incompressible regime.
6 Conclusion
**The** aim of this paper was to provide **an** **all**-**speed** **scheme** **for** **the** numerical simulation of mixed compressible **and** incompressible fluid flows. **The** second-order discretization of **the** proposed **Asymptotic** **Preserving** **scheme** shows a very good behavior in both flow regimes. In compressible situations, we obtain good shocks properties as **the** **scheme** is conservative. In **the** low Mach number regime, **the** **Asymptotic** **Preserving** property provides a consistent discretization of **the** incompressible model, **the** divergence-free condition on **the** velocity is respected **and** **the** pressure is solved via **an** elliptic equation. **The** centered spatial discretization of **the** implicit pressure term allows **the** time-step to be based on **the** fluid velocity **and** not on **the** acoustic velocity. **The** time-step can be much larger than with **an** explicit upwind method **and** does not depend on **the** Mach number. **The** proposed **scheme** therefore shows a very good behavior on **the** weakly compressible numerical test-cases such as **the** backward-facing step **and** **the** lid-driven cavity as it provides **the** expected recirculations of **the** fluid, **and** also provides **the** correct solution on **the** heat-driven cavity which uses **the** energy equation.

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3.1. Lagrange-Projection **and** source terms decomposition. **The** objec-
tive of **the** Lagrange-projection is to decompose **the** full system into two sub-systems using a chain rule argument on **the** space derivatives. **The** first one only involves **the** transport wave, **and** **the** second one involves **the** acoustic waves (**and** **the** source terms). This kind of approach can be seen as **an** operator splitting strategy, **and** **the** main interest is to enable **the** use of different solvers **for** each subsystem. In this work, we shall consider explicit-explicit solvers but also implicit-explicit solvers in order to get rid of **the** strong CFL stability restriction imposed by **the** sound **speed**. By implicit-explicit, we mean here implicit on **the** acoustic waves **and** source terms **and** explicit on **the** transport part following **the** same approach as in [18] (see [22]). Note that **the** source term associated with **the** internal energy in **the** last equation of (2.27) being not considered in [18], we shall treat it separetely here using again a splitting strategy. Therefore, we shall end with three sub-systems to be treated numerically. Using **the** property ∂ x ρuX = u∂ x ρX + ρX∂ x u **for** X = {1, u, E} in **the** full system (2.27), we get **the** following transport system:

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Figure 3. L 1 -error (in logarithmic scale) between **the** exact solution **and** **the** numerical solution as a function of ∆x at **the** time T = 0.01 **for** ε ∈ {1, 10 −2 , 10 −4 }. **The** CFL number is 0.1.
7. Conclusion
We considered **the** numerical discretization of **the** quasi-neutral **Euler**–Boltzmann **equations**. We proposed a non linear implicit **scheme** based on staggered grids that was proven to be unconditionally stable **and** to provide some uniform bounds with respect to ε. Because of its non linearity, we proposed **an** iterative linear implicit **scheme** to solve it. **The** iterative **scheme** was proven to preserve **the** positivity **and** to be L 2 linearly stable under a CFL condition that does not involve ε. We test **the** ability of **the** **scheme** to compute **the** correct shock **speed** through a Riemann problem **and** we showed that **the** **scheme** is stable when **the** CFL number is larger than √

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Keywords: Low Mach number limit, **Asymptotic** **preserving** schemes, **Euler** sys- tem, stability analysis.
1. Introduction. Almost **all** fluids can be said to be compressible. However, there are many situations in which **the** changes in density are so small to be considered negligible. We refer to these situations saying that **the** fluid is in **an** incompressible regime. From **the** mathematical point of view, **the** difference between compressible **and** incompressible situations is that, in **the** second case, **the** equation **for** **the** conservation of mass is replaced by **the** constraint that **the** divergence of **the** velocity should be zero. This is due to **the** fact that when **the** Mach number tends to zero, **the** pressure waves can be considered to travel at infinite **speed**. From **the** theoretical point of view, researchers try to fill **the** gap between those two different descriptions by determining in which sense compressible **equations** tend to incompressible ones [2, 20, 21, 22, 33]. In this article we are interested in **the** numerical solution of **the** **Euler** system when used to describe fluid flows where **the** Mach number strongly varies. This causes **the** gas to pass from compressible to almost incompressible situations **and** consequently it causes most of **the** numerical methods build **for** solving compressible **Euler** **equations** to fail. In fact, when **the** Mach number tends to zero, it is well known that classical Godunov type schemes do not work anymore. Indeed, they lose consistency in **the** incompressible limit. This means that when close to **the** limit, **the** accuracy of theses schemes is not sufficient to describe **the** flow. Many efforts have been done in **the** recent past in order to correct this main drawback of Godunov schemes, **for** instance by using preconditioning methods [34] or by splitting **and** correcting **the** pressure on **the** collocated meshes [5], [9, 10], [12], [13, 14, 30], [15], [23, 24], [26, 27, 28], or instead by using staggered grids like in **the** famous MAC **scheme**, see **for** instance [3], [16], [17], [18], [19], [31]. Unfortunately, even if these approaches permit to bypass **the** consistency problem of Godunov methods, they **all** need to resolve **the** scale of

