that can be found in the schoolbook (Fig. 2 & Fig. 3). According to these definitions, there are no inflection points on graph 1 but infinitely many extremums on graph 2 (cf. Fig. 1).

Examples of 'peculiar' student responses to the task

In 2019, a student group presented an exercise solution and explained correctly why the schoolbook theorems do not yield useful information, but could still not answer questions 1) and 2) because they had not thought of looking into the definition as a viable course of action. In 2021 a student, who wanted to write an essay about the topic, tried to apply the strategy of approximating the functions pertaining to graphs 1 and 2 with step functions and to transfer knowledge about properties of the step functions onto the limit functions, before we directed him towards the definitions. Overall, we could observe in many students' approaches a peculiar absence of the strategy of consulting the (formal) definitions of the concepts in question, and a replacement of this strategy by a variety of 'unproductive' and 'unusual' strategies.

Description of the task's economy and ecology

In German school mathematics, the curve sketching exercise can be considered 'unusual' because exercises within the topic of curve sketching typically ask for the application of algorithmic procedures derived from schoolbook theorems which give criteria for the existence of extremums and inflection points. These algorithms constitute the institution's 'official answer' A ◊ School for virtually all schoolbook exercises which ask to find extremums or inflection points on functions, which are typically given in algebraic form. A ◊ School thus constitutes the dominant epistemological model (DEM) in the institution of school mathematics regarding the topic of curve sketching. In the curve sketching exercise, however, the standard procedures for determining extrema and inflection points fail: The schoolbook theorems are formulated as unidirectional conditional statements (cf. Fig. 4), but on the straight segments of graph sketches 1 and 2 (Fig. 1) the sufficient conditions do not hold.

In university mathematics, a standard technique to approach any mathematical problem is the verification of one's own knowledge of the involved definitions, which we will call A ◊ University . In the case of the curve sketching exercise, making sensible use of the schoolbook definitions will lead to an answer to questions 1) and 2).

The curve sketching exercise was designed in such way, that the media provided strongly evoke the DEM of the institution of school mathematics: it is introduced by a fictional conversation between two school students and provides a schoolbook excerpt as material (W School ). A ◊ University , on the other hand, becomes less associated with the task (→ A ◊ University ). One aim of evoking the DEM was to make our students experience its mathematical limitations and provoke subsequent questioning of the DEM.

The milieu actively provided was a page on an online learning platform that contained the media and some instructions. However, as the page is non-responsive and does not give feedback on student's activities (apart from providing the media), it is not included in the Herbartian schema. The teaching team Y offered to respond to questions at any time and would add further elements to the milieu upon request (in the form of replies). An 'invisible' but nonetheless crucial part of the milieu are the results of an application of logical/mathematical reasoning (D math ). A ♥ was not included in the milieu (e.g. in the form of a sample solution for self-check).

Question-answer-dialectic, chronogenesis and media-milieu-dialectic

Students are confronted with Q ♥ (cf. Fig. 1) and many first try A ◊ School to solve it. A correct(!) application of mathematical reasoning (i. e. interaction with the milieu) should yield D math , i.e. the 'feedback' that A ◊ School cannot answer Q ♥ , which should give rise to new questions Q i . This coincides with the broader intention of the curve sketching exercise to initiate a questioning of A ◊ School with regard to its "validity and limitations …, its adequacy to Q[ ♥ , the authors], the adaptations required, etc." (Bosch, 2019, p. 4040).

The original curve sketching exercise can be depicted schematically as:6 [S(X; Y; Q ♥ ) → M={A ◊ School , A ◊ University , W School , D math , Q i , …}] => A ♥ However, problems with the chronogenesis of the inquiry can arise and students can get stuck in a particular way: as the only part of the (initially provided) milieu that can give feedback on the correctness of students' answers is Y or D math , which are both spatially and/or temporarily displaced (hence the grey font), it can happen that students end up believing, e.g. due to a logical error in the interpretation of the schoolbook theorems (taking unidirectional conditional statements for equivalences), that they have answered the question Q ♥ correctly even though they haven't. We suspect, that the DEM is being evoked too strongly and that the initial milieu doesn't provide enough feedback to enable (at least part of) our students to overcome and question the DEM.

As a consequence, in subsequent implementations of the online course, the milieu was enriched by adding new materials (…) i to the learning platform which consisted in fictional dialogues W Student between students who try to solve the curve sketching exercise but who run into the same dead ends as some actual students did and give an incorrect answer. The materials then pose the question Q DM , why the displayed reasoning is incorrect, and include an official answer A ◊ DM to Q DM by the institution that is our didactics course in the form of a sample explanation.

The modified curve sketching exercise can thus be schematically represented as:

[S(X; Y; Q ♥ ) → M={A ◊ School , A ◊ University , (W Student , Q DM , A ◊ DM ) 1 , (…) 2 , …, W School , D math , Q i , …}] => A ♥

THE REFERENCE EPISTEMOLOGICAL MODEL

In order to better understand the potential of the exercise to induce question-answerand media-milieu-dialectics and to enable more in-depth analyses of students' solutions we started to work out a REM with reference to the institution of mathematics education constituted by our working group and through our courses. 7 The REM contains all praxeological elements which occur in a set of solutions to the curve sketching exercise that we created on the basis of W School and our knowledge of mathematics (cf. D math ) and which were deemed the most 'obvious', 'plausible' and 'expected' (i.e. 'normal') solutions from the point of view of reference.8 A visualisation of this set of solutions is given in the form of a flowchart (Fig. 5), which was inspired by the tool of questions-answers map (Bosch, 2019, p. 4041). The chart is not meant to be interpreted strictly chronologically: drawing a 'solution path' into the chart is supposed to indicate which praxeological elements occur and don't occur, but not necessarily at which stage of the solution they are employed (they might be employed several times at different stages in the solution). The detailed REM can be accessed by consulting tables 1, 2 and 3 in the supplementary material in combination with the flowchart. The tables specify three regional praxeologies that will now be explained.

Regional praxeology Differentiation (cf. Table 1)

We regard the differentiation of real functions as a praxeology in its own right, because it is taught in school as separate topic before curve sketching is introduced in later school years. In the context of our course, differentiation is also treated differently from curve sketching as our didactic contract demands that the schoolbook be cited when curve sketching theorems are used, while justifications for the application of differentiation techniques are not necessary. The praxeology differentiation includes all praxeological elements that were relevant in our solutions, even if some of them are likely not commonly taught in German schools but in introductory Analysis courses at university (e.g. θ(diff).1 and 2). In this sense, the resulting praxeology is specific to the institution of mathematics education, whose praxeological equipment intersects with that of both school and university mathematics (as well as that of other disciplines).

Regional praxeology Curve Sketching (cf. Table 2)

We consider the praxeologies extrema (E) and inflection points (IP) to be two subpraxeologies of the regional praxeology curve sketching. The two praxeologies possess largely similar logos-blocks (monotony is relevant for both; justifications of theorems are based on the same ideas) and are consolidated on page 67 of [START_REF] Freudigmann | Lambacher Schweizer -Mathematik für Gymnasien[END_REF] into a sequence of activities commonly called "curve sketching". However, the book's presentation assigns the two praxeologies to two different subchapters with their own specific definitions, theorems, examples and tasks. Table 2 represents the praxeology extrema (E); the respective praxeology inflection points (IP) is analogous, but "E" is swapped with "IP" and other schoolbook theorems are relevant.

Regional praxeology Solving Tasks (cf. Table 3)

After we had coded all steps of each solution using the above praxeologies as coding manual9 , there remained passages in our solutions to which no praxeological elements had yet been assigned. For this reason, we defined an additional praxeology solving tasks that includes general strategies for solving tasks. These strategies are commonly introduced in school, applied in all subjects (not just mathematics), and are relevant for university mathematics as well, where they undergo a mathematical specialisation as stronger emphasis is placed on mathematical rigour (e.g. in mathematical case differentiations at university, the cases have to cover all logical possibilities and any two cases have to be disjunct from each other).

DISCUSSION

The design of the curve sketching exercise addresses two circumstances: First, it is a reaction to mathematical difficulties students typically experience at the transition from school to university. 10 By posing a mathematical problem which cannot be solved within a well-known DEM from school mathematics, we intend to initiate questioning and further analysis of this DEM. This idea is especially relevant in the education of teacher students, as this group can not only benefit from questioning and increasing their mathematical knowledge, but also from using didactic theory to analyse and reflect upon the teaching of the DEM in school, its goals and effects. Secondly, exercise modifications intend to mitigate teaching difficulties arising from the nonresponsiveness of our digital learning environment and the lack of feedback by the milieu. These difficulties are in fact not only a problem of our specific learning platform but also, more generally, a feature of the type of task proposed: the idea for the task is based on the expectation (and experience) that many students are fixated on the DEM so strongly that they overlook solution strategies that are very elementary from the point of view of university mathematics. The potential of the exercise lies in its capacity to incentivise reflection on these issues, i. e. on biases in one's own mathematical thinking and their possible origins. With regard to the learning platform, the question arises of how to provide feedback that is standardised but will nonetheless relate to individual mistakes to some degree in order to help students who are 'stuck in the DEM' and to encourage them to continue their investigation into Q ♥ . Due to worries that the presence of a sample solution A ♥ might discourage further investigation and bar access to certain learning experiences, we did not provide A ♥ on the page of the learning platform that contained the exercise. The option of enriching the milieu with discursive material in the way explained, on the other hand, has worked well for us: some students became aware of mistakes they had made themselves while all of them were provided an opportunity to analyse typical learner mistakes using didactic theory.

In order to create and refine an exercise that targets DEM bias in the described way, it is necessary to gather knowledge about typical student difficulties. Consultation of literature may guide towards promising mathematical topic areas and can generate ideas for potential exercises at the beginning of a design process. Once concrete exercises have been formulated, the potential of an exercise can be illuminated by creating a questions-answers map (our flowchart being a variation thereof). An initial questions-answers map functions as a reflection tool that gives an overview of the knowledge involved in various ways of dealing with the task and may generate (on the basis of knowledge about the DEM) a priori hypotheses about possible student difficulties. After a first trial of the exercise, materials that address mathematical difficulties with the task can be created on the basis of (naïve) observations of typical student mistakes and with the help of the map, which can facilitate a systematic covering of all difficulties of interest.

The questions-answers map can be refined in connection with the creation of a detailed REM, which has happened in our case. The REM does not only help deepen reflection on an exercise's potential and the involved institutional knowledge(s), but can be employed as analytical tool in ATD-based qualitative analyses of (various types of) data documenting student solutions or solution processes. In such analyses, it will serve as a normative reference against which deviations in solutions (or solution processes) can be systematically identified and characterised. This may be useful in cases where student difficulties are yet unknown or not documented sufficiently in existing research. We plan an exploratory investigation into our students' solution strategies on the basis of our REM in order to see if we can learn more about the types of difficulties that occur and how to address them in our teaching. (There is a possibility, too, that unexpected solution strategies which are viable in certain milieus will inform the development of further materials.) Additionally, our REM constitutes a novelty in that it mixes praxeological elements from university and school mathematics to create a representation of a praxeological organisation that stems from the institution of mathematics education. This may open up new possibilities of investigating questions of teacher education more broadly. E.g.: Can the necessity to use praxeological elements from university mathematics in tasks that 'could possibly occur in school' provide a profession-oriented raison-d'être for these elements in the study curriculum?

Fig. 1 :

 1 Fig. 1: Task instruction of the curve sketching exercise

Fig. 2 :

 2 Fig. 2: Definition of extremum (Freudigmann et al., 2012, p.46, translation by authors) Definition: Let the function 𝑓 be defined on an interval 𝐼, differentiable and let 𝑥 0 be an inner point of the interval. A point 𝑥 0 at which the graph of 𝑓 changes from being a left-hand 5 curve to being a righthand 4 curve or the other way around is called inflection point of 𝑓. The respective point 𝑊(𝑥 0 |𝑓(𝑥 0 )) is called inflection point of the respective graph.

Fig. 3 :

 3 Fig. 3: Definition of inflection point (Freudigmann et al., 2012, p.56, translation by authors)

Fig. 4 :

 4 Fig. 4: Schoolbook theorem: criteria for inflection points (Freudigmann et al., 2012, p. 56, transl. by authors)

Course development happens as part of the BMBF-funded teacher education project Leibniz-Prinzip (cf. LSE,

2021)

Due to limited space, we cannot really explain all relevant concepts of the ATD in detail and therefore have to refer the reader to the existing literature. Selecting a few works, we refer to[START_REF] Chevallard | Introducing the anthropological theory of the didactic: An attempt at a principled approach[END_REF] for an introduction to ATD and an overview.Lucas, Fonseca, Gascón and Schneider (2019) focus on the REM and the DEM as important concepts for research in ATD. For a concise introduction to the Herbartian schema we refer to[START_REF] Bosch | Study and Research Paths: A Model for Inquiry[END_REF].

The ♥ is a metaphor used within the ATD to indicate that this question is "at the heart" of the study process. A ♥ is then the answer to the question Q ♥ .

For a more detailed description of the 2020 implementation cf. https://platinum.uia.no/download/io3-teaching-unitsfor-student-inquiry/luh-teaching-units/ (last accessed 12.08.2022).

5 'left-hand curve' = strictly convex curve; 'right-hand curve' = strictly concave curve [the authors]

The symbols stand for: X = students, Y = teaching team, Q ♥ = questions 1) and 2) from Fig.1, A◊ School = algorithms for checking criteria for the existence of extremums and inflection points, A ◊ University = verification of one's own knowledge of the involved definitions, WSchool = provided schoolbook excerpt, Dmath = results of (correct) mathematical reasoning

It is prominently characterised by the assessment criteria we apply in our exams and which we communicate to our students, by the knowledge and practices we teach in our courses, etc.

Naturally, included solutions had to be mathematically sound.

Coding is understood here in the sense of qualitative content analysis; the units that were coded were the elements of the 4-T-model, i. e. (the application of) techniques, reasons given for applying a technique, more rarely: intermediate questions/tasks.

In German mathematics education, such exercises are referred to as "Schnittstellenaufgaben".


Frédéric Coquel

Khaled Saleh

Nicolas Seguin

A Relaxation Approach for Simulating Fluid Flows in a Nozzle

Keywords: Relaxation scheme, Godunov-type scheme, resonant transonic flows. MSC2010: 76M12, 76H05, 76S05, 65M12

We present here a Godunov-type scheme to simulate one-dimensional flows in a nozzle with variable cross-section. The method relies on the construction of a relaxation Riemann solver designed to handle all types of flow regimes, from subsonic to supersonic flows, as well as resonant transonic flows. Some computational results are also provided, in which this relaxation method is compared with the classical Rusanov scheme and a modified Rusanov scheme.

Introduction

In this paper, we are interested in the numerical approximation of the solutions of a model describing one-dimensional barotropic flows in a nozzle. In this model, ρ and w are respectively the density and the velocity of the fluid while α stands for the cross-section of the nozzle, which is assumed to be constant in time. Under the classical assumption that α is small with respect to a characteristic length in the mainstream direction, the flow can be supposed to be one-dimensional and described by the following set of partial differential equations:

∂ t α = 0, ∂ t (αρ) + ∂ x (αρw) = 0, t > 0, x ∈ R, ∂ t (αρw) + ∂ x (αρw 2 + α p(τ)) -p(τ)∂ x α = 0, (1)
Frédéric Coquel: CMAP, UMR 7641, Ecole Polytechnique, route de Saclay, F-91128 Palaiseau (frederic.coquel@cmap.polytechnique.fr) Khaled Saleh: EDF R&D, MFEE, 6 Quai Watier, F-78400 Chatou (khaled.saleh@edf.fr), and: UPMC & CNRS, UMR 7598, LJLL, F-75005 Paris (saleh@ann.jussieu.fr) Nicolas Seguin: UPMC & CNRS, UMR 7598, LJLL, F-75005 Paris (seguin@ann.jussieu.fr)

1
where τ = ρ -1 is the specific volume and τ → p(τ) is a barotropic pressure law (satisfying p ′ (τ) < 0 and p ′′ (τ) > 0). System (1) takes the condensed form:

∂ t U + ∂ x f(U) + c(U)∂ x U = 0, (2)
where the state vector is U = (α, αρ, αρw) T . The solutions are sought in the phase space of positive solutions defined as

Ω = {U = (α, αρ, αρw) T ∈ R 3 , α > 0, αρ > 0}. (3)
We recall the properties of this model:

• Property 1.1 (Hyperbolicity) System (1) admits, for U in Ω , the following eigenvalues

λ 0 (U) = 0, λ 1 (U) = w -c(τ), λ 2 (U) = w + c(τ), ( 4
)
where c(τ) = τ -p ′ (τ). The system is hyperbolic (i.e. the corresponding eigenvectors span R 3 ) if and only if |w| = c(τ). Besides, the fields associated with the λ 1 and λ 2 eigenvalues are genuinely non-linear while the field associated with λ 0 is linearly degenerate.

• Property 1.2 (Entropy) The entropy solutions of system (1) satisfy the following inequality in the weak sense

∂ t (αρE ) + ∂ x (αρE w + α p(τ)w) ≤ 0 ( 5
)
where E = w 2 2 + e(τ) is the total energy and where the function τ → e(τ) is given by e ′ (τ) = -p(τ).

The Godunov scheme for this model is difficult to implement because the Riemann problem for system (1) is hard to solve due to the non linearities of the pressure law (giving rise to the genuinely non-linear acoustic fields), to the absence of a satisfactory definition of the non-conservative product p(τ)∂ x α and to the resonance phenomenon that appears for transonic flows causing the model to lose hyperbolicity [START_REF] Goatin | The Riemann problem for a class of resonant hyperbolic systems of balance laws[END_REF]. For these reasons, we rather follow the classical approach of [7] and design an approximate Riemann solver, relying on a relaxation method. With this end in view, the solutions of system (1) are approximated by the solutions of the following enlarged relaxation system in the limit of a vanishing positive parameter ε:

∂ t α ε = 0, ∂ t (αρ) ε + ∂ x (αρw) ε = 0, t > 0, x ∈ R, ∂ t (αρw) ε + ∂ x (αρw 2 + απ(τ, T )) ε -π(τ, T ) ε ∂ x α ε = 0, ∂ t (αρT ) ε + ∂ x (αρT w) ε = 1 ε (αρ) ε (τ -T ) ε , (6)
with a linearization of the pressure law given by π(τ, T ) = p(T ) + a 2 (T -τ). The variable T is an additionnal unknown relaxing towards the specific volume τ in the limit ε ց 0, and the constant a is a numerical parameter that must be taken large enough so as to guarantee the non-linear stability of the numerical approximation. The state vector for the relaxation system is W = (α, αρ, αρw, αρT ) T and the solutions are sought in the phase space

Ω r = {W = (α, αρ, αρw, αρT ) T ∈ R 4 , α > 0, αρ > 0, αρT > 0}. ( 7
)
The following property motivates the introduction of this relaxation system Property 1.3 (Hyperbolicity) The convective part of ( 6) admits, for W in Ω r , the following eigenvalues

σ 0 (W) = 0, σ 1 (W) = w -aτ, σ 2 (W) = w, σ 3 (W) = w + aτ. ( 8
)
The system is hyperbolic (i.e. the corresponding eigenvectors span R 4 ) if and only if |w| = aτ, and all the fields are linearly degenerate.

The Riemann problem for the relaxation system

In this section, we give the main ideas leading to the construction of solutions to the Riemann problem for the convective part of the relaxation system [START_REF] Godlewski | Numerical Approximation of Hyperbolic Systems ofConservation Laws[END_REF]. Being given W L and W R two states in Ω r , we look for solutions of

∂ t W + ∂ x g(W) + d(W)∂ x W = 0, W(x, 0) = W L if x < 0 and W R if x > 0. (9)
As all the characteristic fields are linearly degenerate, the solution turns out to be simpler to construct than a solution of the Riemann problem for the equilibrium system [START_REF] Bouchut | Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws[END_REF]. Indeed, the solution is sought in the form of a self-similar function consisting in constant intermediate states separated by contact discontinuities. The linear degeneracy of the fields provides natural jump relations across each discontinuity and yields a set of equations eventually leading to the expessions of the wave speeds and intermediate states. However, some issues related to the resonance phenomenon still need to be handled with care (see [START_REF] Coquel | Relaxation and numerical approximation for fluid flows in a nozzle[END_REF] for details).

We show that the solutions can be expressed in terms of the physical data

V L = (ρ L , w L , T L ) and V R = (ρ R , w R , T R ) (i.
e. all the initial data excluding the crosssection α) and of the ratio of left and right initial sections ν := α L α R . More precisely, we introduce the following quantities depending only on (V L , V R )

w ♯ := 1 2 (w L + w R ) - 1 2a (π R -π L ), (10)
τ ♯ L := τ L + 1 a (w ♯ -w L ) = τ L + 1 2a (w R -w L ) - 1 2a 2 (π R -π L ), (11)
τ ♯ R := τ R - 1 a (w ♯ -w R ) = τ R + 1 2a (w R -w L ) + 1 2a 2 (π R -π L ), (12)
where w ♯ has the dimension of a speed and τ ♯ L , τ ♯ R the dimension of specific volumes. These quantities appear in the explicit expressions of the solutions and it can be proved that these specific volumes need to be positive in order to guarantee the positivity of the solutions. In the numerical applications however, a will be chosen large for stability matters (see section 4) and it will always be possible to impose the positivity of τ ♯ L and τ ♯ R by taking a large enough.

The main result of this section is the existence theorem for the Riemann problem.

Theorem 2.1 Let W L and W R be two positive states in Ω r . Assume that a is such that τ ♯ L > 0 and τ ♯ R > 0. Then the Riemann problem (9) admits a positive self-similar solution whatever the ratio ν = α L α R is.

Sketch of the proof (see [START_REF] Coquel | Relaxation and numerical approximation for fluid flows in a nozzle[END_REF] for details). The proof consists in the effective construction of a solution. For the relaxation system, the eigenvalues are not naturally ordered because of the existence of a standing wave, and a resonance phenomenon does appear for transonic flows. Therefore, in order to construct solutions, we investigate all admissible wave configurations (including sonic and supersonic ones) and for each admissible ordering of the eigenvalues, we determine sufficient conditions on the initial states W L and W R for the solution to have this particular ordering.

Eventually, we check a posteriori that the determined conditions totally cover the whole space of initial conditions Ω r × Ω r . ⊓ ⊔ Figure 1 represents the map of the admissible solutions given by Theorem 2.1 with respect to the initial states W L and W R . The right part of the chart corresponds to the solutions with positive material speed, while the left part depicts the symmetric configurations with negative material speed.

Numerical approximation

In this section, we derive a numerical scheme from the relaxation approximation introduced in section 1, the aim being to approximate the weak solutions of a Cauchy problem associated with system (1):

∂ t U + ∂ x f(U) + c(U)∂ x U = 0, U(x, 0) = U 0 (x). (13)
Let ∆ x be a space step and ∆t a time step. The space is partitioned into cells

R = j∈Z C j with C j = [x j-1 2 , x j+ 1 2
[, where x j+ 1 2 = ( j + 1 2 )∆ x are the cell interfaces. At the discrete times t n = n∆t, the solution of ( 13) is approximated on each cell C j by a constant value denoted by U n j = α n j , (αρ) n j , (αρw) n j T

. We now describe the twostep splitting method associated with the relaxation system (6) in order to calculate the values of the approximate solution at time t n+1 (U n+1 j ) j∈Z from those at time t n .

Step 1: Time evolution (t n → t n+1,-)

We first introduce the piecewise constant approximate solution of the relaxation system at time t n : x → W(x,t n ) = W n j in C j with W n j = α n j , (αρ) n j , (αρw) n j , (αρT ) n j , where T n j := τ n j , i.e. W n j is at equilibrium. Then, the following Cauchy problem is exactly solved for t ∈ [0, ∆t] with ∆t small enough (see condition (15) below)

∂ t W + ∂ x g( W) + d( W)∂ x W = 0, W(x, 0) = W(x,t n ). (14)
Since the initial condition x → W(x,t n ) is piecewise constant, the exact solution of ( 14) is obtained by gluing together the solutions of the Riemann problems set at each cell interface x j+ 1 2 , provided that these solutions do not interact during the period ∆t, i.e. provided the following classical CFL condition ∆t ∆ x max

W |σ i (W)| < 1 2 , i ∈ {0, ..., 3}, (15)
for all W under consideration. More precisely, if

(x,t) is in [x j , x j+1 ] × [0, ∆t], then W(x,t) = W r x -x j+1/2 t ; a j+1/2 , W n j , W n j+1 , (16)
where (x,t) → W r x t ; a, W L , W R is the self-similar solution of the Riemann problem constructed in section 1, which clearly depends on the local choice of the parameter a. Then, in order to define a piecewise constant approximate solution at time t n+1,-, the solution W(x,t) is averaged on each cell C j at time ∆t:

W(x,t n+1,-) = W n+1,- j := 1 ∆ x x j+ 1 2 x j-1 2 W(x, ∆t)dx, ∀x ∈ C j , ∀ j ∈ Z. (17)
Step 2: Instantaneous relaxation (t n+1,-→ t n+1 )

The second step consists in sending ε to zero instantaneously in the piecewise constant function W(x,t n+1,-) obtained at the end of the first step. This amounts to imposing T n+1 j := τ n+1 j , thus we have

W n+1 j = α n+1,- j , (αρ) n+1,- j , (αρw) n+1,- j , α n+1,- j T . (18)
Finally, the new cell value at time t n+1 of the approximate solution reads

U n+1 j = α n+1,- j , (αρ) n+1,- j , (αρw) n+1,- j T . (19)
We can prove that this two-step relaxation method can be equivalently rewritten in the form of a Godunov-type finite volume scheme [7].

Non-linear stability of the scheme

Non-linear stability issues are usually dealt with through a so-called discrete entropy inequality, which is the discrete counterpart of the entropy inequality (5) satisfied by the weak solutions of the model. We have the following definition: Definition 4.1 We say that a numerical scheme satisfies a discrete entropy inequality if there exists a numerical entropy flux G(U L , U R ) which is consistent with the exact entropy flux G = αρE w + α p(τ)w (in the sense that G(U, U) = G (U) for all U) such that, under some CFL condition, the discrete values (U n j ) j∈Z,n∈N computed by the scheme automatically satisfy

(αρE )(U n+1 j ) -(αρE )(U n j ) + ∆t ∆ x (G(U n j , U n j+1 ) -G(U n j-1 , U n j )) ≤ 0. ( 20
)
As seen in section 3, under the CFL condition (15), the different Riemann problems at each interface do not interact and the parameter a = a j+ 1 2 can be chosen locally interface by interface. Usually, if a j+ 1 2 is large enough, so as to satisfy a so-called Whitham condition (see [START_REF] Bouchut | Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws[END_REF]), then a discrete entropy inequality (20) is guaranteed. In order to define a j+ 1 2 , we propose a weak Whitham-like condition that handles the resonance phenomenon and still guarantees a discrete entropy inequality under the CFL condition (15) (see [START_REF] Coquel | Relaxation and numerical approximation for fluid flows in a nozzle[END_REF] for details).

Numerical tests

In this section, we run the relaxation scheme described in section 3 on a Riemann problem that contains the standing wave associated with the constant cross-section α, a left-going λ 1 -rarefaction wave, a sonic right-going λ 1 -rarefaction wave and a right-going λ 2 -shock. The chosen pressure law is an ideal gas barotropic pressure law p(τ) = τ -γ , with γ = 3. The left and right initial conditions are given by α L = 3.0, ρ L = 1.0, w L = 0, α R = 1.0, ρ R = 0.1, and w R = 0. The outcome of the relaxation method is compared with two other numerical schemes. The first one is the classical Rusanov scheme where the cross-section α is preserved throughout time:

α n+1 j := α n j . (21)
The second one is a modification of the Rusanov scheme that consists in applying the scheme to the whole state vector U (including the cross-section α) causing α to be dissipated:

α n+1 j := α n j - ∆t ∆ x q n j+ 1 2 -q n j-1 2 , ( 22
) with q n j+ 1 2 = -r(U n j )(α n j+1 -α n j )
where the scalar r(U n j ) is the maximal value of the spectral radius of the Jacobian matrices (∇f + c)(U n k ) for k = j, j + 1. In Figure 2, we can see that, due to a smoothing effect, the dissipation of the cross-section α provides a notable improvement for the Rusanov scheme (see [START_REF] Girault | A two-fluid hyperbolic model in a porous medium[END_REF] and [8] for different approaches to improve the Rusanov scheme). The L 1 -norm of the error on α, at the final time T , vanishes as the space step ∆ x goes to zero (with ∆t/∆ x constant) with the order O(∆ x 1/2 ). The paper is in final form and no similar paper has been or is being submitted elsewhere.

Fig. 1

 1 Fig. 1 Wave configuration of the solution of the Riemann problem (9) with respect to W L and W R . M L = w L aτ L and M R = w R aτ R are the Mach numbers of the initial left and right states W L and W R . The material wave is represented by a dashed line.

Fig. 2

 2 Fig. 2 Solution of the Riemann problem at time T = 0.2. Space step ∆ x = 10 -5 . Straight line: relaxation scheme, circles: classical Rusanov scheme, triangles: Rusanov scheme with dissipation of the cross-section.

Acknowledgements

The second author receives a financial support by ANRT through an EDF-CIFRE contract 529/2009. Computational facilities were provided by EDF. The third author is partially supported by the LRC Manon (Modélisation et Approximation Numérique Orientées pour l'énergie Nucléaire -CEA/DM2S-LJLL).


Fabrice Babik

Jean-Claude Latché

Bruno Piar

Khaled Saleh

A staggered scheme with non-conforming refinement for the Navier-Stokes equations

We propose a numerical scheme for the incompressible Navier-Stokes equations. The pressure is approximated at the cell centers while the vector valued velocity degrees of freedom are localized at the faces of the cells. The scheme is able to cope with unstructured non-conforming meshes, involving hanging nodes. The discrete convection operator, of finite volume form, is built with the purpose to obtain an L 2 -stability property, or, in other words, a discrete equivalent to the kinetic energy identity. The diffusion term is approximated by extending the usual Rannacher-Turek finite element to non-conforming meshes. The scheme is first order in space for energy norms, as shown by the numerical experiments.

Introduction

Let Ω be an open bounded connected subset of R d , with d ∈ {2, 3}, which is supposed to be polygonal if d = 2 and polyhedral if d = 3. Let T ∈ R + . We address in this paper the system of incompressible Navier-Stokes equations: ∂ t u + div(u ⊗ u) -µ∆ ∆ ∆u + ∇ ∇ ∇p = 0, on Ω × (0, T ), (1a) divu = 0, on Ω × (0, T ), (1b)

u| ∂ Ω = u ∂ Ω , u| t=0 = u 0 . (1c)
The variables u ∈ R d and p ∈ R are the velocity and the pressure in the flow, and µ is a positive constant viscosity. The initial condition u 0 is supposed to be divergencefree, and the integral of u ∂ Ω • n ∂ Ω over ∂ Ω vanishes, where n ∂ Ω stands for the normal vector to ∂ Ω outward Ω .

We develop in this paper a projection scheme to approximate the solution of (1), based on a staggered space discretization and able to cope with non-conforming mesh refinement. During the last years, a research program has been undertaken to develop staggered schemes satisfying a discrete kinetic energy balance [START_REF] Ansanay-Alex | An L 2 -stable approximation of the Navier-Stokes convection operator for low-order non-conforming finite elements[END_REF][START_REF] Gastaldo | Staggered discretizations, pressure correction schemes and all speed barotropic flows[END_REF]. This point is crucial with respect to many issues: it readily provides stability estimates, Fabrice Babik, Jean-Claude Latché, Bruno Piar and Khaled Saleh Institut de radioprotection et de sûreté nucléaire (IRSN), e-mail: [fabrice.babik, jean-claude.latche, bruno.piar, khaled.saleh]@irsn.fr a property which is a prerequisite for LES applications, and, last but not least, it is a starting point for the extension of the schemes to compressible flows (shallow water, compressible Navier-Stokes and Euler equations). The difficulty lies in the definition of the velocity convection operator, which must be in some sense consistent with the discrete mass balance; the definition of this operator is thus intricate and, to our knowledge, novel, at least for density variable flows. The objective of the present paper is to show how to extend this definition to non-conforming meshes. We first define admissible meshes (Section 2), then describe the scheme (Section 3) and finally present some numerical experiments to assess its behavior (Section 4).

Definition of the meshes

Let M be a decomposition of the domain Ω either in convex quadrilaterals (d = 2) or hexahedra (d = 3). The mesh M is supposed to be obtained from a usual finite element regular discretization (e.g. [START_REF] Ciarlet | Basic error estimates for elliptic problems[END_REF]) by recursively splitting some cells in 2 d subcells obtained by joining every two opposite face centers (Fig. 1). We allow at most one hanging node at the mass center of a cell face, which means that the maximum level of refinement between two adjacent cells is one. We denote by E (K) the set of the faces of an element K ∈ M . We exclude the presence of a node in the interior of a face, i.e. we split an initial face in 2 d-1 faces if one of the cells adjacent to the face is split. The number of faces, N E K , of a cell K thus ranges between 2d and 2 d d.

Let E = ∪ K∈M E (K), E ext = {σ ∈ E , σ ⊂ ∂ Ω } and E int = E \ E ext .
A face σ ∈ E int separating the cells K and L is denoted by K|L. For σ ∈ E (K), n K,σ is the unit normal vector to σ outward K. Hereafter, | • | stands for the d-or (d -1)-dimensional measure of a subset of R d or R d-1 respectively.

We define a dual mesh associated with the faces E as follows. When K ∈ M is a rectangle or a cuboid, for σ ∈ E (K), we define the half-diamond cell D K,σ as the cone with basis σ and with vertex the mass center of K (see Fig. 2). We thus obtain a partition of K in N E K sub-volumes, each sub-volume having a measure |D K,σ | equal to |K|/(2d), when σ has not been split, or |K|/(2 d d) otherwise. We extend this definition to general quadrangles and hexahedra, by supposing that we have built a partition with the same connectivities and the same ratio between the volumes of the half-diamonds and of the cell. For σ ∈ E int , σ = K|L, we now define the dual (or diamond) cell D σ associated with σ by

D σ = D K,σ ∪ D L,σ . For σ ∈ E (K) ∩ E ext , we define D σ = D K,σ .
We denote by Ẽ (D σ ) the set of faces of D σ , and by ε = D σ |D σ ′ the face separating two dual cells D σ and D σ ′ (see Fig. 2). 3 The pressure correction scheme

D σ D σ ′ σ ′ =K |M K σ =K |L L M ε= D σ |D σ ′

General form of the scheme

The space discretization is staggered in the sense that the pressure and the velocity are piecewise constant functions respectively on the primal and dual mesh. The initial discrete velocity is defined on a dual cell D σ , σ ∈ E int , by the mean value u 0 σ of the function u 0 over the face σ . The Dirichlet boundary condition is taken into account by setting u n σ to the mean value of u ∂ Ω over σ , for all σ ∈ E ext and all n ≥ 0. We consider a constant time step δt. As usual [START_REF] Temam | Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires II[END_REF][START_REF] Chorin | Numerical solution of the Navier-Stokes equations[END_REF][START_REF] Guermond | An overview of projection methods for incompressible flows[END_REF], the projection scheme is a two-step algorithm:

Prediction step -Find (u ⋆ σ ) σ ∈E int such that: 1 δt (u ⋆ σ -u n σ ) + 1 |D σ | ∑ ε∈ Ẽ (D σ ) F n σ ,ε u ⋆ ε -µ (∆ ∆ ∆ u) ⋆ σ + (∇ ∇ ∇p) n σ = 0, σ ∈ E int , (2a)
Correction step -Find (u n+1 σ ) σ ∈E int and (p n+1 K ) K∈M such that: 1 δt (u n+1 σ -u ⋆ σ ) + (∇ ∇ ∇p) n+1 σ -(∇ ∇ ∇p) n σ = 0, σ ∈ E int , (2b)
∑ σ ∈E (K) F n+1 K,σ = 0, with F n+1 K,σ = |σ | u n+1 σ • n K,σ , K ∈ M . ( 2c
)
The pressure gradient is built as the dual operator of the discrete divergence:

(∇ ∇ ∇p) σ = |σ | |D σ | (p L -p K ) n K,σ , σ = K|L.
The discretization of the diffusion term relies on the so-called "rotated bi-linear element" introduced by Rannacher and Turek [START_REF] Rannacher | Simple nonconforming quadrilateral Stokes element[END_REF]. The reference element K is the unit d-cube (0, 1) d , and the discrete functional space is:

Q1 ( K) = span 1, (x i ) i=1,...,d , (x 2 i -x 2 i+1 ) i=1,...,d-1 .
When there is no hanging node on a face σ , we impose the jump through the face to have a zero mean value. When there is a hanging node, we only impose to zero the integral of the jump through the initial face. Hence, the set {ζ σ , σ ∈ E int } of nodal functions associated with the Rannacher-Turek element is defined as follows.

When no vertices of σ = K|L is a hanging node, we define ζ σ such that supp(ζ σ ) ⊂ K ∪ L, for all K ∈ M , ζ σ | K belongs to the Rannacher-Turek local discrete space of K (i.e. the image of the space Q1 ( K) by the Q 1 mapping) and:

1 |σ | σ ζ σ = 1 and, for all σ ′ ∈ E , σ ′ = σ , σ ′ ζ σ = 0. ( 3
)
When one of the vertices of σ = K|L is a hanging node, it means that σ separates a cell obtained by splitting the mesh, say L, from an unsplit one, say K. The support of ζ σ is still K ∪ L and on L, ζ σ is still given by (3). Let Σ be the initial face of K including σ , and let ζ Σ be the Rannacher-Turek usual shape function (i.e. the function satisfying an analogue of (3) on the initial mesh). Then, we define

ζ σ on K by ζ σ (x) = |σ | |Σ | ζ Σ (x). K L σ Σ σ Σ |σ | |Σ | ζ Σ ζ σ Fig. 3 Piecewise definition of ζ σ .
Finally, dropping the time index, the discretization of the diffusion term reads:

-(∆ ∆ ∆ u) σ = 1 |D σ | ∑ K∈M K ∑ σ ′ ∈E (K) u σ ′ (∇ ∇ ∇ζ σ ′ • ∇ ∇ ∇ζ σ ). (4)
In the convection term, the velocity interpolates at the internal dual faces u ⋆ ε is chosen centered:

u ε = (u σ + u σ ′ )/2, for ε = D σ |D ′ σ .
To make the description of the scheme complete, we now only need to define the mass fluxes through the dual faces (F n σ ,ε ) ε∈ Ẽ : this is the purpose of the next section.

Discrete kinetic energy and mass fluxes

The discrete mass fluxes through the faces of the dual mesh are built so that a finite volume discretization of the divergence constraint (1b) holds over the dual cells:

∑ ε∈ Ẽ (D σ ) F n σ ,ε = 0, σ ∈ E int . (5)
This is crucial in order to reproduce, at the discrete level, the derivation of a kinetic energy balance equation, thus ensuring discrete analogues of the usual L ∞ (L 2 )and L 2 (H 1 )stability estimates for the velocity. It may be shown that Relation (5) holds if the dual fluxes are computed from the primal ones (F n K,σ ) σ ∈E (K) at the previous time-step so as to satisfy the following three constraints (see [START_REF] Ansanay-Alex | An L 2 -stable approximation of the Navier-Stokes convection operator for low-order non-conforming finite elements[END_REF][START_REF] Gastaldo | Staggered discretizations, pressure correction schemes and all speed barotropic flows[END_REF] for details):

• (H1) -For all primal cell K in M , the set (F σ ,ε ) ε⊂K of dual fluxes through faces included in K satisfies the following linear system, with ξ σ K = |D K,σ |/|K|:

F K,σ + ∑ ε∈ Ẽ (D σ ), ε⊂K F σ ,ε = ξ σ K ∑ σ ′ ∈E (K) F K,σ ′ , ∀σ ∈ E (K). (6)
• (H2) -The dual fluxes are conservative:

F σ ,ε = -F σ ′ ,ε for all ε = D σ |D ′ σ .
• (H3) -The dual fluxes are a bounded function of the primal ones (F K,σ ) σ ∈E (K) :

|F σ ,ε | ≤ C max {|F K,σ |, σ ∈ E (K)} , K ∈ M , σ ∈ E (K), ε ∈ Ẽ (D σ ), ε ⊂ K.

Dual fluxes for non-refined meshes

The system of equations ( 6) has an infinity of solutions, which makes necessary to impose in addition the constraint (H3). Since ( 6) is linear with respect to the 6) may thus be expressed as:

F σ ,ε , σ ∈ E (K), ε ∈ Ẽ (D σ ), ε ⊂ K, a solution of (
F σ ,ε = ∑ σ ′ ∈E (K) (α K ) σ ′ σ F K,σ ′ , σ ∈ E (K), ε ∈ Ẽ (D σ ) and ε ⊂ K,
and (H3) is equivalent to requiring bounded coefficients

((α K ) σ ′ σ ) σ ,σ ′ ∈E (K)
. In addition, since ξ σ K = 1/(2d) for all K ∈ M and σ ∈ E (K), system (6) is completely independent from the cell K under consideration. We may thus consider a particular geometry for K, let us say K = (0, 1) d , and find an expression for the coefficients ((α K ) σ ′ σ ) σ ,σ ′ ∈E (K) which we will apply to all the cells, thus automatically satisfying the constraint (H3). A technique for this computation is described in [1, Section 3.2]. The idea is to build a momentum field w with a constant divergence and such that σ w • n K,σ dσ (x) = F K,σ , for all σ ∈ E (K). Then, an easy computation shows that the definition F σ ,ε = ε w • n σ ,ε dσ (x) satisfies [START_REF] Guermond | An overview of projection methods for incompressible flows[END_REF]. The set of coefficients

((α K ) σ ′ σ ) σ ,σ ′ ∈E (K)
obtained for a quadrangle is given in [1, Section 3.2]; extension to the three-dimensional case is straightforward.

Dual fluxes for 2D-refined meshes

Here again, we may restrict the computation to square cells. In 2D, if a primal cell is surrounded with four refined cells, the half-diamond cells are obtained by splitting the cell in four sub-squares, each one being split in two triangles. Hence, eight dual fluxes must be computed; if some of the neighboring cells are not refined, one uses a coarsening procedure. We begin with computing the dual fluxes across the four subsquares faces (solid gray color in Fig. 4) so that (6) holds, with (F K,σ ) σ ∈E (K) denoted here by F i (4 ≤ i ≤ 11) and F σ ,ε , σ ∈ E (K), ε ⊂ K denoted here by F i (4 ≤ i ≤ 7). The linear system to solve has a one dimensional kernel and a particular solution satisfying (H3) is given in Fig. 4. Then, the dual fluxes across the diagonal faces Fi (0 ≤ i ≤ 3) (dashed gray color in Fig. 4) are computed by isolating the sub-squares and applying the procedure described above for the non-refined case.

F 11 F 10 F 6 F 7 F 4 F 5 F 9 F 8 F 7 F 5 F 4 F 6 F3 F0 F1 F2 F 4 = 3 8 ( F 5 + F 6 -F 11 -F 4 ) + 1 8 ( F 7 + F 8 -F 9 -F 10 ) F 5 = 3 8 ( F 7 + F 8 -F 5 -F 6 ) + 1 8 ( F 9 + F 10 -F 11 -F 4 ) F 6 = 3 8 ( F 9 + F 10 -F 7 -F 8 ) + 1 8 ( F 4 + F 11 -F 5 -F 6 ) F 7 = 3 8 ( F 4 + F 11 -F 9 -F 10 ) + 1 8 ( F 5 + F 6 -F 7 -F 8 )
Fig. 4 Dual fluxes for the neighboring cell of refined cells (2D case).

Dual fluxes for 3D-refined meshes

The procedure is the same as in the 2D-case. The first step consists in splitting the cube in eight sub-cubes and computing the dual fluxes across the faces of these sub-cubes. The formula of one of these intermediate fluxes F is given in Fig. 5. The computation of the other fluxes across the faces separating two sub-cubes is deduced by permutations of the indices.

In the second step, each sub-cube is split in 3 half-diamonds of equal volumes. One obtains 24 half-diamonds and 48 internal half-diamond faces of two possible types (see Fig. 6). The dual fluxes across these faces are obtained by isolating the sub-cubes and applying the procedure described above for the non-refined case.

Numerical test

We assess the behavior of the proposed numerical scheme on an exact analytical solution to the stationary Navier-Stokes equations known as the Kovasznay flow [START_REF] Kovasznay | Laminar flow behind a two-dimensional grid[END_REF]. Computations are performed with the free software CALIF 3 S developed at IRSN [START_REF]CALIF 3 S: A software components library for the computation of reactive turbulent flows[END_REF]. The velocity and pressure fields are given by:

F 24 F = 7 ( F 14 + F 21 + F 27 ) -7 ( F 15 + F 19 + F 25 ) + 2 ( F 7 + F 20 + F 26 ) -2 ( F 6 + F 18 + F 24 ) + 2 ( F 11 + F 16 + F 29 ) -2 ( F 13 + F 17 + F 23 ) + ( F 9 + F 10 + F 28 ) -( F 8 + F 12 + F 22 )
Fig. 5 Intermediate dual fluxes for the neighboring cell of refined cells (3D case).

   , p = 1 2 (1 -e 2λ x ), λ = 1 2µ - 1 4µ 2 + 4π 2 1/2 ,
where µ stands for the viscosity of the flow, taken here as µ = 1/40. The computational domain is Ω = (-0.5, 1) × (-0.5, 1.5). The mesh is built from a regular n × n grid, where we refine the sub-domain Ω f = (-0.5, 0.5) × (-0.5, 0.5) ∪ (0.5, 1) × (0.5, 1.5) by splitting each (square) cell included in Ω f in four sub-squares. The solution is computed by the projection scheme, by letting a fictitious transient tend to the desired steady state. Boundary conditions are given by the analytical solution. The obtained numerical errors for various values of n are gathered in the following table, where u exact and p exact stand for the exact velocity and pressure, respectively. The observed order of convergence in L 2 -norm is approximately 2 for the velocity and 1 for the pressure. The contour lines of the first component of the velocity are drawn on Fig. 7. We may check that no spurious perturbation appears along the lines separating the refined and non-refined parts of the computational domain (in other words, the lines composed by the union of the faces including a hanging node). The theoretical study of this scheme is underway, and the error analysis confirms these experiments.

Fig. 1

 1 Fig.1An exemple of admissible mesh refinement.

Fig. 2

 2 Fig. 2 Notations for control volumes and diamond cells.

FFFig. 6 1 -

 61 Fig. 6 Two possible types of internal half-diamond faces (3D case).

n u -u exact L 2 (Fig. 7

 27 Fig. 7 Contour lines of the field u 1 . The dashed lines materialize the boundary of the refined area (bottom-left and top-right sub-domains).


Khaled Saleh

Arnaud Duran

Christophe Berthon

An easy control of the artificial numerical viscosity to get discrete entropy inequalities when approximating hyperbolic systems of conservation laws

An easy control of the artificial numerical viscosity to get discrete entropy inequalities when approximating hyperbolic systems of conservation laws

Main motivations

The present work is devoted to the numerical approximation of the weak solutions of hyperbolic systems made of N ≥ 1 conservation laws in the form

∂ t w + ∂ x f (w) = 0, x ∈ R, t > 0, (1)
where w ∈ R N stands for the unknown vector and f : R N → R N denotes the flux function. According to some physical restriction, the solution may be imposed to belong to an invariant admissible domain Ω ⊂ R N , so that we have w ∈ Ω. The model is complemented with an initial data w(x, t = 0) = w 0 (x) where w 0 : R → R N is given.

Since the system (1) is assumed to be hyperbolic, in a finite time the solution may contain discontinuities. Such discontinuities are governed by the well-known Christophe Berthon Laboratoire de Mathématiques Jean Leray, CNRS UMR 6629, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes, France, e-mail: christophe.berthon@univ-nantes.fr Arnaud Duran and Khaled Saleh Institut Camille Jordan, CNRS UMR 5208, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France e-mail: duran@math.univ-lyon1.fr and e-mail: saleh@math.univ-lyon1.fr Rankine-Hugoniot relations (for instance, see [START_REF] Toro | Riemann solvers and numerical methods for fluid dynamics[END_REF]). Unfortunately, by only adopting theses relations, the uniqueness of the solution is lost. In order to rule out nonphysical discontinuities, the system is endowed with entropy inequalities given by

∂ t η(w) + ∂ x G(w) ≤ 0, (2)
where η : Ω → R is a convex function, called the entropy function, and G : Ω → R is the entropy flux function such that t ∇ w f (w) = t ∇ w η(w)∇ w G(w). A solution of ( 1) is said entropy satisfying if the entropy inequalities (2) are verified for all entropy pairs (η, G). Now, let us consider the numerical approximation of w. To address such an issue, we first introduce a discretization of the space by adopting a uniform mesh made of cells (x i-1/2 , x i+1/2 ) of constant size ∆x > 0 so that x i+1/2 = x i-1/2 + ∆x for all i in Z. We denote x i = (x i-1/2 + x i+1/2 )/2 for i in Z. Concerning the time discretization, we fix t n+1 = t n + ∆t with ∆t > 0 the time increment restricted according to a suitable CFL condition (for instance, see [START_REF] Toro | Riemann solvers and numerical methods for fluid dynamics[END_REF]).

At time t n , we consider an approximation of the solution given by a piecewise constant function as follows:

w ∆ (x, t n ) = w n i if x ∈ (x i-1/2 , x i+1/2 ). (3)
In order to evolve with respect to time this approximation, the updated states (w n+1 i ) i ∈Z are evaluated by the following numerical scheme:

w n+1 i = w n i - ∆t ∆x f ∆ (w n i , w n i+1 ) -f ∆ (w n i-1 , w n i ) , (4)
where f ∆ (w L , w R ) is the numerical flux function. From now on, we underline that this numerical flux function must satisfy f ∆ (w, w) = f (w) for consistency reasons.

During the fifty last years, numerous works were devoted to propose relevant formulas to define f ∆ (w L , w R ). Of course, it is not possible to give an exhaustive list of the numerical methods available to approximate the weak solutions of (1), but the reader is referred to [START_REF] Toro | Riemann solvers and numerical methods for fluid dynamics[END_REF][START_REF] Godlewski | Numerical approximation of hyperbolic systems of conservation laws[END_REF] for an overview of some well-known numerical techniques.

For stability reasons, the scheme (4) is complemented with a CFL condition (for instance, see [START_REF] Toro | Riemann solvers and numerical methods for fluid dynamics[END_REF][START_REF] Godlewski | Numerical approximation of hyperbolic systems of conservation laws[END_REF]) to restrict the time step by ∆t ∆x Λ ≤ 1 2 , where Λ ≥ 0 is determined according to the numerical flux function definition.

At this level, according to the celebrated Lax-Wendroff Theorem [START_REF] Lax | Systems of conservation laws[END_REF], we may expect a convergence of w ∆ (x, t n ) to a weak solution w(x, t) of system (1) as ∆t and ∆x tend to zero. In order to avoid some possible nonphysical solutions, the scheme must be enriched with discrete entropy inequalities. As a consequence, the updated states (w n+1 i ) i ∈Z , given by ( 4), must satisfy the following estimation:

η(w n+1 i ) -η(w n i ) + ∆t ∆x G ∆ (w n i , w n i+1 ) -G ∆ (w n i-1 , w n i ) ≤ 0, (5)
where G ∆ (w L , w R ) denotes the numerical entropy flux function. From now on, we underline that this numerical flux function must satisfy G ∆ (w, w) = G(w) for all w in Ω.

If the discrete entropy inequalities [START_REF] Harten | On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[END_REF] are satisfied by all entropy pairs (η, G), the scheme is said entropy stable or entropy preserving. The establishment of ( 5) may be very difficult to obtain as soon as N > 1. The Godunov scheme [START_REF] Godunov | A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[END_REF] or the HLL scheme [START_REF] Harten | On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[END_REF] are known to be entropy preserving. Next, considering the isentropic gas dynamics or the Euler model, some schemes such as the HLLC scheme [START_REF] Toro | Restoration of the contact surface in the HLL-Riemann solver[END_REF] or the Suliciu relaxation scheme [START_REF] Chalons | Godunov-type schemes for hyperbolic systems with parameter-dependent source. The case of Euler system with friction[END_REF] are also proved to be entropy preserving. But, in general, the proof of ( 5) is not reachable or, eventually, is violated. For instance, the Roe scheme [START_REF] Roe | Approximate Riemann solvers, parameter vectors, and difference schemes[END_REF] or the VF-Roe scheme [START_REF] Gallouët | Some recent finite volume schemes to compute Euler equations using real gas EOS[END_REF] are known to be entropy violating and suitable entropy fixes must be adopted (for instance, see [START_REF] Helluy | A simple parameter-free entropy correction for approximate Riemann solvers[END_REF]).

In this paper, we introduce a simple extension of the artificial viscosity technique [START_REF] Tadmor | Numerical viscosity and the entropy condition for conservative difference schemes[END_REF][START_REF] Tadmor | Entropy stable schemes. Handbook of Numerical Methods for Hyperbolic Problems: Basic and Fundamental Issues[END_REF] in order to easily recover the required entropy stability given by ( 5). In the next section, we propose a reformulation of the artificial viscosity within the framework of Godunov-type schemes. In Section 3, we give a direct control of the artificial viscosity to get (5) for a given entropy pair. The last section is devoted to illustrate the relevance of the suggested numerical procedure.

Godunov-type scheme with artificial viscosity

In order to enforce the required entropy stability, we introduce artificial viscosity into the scheme (4). As a consequence, the improved numerical method of interest now reads

w n+1 i = w n i - ∆t ∆x f ∆ (w n i , w n i+1 ) -f ∆ (w n i-1 , w n i ) + γ 2 ∆t ∆x w n i+1 -2w n i + w n i-1 , (6)
where γ ≥ 0 is a parameter to govern the artificial viscosity. Now, we propose to reformulate (6) as a Godunov-type scheme. For this purpose, we introduce the following approximate Riemann solver:

w R x t ; w L , w R =                        w L if x < -(Λ + γ)t, wL if -(Λ + γ)t < x < -Λt, w ⋆ L if -Λt < x < 0, w ⋆ R if 0 < x < Λt, wR if Λt < x < (Λ + γ)t, w R if x > (Λ + γ)t, (7)
where we have set

wL = w L + 1 2 (w R -w L ) = 1 2 (w L + w R ), wR = w R - 1 2 (w R -w L ) = 1 2 (w L + w R ), w ⋆ L = w L - 1 Λ ( f ∆ (w L , w R ) -f (w L )) , w ⋆ R = w R + 1 Λ ( f ∆ (w L , w R ) -f (w R )) . (8)
To simplify the notations, we set w = wL = wR . Next, equipped with this approximate Riemann solver, we evolve the approximate solution (3) in times as follows:

w ∆ (x, t n + t) = w R x -x i+1/2 t ; w n i , w n i+1 for (x, t) ∈ [x i , x i+1 ) × (0, ∆t),
where ∆t is restricted by

∆t ∆x (Λ + γ) ≤ 1 2 . ( 9
)
Such a time evolution coincides to a juxtaposition of approximate Riemann solvers w R stated at each interface x i+1/2 . Because of the CFL condition ( 9), two successive approximate Riemann solvers never interact. After a straightforward computation, we notice that the updated states w n+1 i , given by ( 6), reformulate as follows:

w n+1 i = 1 ∆x ∫ x i+1/2 x i-1/2 w ∆ (x, t n + ∆t)dx, = 1 ∆x ∫ ∆x/2 0 w R x ∆t ; w n i-1 , w n i dx + 1 ∆x ∫ 0 -∆x/2 w R x ∆t ; w n i , w n i+1 dx, (10)
Because of the Godunov-type reformulation, we are now able to control the artificial viscosity, parameterized by γ, order to obtain the expected discrete entropy inequality (5).

Discrete entropy inequality

To establish the required entropy stability (5), we first recall a result stated by Harten, Lax and van Leer in [START_REF] Harten | On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[END_REF]. In fact, this statement gives a local sufficient condition, interface per interface, that implies [START_REF] Harten | On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[END_REF]. The reader is referred to [START_REF] Harten | On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[END_REF] for a proof.

Lemma 1 (Harten, Lax and van Leer [5])

Let w L and w R be given in Ω. Assume that the approximate Riemann solver satisfies

1 ∆x ∫ ∆x/2 -∆x/2 η w R x ∆t ; w L , w R dx ≤ 1 2 (η(w L ) + η(w R )) - ∆t ∆x (G(w R ) -G(w L )) , (11
) for a given entropy pair (η, G). Then, the Godunov-type scheme [START_REF] Tadmor | A minimum entropy principle in the gas dynamics equations[END_REF] satisfies a discrete entropy inequality [START_REF] Harten | On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[END_REF] where the numerical entropy flux function reads as follows:

G ∆ (w L , w R ) = G(w R ) - ∆x 2∆t η(w R ) + 1 ∆x ∫ ∆x/2 0 η w R x ∆t ; w L , w R dx.
From now on, let us underline that, as soon as the approximate Riemann solver is given by ( 7), the numerical entropy flux function rewrites

G ∆ (w L , w R ) = G(w R ) + Λη(w ⋆ R ) + (Λ + γ)(η( w) -η(w R )).
Now, arguing Lemma 1, it is sufficient to satisfy [START_REF] Tadmor | Entropy stable schemes. Handbook of Numerical Methods for Hyperbolic Problems: Basic and Fundamental Issues[END_REF] to obtain [START_REF] Harten | On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[END_REF]. After a straightforward computation, the inequality ( 11) reads as follows:

E 0 + γD ≤ 0, ( 12
)
where we have set

E 0 = Λ η(w ⋆ L ) + η(w ⋆ R ) -η(w L ) -η(w R ) + (G(w R ) -G(w L )) , ( 13
) D = 2η( w) -η(w L ) -η(w R ). ( 14
)
Let us emphasize that E 0 coincides with the entropy dissipation rate for the initial scheme (4) when the artificial viscosity γ vanishes. Of course E 0 turns out to be negative as long as the scheme ( 4) is entropy preserving. But E 0 may become positive for an entropy violating scheme. Next, since η is a convex function, we immediately get D ≤ 0 with equality to zero if and only if w L = w R . Moreover, we remark that neither E 0 nor D depend on γ. As a consequence, as soon as D 0, it is possible to fix γ ≥ 0 large enough such that the local entropy inequality (12) holds.

Lemma 2 Let w L and w R be given in Ω. Then there exists γ ≥ 0 such that [START_REF] Toro | Riemann solvers and numerical methods for fluid dynamics[END_REF] holds. Moreover, assume that the matrix ∇ 2 w η(w) is positive definite then γ is bounded. Proof The existence of γ ≥ 0 is obvious. Now, we establish that γ is bounded in a neighborhood, denoted V, of w L = w R . Indeed, since D vanishes only into V, γ is immediately bounded far away from V. First, we give the expansion with respect of w of D, given by ( 14), in V. A direct evaluation gives D = O( w Rw L 2 ) with optimal order since ∇ 2 w η( w) is positive definite. Because of the convexity of η, never this quantity vanishes in V as long as w L w R . Concerning the expansion of E 0 , we first introduce

w H LL = 1 2 (w L + w R ) - 1 2Λ ( f (w R ) -f (w L )).
Following [START_REF] Harten | On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[END_REF], we have 2η(w

H LL ) ≤ η(w L ) + η(w R ) -G(w R ) + G(w L ), with equality if and only if w L = w R . Then we obtain E 0 /Λ ≤ η(w ⋆ L ) + η(w ⋆ R ) -2η(w H LL ). Next, we notice that w ⋆ L + w ⋆ R = 2w H LL , then we have η(w ⋆ L ) + η(w ⋆ R ) -2η(w H LL ) = O( w R -w L
2 ), with equality to zero if and only if w L = w R . As a consequence, we

Numerical results

To illustrate the procedure we adopt the Euler model, defined by w = t (ρ, ρu, E) and f (w) = t (ρu, ρu 2 + p, (E + p)u), where p is defined by the usual perfect gas law [START_REF] Toro | Riemann solvers and numerical methods for fluid dynamics[END_REF] with adiabatic coefficient fixed equal to 1.4. Here, the entropy function is given by η(w) = ρϕ(p/ρ 1.4 ), where ϕ is a smooth function which satisfies the restrictions determined in [START_REF] Tadmor | A minimum entropy principle in the gas dynamics equations[END_REF].

In Fig 1, we display the numerical results obtained when simulating a Riemann problem with initial data given by w 0 (x) = t (1, 0, 2.5) if x < 0 and w 0 (x) = t (0.125, 0, 0.025) if x > 0. The adopted initial scheme is given by the VF-Roe method [START_REF] Gallouët | Some recent finite volume schemes to compute Euler equations using real gas EOS[END_REF], known to be entropy violating. Then, we notice a nonphysical shock wave at the sonic point within the rarefaction. Next, we introduce the numerical artificial viscosity governed by γ fixed according to [START_REF] Toro | Riemann solvers and numerical methods for fluid dynamics[END_REF]. We remark that the entropy violating shock wave no longer remains. In addition, Fig 1 displays the values of γ versus time. As expected, γ is bounded.

To conclude this numerical experiment, let us underline that the control of the artificial viscosity is performed according to a single entropy pair. Here, we have adopted the entropy defined with ϕ 1 (θ) = ln(θ). As a consequence, a natural question arising concerns the behavior of the other entropies. In Tab 1, we present the value of L 1 -norm of the positive part of the entropy budget, defined as the left hand side of inequality [START_REF] Harten | On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[END_REF], obtained for two other entropies defined by ϕ 2 (θ) = -θ 1/2.4 and ϕ 3 (θ) = θ -2/1.4 when considering the original VF-Roe scheme. Since we expect negative entropy dissipation rate, the value of the positive part must be zero. With the original VF-Roe scheme, we get positive values of the entropy dissipation rate (see Tab 1) while we obtain values less than 10 -16 with the derived single entropy preserving scheme.

Fig. 1

 1 Fig.1Riemann solution for the density ρ, the velicity u, the pressure pand volution of γ.

Table 1

 1 Evaluation of L 1 -norm of the positive part of the entropy budget. = O( w Lw L 2 ) once again with optimal order since ∇ 2 w η(w H LL ) is positive definite. It results γ = O(1) and the proof is achieved.

	Cells	ϕ 1	ϕ 2	ϕ 3
	100	1.05291E-6	4.40092E-7	1.48806E-6
	800	5.53458E-8	2.30603E-8	7.90713E-8
	1600	3.42990E-8	1.42867E-8	4.90515E-8
	12800	8.27230E-9	3.44473E-8	1.18418E-8
	102400	1.23514E-9	5.14317E-9	1.76836E-9
	have E 0


Martino Trassinelli

Ajay Kumar

Heinrich F Beyer

Paul Indelicato

Renate Märtin

Regina Reuschl

Yuri S Kozhedub

Carsten Brandau

Harald Bräuning

Sabrina Geyer

A Gumberidze

S Hess

P Jagodzinski

C Kozhuharov

D Liesen

U Spillmann

S Trotsenko

G Weber

D F A Winters

Th Stöhlker

Observation of the 2p3/2 -> 2s1/2 intra-shell transition in He-like uranium

Keywords: 32.30.Rj -X-ray spectra 31.30.J--Relativistic and quantum electrodynamic effects in atoms, molecules, and ions 12.20.Fv -Experimental tests 07.85.Nc -X-ray and γ-ray spectrometers 32.10.Fm -Fine and hyperfine structure

He-like ions are the simplest multi-body atomic systems. Investigations of these ions along their isoelectronic sequence up to the heaviest species uniquely probe our understanding of correlation, relativistic, and Quantum Electrodynamic (QED) effects. In recent years, substantial progress in the investigations of these fundamental systems has been achieved in the high-Z region, in which the nucleus Coulomb field strength and electron velocity, both proportional to Zα, are very high. In theory, benchmark calculations have been reported where even second order QED effects were considered in a rigorous way for both the ground-state as well as for the first excited states [1][2][3]. Experimentally, progress has been mainly achieved by a technique recently introduced [4], which allows for the iso-(a) E-mail: martino.trassinelli@insp.jussieu.fr lation of the two-electron contributions to the ionization potential of the ground-state [4][5][6]. Here, the achieved uncertainty already approaches the expected size of higherorder QED contributions [5,6]. However, for the excited levels at high-Z (Z > 54) virtually no experimental data on binding or transition energies are available. Besides the general importance of such data for atomic structure investigations, the great interest in more detailed information about the excited levels in high-Z He-like ions is motivated by their relevance for the study of the influence of the electroweak interaction on the atomic structure. Since many years, high-Z He-like ions are under the discussion as an ideal candidates for the study of atomic parity violation effects [7,8]. The ability to observe parity-violating transitions critically depends on the relative binding energies of the excited states. A level crossing of two states with different parity, namely the 1s2p 3 P 0 and the 1s2s 1 S 0 levels, is predicted to occur close to Z = 66 and Z = 92. However, a benchmark test of the atomic structure theory for the excited states in He-like ions is still pending. Therefore, a precise measurement of the ∆n = 0 1s2p 3 P 2 → 1s2s 3 S 1 intra-shell transition energy is of particular importance. (a partial level scheme of a high-Z He-like ion is shown in Fig. 1). An attempt of an energy measurement was reported for the 1s2p → 1s2s intra-shell transition for He-like uranium using the electron-beam ion trap (EBIT) at Lawrence Livermore National Laboratory (LLNL) [9]. There, the desired 1s2p 3 P 2 → 1s2s 3 S 1 transition could not be identified unambiguously. This is in contrast to lighter ions up to Z = 54, where energy measurements of the 1s2p → 1s2s transitions have been performed by spectroscopy in the visible to the far UV region [3,10,11]. For higher values of Z, information on excited states has been obtained only via lifetime measurements of the 1s2p 3 P 0 level [12][13][14].

[1s 1/2 ,2s 1/2 ] 1 S 0 M1 2E1 E1M1 E1 E1 E1 E1 M2 [1s 1/2 ,2s 1/2 ] 3 S 1 [1s 1/2 ,2p 3/2 ] 1 P 1 [1s 1/2 ,2p 3/2 ] 3 P 2 [1s 1/2 ,2p 1/2 ] 3 P 0 [1s 1/2 ,2p 1/2 ] 3 P 1 [1s 1/2 ,1s 1/2 ] 1 S 0
In this Letter we present the first clear identification of the 1s2p 3 P 2 → 1s2s 3 S 1 intra-shell transition in He-like uranium and its energy measurement. For the purpose of the experiment, we used a standard Ge(i) solid-state detector and a new Bragg spectrometer specially designed for accurate spectroscopy of fast ions. The two instruments are complementary: The Ge(i) detector has a high detection efficiency and covers a wide spectral range with a moderate spectral resolving power. The focussing crystal spectrometer serves as an accurate wavelength comparator in a narrow wavelength interval. In concert, the present measurements allow for an unambiguous identification of the spectral lines observed plus an accurate determination of the transition wavelength or energy.

The experiment was performed at the ESR storage ring at the GSI in Darmstadt. Here, a beam of ∼ 4 × 10 7 hydrogen-like uranium ions was stored, cooled and decelerated to an energy of 43.57 MeV/u. The ion beam's momentum spread was close to ∆p/p ≈ 10 -5 , and its width Fig. 2: The x-ray spectrum (color online), originating from 43.57 MeV/u He-like uranium ions, as recorded by the Ge(i) detector. The energies correspond to the emitter frame. The inset shows a zoom around the 3 P2 → 3 S1 transition with the vertical bars and the continuous (red) curve representing the results of a full cascade calculation [START_REF] Stöhlker | [END_REF].

was about 2 mm. He-like excited ions were formed by electron capture during the interaction of the ion beam with a supersonic nitrogen gas-jet. The gas-jet had a width of about 5 mm and a typical areal density of 10 12 particles/cm 2 , which guaranteed single-collision conditions in the ion-target interaction. At the selected velocity, electrons are primarily captured into the shells with principal numbers n ≤ 20 [15][16][17]. This allows for an efficient population of the 1s2p 3 P 2 level via cascade feeding.

The 1s2p 3 P 2 excited state mainly decays in two ways (Fig. 1): to the ground state via a magnetic quadrupole (M2) transition, with a branching ratio of 70% and a decay time of 4.9 × 10 -15 sec, and to the 1s2s 3 S 1 state by an electric dipole (E1) intra-shell transition, with a branching ratio of 30% and a decay time of 1.2 × 10 -14 sec [START_REF] Indelicato | a MultiConfiguration Dirac Fock and general matrix elements program[END_REF]. In the reference frame of the projectile, the x-ray photons arising from the 2 3 P 2 → 2 3 S 1 transition have an energy of 4510 eV.

The Ge(i) solid state detector and the Bragg crystal spectrometer were mounted under observation angles of 35 • and 90 • , respectively. Both instruments were separated from the ultra-high vacuum of the gas target chamber by 100µm-thick beryllium windows transparent for the low-energy x rays.

The Ge(i) detector crystal has a diameter of 16 mm and therefore provides a relatively large solid angle. An xray collimator was mounted in front of the Ge(i) in order to limit the horizontal acceptance angle thereby reducing the Doppler broadening to match the corresponding Doppler width with the intrinsic line width of the detector, amounting to 250 eV at 5.9 keV photon energy.

A survey spectrum recorded by the Ge(i) detector is displayed in Fig. 2 with prominent lines originating from n ′ → n transitions, with n = 2 -4. The existence of lines originating from high n ′ to n = 2 indicates that the 3 P 2 state is mainly populated by cascade feeding which is also supported by a theoretical study [15]. The inset in Fig. 2 shows a zoom of the low-energy region of the spectrum. The vertical lines in the inset mark the line energies and intensities obtained by a full cascade calculation [START_REF] Stöhlker | [END_REF] whereas the continuous (red) line represents a fit of the theoretical model taking into account the experimental energy resolution. The good over-all correspondence between experiment and model gives confidence for the correct identification of the strong line at the expected energy of the 2 3 P 2 → 2 3 S 1 transition.

The measurement using the crystal spectrometer provided a much higher accuracy for the spectral line position. In this case, the observable energy range, principally limited by the ion beam diameter and its distance from the crystal, was in the order of 4308 ± 40 eV -much narrower than the linewidth observed with the Ge(i). For an unambigous identification of the He-like uranium intra-shell transition, the experimentally well known 1s 2 2p 2 P 3/2 → 1s 2 2s 2 S 1/2 transition [20,21] in Li-like uranium has been used as reference. Similar to the He-like system, Li-like ions were obtained by electron capture into He-like uranium ions. In order to park the Li-like transition close to the He-like line, we used the Doppler effect selecting a kinetic energy of 32.63 MeV/u for Li-like ions (to be compared with 43.57 MeV/u for He-like ones). This way both Li-and He-like uranium transitions appeared at nearly the same Bragg angle and consequently in the same narrow spatial region of the position-sensitive detector.

The crystal spectrometer was mounted in the Johann geometry for the detection of x rays with a corresponding Bragg angle Θ B around 46.0 • . A cylindrically bent germanium (220) crystal with size of 50 × 25 mm 2 and a radius of curvature of 800 mm was installed. The spectrometer did not need any collimation because the imaging properties of the curved crystal were used to resolve spectral lines from fast x-ray sources nearly as well as for stationary sources [22]. For this purpose it was necessary to place the Rowland-circle plane of the spectrometer perpendicular to the ion-beam direction. In such a configuration the spectral lines appear slanted in the image plane of the spectrometer with their slope proportional to the ionbeam velocity [22][23][24].

Photons diffracted off the crystal were detected with a windowless x-ray charge-coupled device (CCD), Andor DO420, placed at D = 575.75±0.65 mm away from the crystal, corresponding to a Bragg-angle setting of Θ B = 46.0 • . The spectrometer was kept under vacuum conditions (residual gas pressure ∼ 10 -5 mbar) to assure proper working conditions for the CCD camera and to keep x-ray absorption low. The interaction volume between the ion beam and the nitrogen jet was located 758.05 ± 0.46 mm away from the crystal, i.e. about 182 mm outside of the Rowland circle. The ion velocity was selected to observe the x-ray radiation under study with a photon energy of about 4308 eV in the laboratory frame, i.e. in the vicinity of the 8.6 keV Kα 1,2 lines of zinc observed in the second order of diffraction. At the same time, this matches the spectrometer configured to a Bragg angle of 46 • . The zinc lines, produced by fluorescence with a commercial x-ray tube and a zinc target, were used as calibration and for stability controls. For this purpose a zinc plate was mounted on a removable support between the target chamber and the crystal. In this configuration, the spectrometer had an efficiency of ∼ 7 × 10 -7 and a resolution of ∼ 2 eV for the first order Bragg reflection.

The data were acquired during a total period of about 4.5 days. To check for the stability of the spectrometer, daily calibrations with the zinc target were performed. During each accelerator cycle -consisting of injection into the ESR, cooling, deceleration, cooling, measurementthe CCD acquired data for 25 and 50 seconds for Li-and He-like U, respectively. Over the whole experimental run, the spectrometer was very stable, with a shift of ±3 µm of the zinc-line reflections on the CCD (corresponding to ±0.2 pixels). For the transition in He-like uranium, a total number of about 300 counts in a net-time of ∼ 24 hours was accumulated. For the Li-like ions, about 160 counts in a net-time of ∼ 5 hours were recorded. As can be observed in Fig. 3, the two intra-shell transitions could be identified unambiguously. These spectra are characterized by a very low background constrained only by the energy cuts and cluster analysis of the CCD raw data. No particular shields, other than the spectrometer stainless steel and aluminum structure, were used. We note that the shape of the lines corresponding to the fast ion emission is slightly asymmetric. Such an asymmetry is, at present, not well understood. No relevant satellite transitions, due to a possible but improbable electron double capture [25], or parasitic fluorescence lines could be found in this energy region, not even when higher reflection orders are considered. Atomic cascade effects, like feeding and decaying time of the 1s2p 3 P 2 level, which, due to the Doppler effect, could introduce an asymmetry in the x-ray energy distribution, are also unlikely to happen [15]. How far a non-uniform x-ray reflectivity over the crystal surface1 could cause irregularities in the spectral response of the spectrometer is difficult to judge.

Using the Li-like intra-shell transition as calibration, with energy E Li = 4359.37 ± 0.21 eV [20,21], the energy of the He-like 2 3 P 2 → 2 3 S 1 transition could be evaluated. This reference line has been chosen instead of the Kα 1,2 zinc lines to reduce the systematic uncertainty. The accuracy of the observation angle, equal to 0.04 • , is due to the spatial uncertainty of the fast-beam x-ray source defined by the gas-jet and ion-beam positions. This caused a systematic uncertainty of about ±0.9 eV on the fast-ion transition-energy measurement when the Zn Kα lines are used as reference. This problem can be circumvented by using a calibration line originating from the fast ion beam rather than from a stationary source.

Starting from Bragg's law in differential form, ∆E ≈ -E cot Θ B ∆Θ B , one obtains an approximate dispersion formula valid for small ∆Θ B . The energy of the He-like 2 3 P 2 → 2 3 S 1 transition E He is given by the simple formula

E He γ He ≈ E Li γ Li 1 + ∆x D tan Θ B , (1)
where ∆x = x He -x Li denotes the distance between the He-and Li-like U line images on the CCD and D the distance between crystal and CCD. γ He = 1.04677 and γ Li = 1.03503 are the Lorentz factors corresponding to the velocities of stored H-and He-like ions, respectively. Their values are determined by the accurately known voltages of the electron cooler. More complicated is the evaluation of the distance ∆x between the two spectral lines from the fast ions, characterized by a slope due to the Doppler effect [23]. Such a slope could not be determined experimentally due to the low statistics and it was calculated from simple geometrical considerations. The measured two-dimensional images have been rotated by the calculated slope before projecting them on the spectrometer dispersion plane and measure the lines relative distance. The result of such projections is presented in Fig 4.

Due to the low statistics, no accurate investigation of the peaks asymmetry could be performed. The systematic effect induced by this asymmetry has been estimated by comparing the line position measurements obtained from different approaches. For this propose, a robust statistical analysis of the data set, like the median distribution, and a series of fit adjustment have been applied. The median value was found in a self-consistent way regarding varying spectral intervals around the line center. The fit adjustments were obtained via a likelihood function maximization for Poisson histograms [26], well adapted for low-statistics data sets, using different distributions such as Gauß-Cauchy curve, Gaussian convoluted with an exponential and a sum of two Gaussian. From this analysis, a scattering of ≈ 110 µm of the ∆x value was measured, with a corresponding statistical uncertainty of about 65 µm. The value of ∆x = +47 µm, obtained via the median position calculation, has been considered as reference for further calculations. The energy of the 1s2p 3 P 2 → 1s2s 3 S 1 transition was measured to be E He = 4509.71 ± 0.48 stat ± 0.83 asym ± 0.24 syst eV, (2) The first uncertainty is statistical, and the second one is

Experiment

4509.71 ± 0.99 Theory Indelicato 2008 [28] 4510. 30 Kozhedub 2008 [29] 4509.86 ± 0.07 Artemyev 2005 [3] 4510.03 ± 0.24 Plante 1994 [2] 4510.46 Chen 1993 [30] 4510.65 Drake 1988 [31] 4510.01 due to the peak asymmetry. The third term takes other sources of systematic errors into account, of which the limited uncertainty of the calibration line (0.21 eV) is the dominant one. The uncertainty of the observation angle gives a contribution of ±0.11 eV, drastically reduced compared to ±0.9 eV when a stationary calibration source is used. Other systematic uncertainties, including those originating from the beam velocity, are negligible (see [27]).

As presented in Table 1, the measured transition energy for He-like uranium agrees well with all theoretical predictions, which, however, reflect different approaches. The value provided by P. Indelicato [28] represents a Multiconfiguration Dirac-Fock (MCDF) calculation, obtained with the code developed by P. Indelicato and J.-P. Desclaux [START_REF] Indelicato | a MultiConfiguration Dirac Fock and general matrix elements program[END_REF]. Radiative QED corrections E QED rad to the inter-electronic interaction, i.e. QED screening effects, are taken into account via the self-consistent treatment of the one-loop vacuum polarization and the Welton approximation for the screened self-energy. The value obtained by Artemyev et al. in [3] is calculated ab initio. In [3] all QED corrections of first and second order in α are taken into account within the rigorous framework of QED. In addition to [28] the non-radiative QED contribution to the interelectronic interaction E QED non-rad , which is the difference between the contribution of the inter-electronic interaction diagrams calculated within the rigorous QED approach and within the Breit approximation, is calculated. Later in [29] the value from [3] has been improved using a new value of the mean-square nuclear-charge radius for uranium [32] and the two-loop one-electron QED corrections for the excited states calculated in [33,34]. Also in [29] the non-QED contribution of three-and more photon exchange diagrams has been calculated using the relativistic configuration-interaction method. Two-electron QED effects contribute to the 3 P 2 → 3 S 1 transition energy with E QED 2 el. = E QED rad + E QED non-rad = 0.76 eV [3], which is close to the present experimental uncertainty.

In addition to the absolute energy measurement of the 2 3 P 2 → 2 3 S 1 transition, the relative measurement of Heand Li-like uranium intra-shell transitions could be evalu- ated. In contrast with the previous case, the experimental systematic uncertainty, peak asymmetry excluded, reduced from 0.24 to 0.11 eV. Theoretical uncertainties due to the finite nuclear size and the one-electron QED effects, are also drastically reduced. A comparison between our value and different predictions is presented in Table 2.

As we can observe, all theoretical values agree with the experimental value within one standard deviation. In summary, we report the first clear identification of the 1s2p 3 P 2 → 1s2s 3 S 1 transition in He-like uranium. In addition we could measure the transition energy of such transition with a relative uncertainty of 2 × 10 -4 , which is currently the most accurate test of many-body and QED contributions in excited levels of very heavy He-like ions. Differential measurements between different charge states of the same fast ion pave the way for increased sensitivity via the reduction of the systematic uncertainty in both experimental and theoretical sides. In the present experiment, the accuracy was principally limited by the statistics and the observed peak asymmetry of the fast-beam spectra. Hence, the improvement of the experimental setup and an increazing of the acquisition time will allow in future experiments for more stringent tests of the twoelectron QED contributions in heavy highly charged ions. * * *

Fig. 1 :

 1 Fig. 1: Level scheme of helium-like uranium and major decay branches. The bold arrow indicates the transition studied.

Fig. 3 :

 3 Fig.3: Reflection of the Li-like (up) and He-like (down) uranium intra-shell transitions on the Bragg spectrometer CCD. The transition energy increases with the increasing of xposition. The slightly negative slope of the line is due to the relativistic velocity of the fast ions.

Fig. 4 :

 4 Fig. 4: X-ray spectra measured with the crystal spectrometer: Zn Kα doublet in second order (top) and the intra-shell transitions of Li and He-like U (middle and bottom, respectively).

Table 1 :

 1 Comparison of the experimental transition energy, in eV, for the 3 P2 → 3 S1 transition in He-like uranium with various theoretical predictions. The experimental uncertainty is obtained from the quadratic sum of the different uncertainties listed in Eq. (2).

Table 2 :

 2 Experimental and theoretical energy difference between He-and Li-like intra-shell transitions in eV. Contributions from one-and two-photon exchange diagram, EBreit, and from two-photon QED, E QED rad and E QED non-rad , are presented. The experimental uncertainty is obtained from the quadratic sum of the different uncertainties.

	EBreit E QED rad	E QED non-rad	Total
	Experiment			50.34 ± 0.96
	Theory
	Indelicato [28] 51.61	-1.66		49.96
	Kozhedub [29] 51.48	-1.14	-0.04	50.30 ± 0.03

After the experiment, the crystal surface was surveyed by the x-ray optics group from the Institute for Optics and Quantum Electronics in Jena.

We thank V. Shabaev, A.N. Artemyev and A. Surzhykov for interesting discussions and theoretical support. We thank O. Wehrhan, H. Marschner and E. Förster for the characterization of the spectrometer crystal. The close collaboration and support by the members of the ESR team, the A. von Humboldt Foundation (M.T.), the DAAD (A.K., No.: A/05/52927) and I3 EURONS (EC contract no. 506065) are gratefully acknowledged. This work was partially supported by Helmholtz Alliance HA216/EMMI. Institut des Nanosciences de Paris and Laboratoire Kastler Brossel are Unité Mixte de Recherche du CNRS n • 7588 and n • 8552, respectively.