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with WKB type initial data
u ε (0, x) = a 0 (x)e iφ 0 (x)/ε , (1.2)
**the** functions a 0 **and** φ 0 being real-valued. **The** aim of this article is to construct **an** **Asymptotic** **Preserving** (AP) numerical **scheme** **for** this equation in **the** semiclassical limit ε → 0. We seek a **scheme** which provides **an** approximation of **the** solution u ε with **an** accuracy, **for** fixed numerical parameters ∆t, ∆x, that is not degraded as **the** scaled Planck constant ε goes to zero. In other terms, such a **scheme** is consistent with (1.1) **for** **all** fixed ε > 0, **and** when ε → 0 converges to a consistent approximation of **the** limit equation, which is **the** compressible, isentropic **Euler** equation (1.4). **The** difficulty here is that, when ε is small, **the** solution u ε becomes highly oscillatory with respect to **the** time **and** space variables, **and** converges to its limit only in a weak sense. In order to follow these oscillations without reducing ∆t **and** ∆x at **the** size of ε, which may be computationally demanding, our construction relies on a fluid reformulation of (1.1), well adapted to **the** semiclassical limit.

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In order to describe **and** simulate complex flows where **the** velocity field cannot be approximated by its vertical mean, multilayer models have been developed [1, 3, 4, 8, 13, 12]. Unfortunately these models are physically relevant **for** non miscible fluids.
In [16, 6, 5, 26], some authors have proposed a simpler **and** more general formulation **for** multilayer model with mass exchanges between **the** layers. **The** obtained model has **the** form of a conservation law with source terms, its hyperbolicity remains **an** open question. Notice that in [5] **the** hydrostatic **Navier**-**Stokes** **equations** with variable density is tackled **and** in [26] **the** approximation of **the** non-hydrostatic terms in **the** multilayer context is studied. With respect to commonly used **Navier**–**Stokes** solvers, **the** appealing features of **the** proposed multilayer approach are **the** easy handling of **the** free surface, which does not require moving meshes (e.g. [14]), **and** **the** possibility to take advantage of robust **and** accurate numerical techniques developed in extensive amount **for** classical one-layer Saint Venant **equations**. Recently, **the** multilayer model developed in [16] has been adapted in [15] in **the** case of **the** µ(I)-rheology through **an** **asymptotic** analysis.

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1.4 A new numerical strategy
In this work, we wish to develop **an** AP procedure by following **the** ideas introduced by J. Haack, S. Jin **and** J.-G. Liu in [20]. **The** stiff **Euler** system is split into two parts: a nonlinear **and** hyper- bolic system of conservation laws that involves only waves propagating with O(1) speeds, **and** a linear system that contains **the** fast acoustic dynamics. **The** former **equations** are treated explic- itly, **the** latter are treated implicitly. Next, in this framework, we propose a space discretization that relies on strategies introduced in [3] **and** that presents **the** following originalities:

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p L = 1.5, p R = 0.1564,
(34)
so that **the** exact solution is made of a continuous 1-rarefaction.
In Tables 1 **and** 2, we give respectively **the** evaluation of E ∆ **and** I ∆ obtained by considering a time first-order **scheme** with several limiter functions. First of **all**, we note that van Leer, MC **and** Superbee are not stable enough **and** a numerical blowup appears with very fine mesh. Concerning minmod **and** van Albada 1 limiter functions, **the** behavior is better because both schemes seem to converge since E ∆ goes to zero as ∆x tends to zero. At this level, we may suspect that **the** blowups are consequences of some compression phenomena, while **the** minmod limiter **and** **the** van Albada 1 limiter seem diffusive enough to avoid such a failure. According to **the** work by Hou **and** LeFloch [32], since **the** converged solution is continuous, **the** entropy dissipation measure I ∆ goes to zero **and** thus **the** measure δ is equal to zero. Figure 1 illustrates **the** results stated in Tables 1 **and** 2.

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grad p is **the** Riemannian gradient of **the** pressure. Note that since u is divergence free, we have also **the** following identity
(13) ∇ u u = div(u ⊗ u).
To define **the** vectorial Laplacian L, we have to make a choice since there is no canonical definition of a Laplacian on vector fields on Riemannian manifolds: there are at least two candidates **for** **the** role of Laplace operator, i.e. **the** Bochner **and** Hodge Laplacians. Following [17], [54] (see also [50], [43]), **the** correct formulation is obtained by introducing **the** stress tensor. Let us recall that on R n , if div u = 0, we have

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INCOMPRESSIBLE **EULER** **EQUATIONS**
THOMAS O. GALLOU ¨ ET **AND** QUENTIN M ´ ERIGOT
Abstract. We approximate **the** regular solutions of **the** incompressible **Euler** **equations** by **the** solution of ODEs on finite-dimensional spaces. Our approach combines Arnold’s interpretation of **the** solution of **the** **Euler** **equations** **for** incompressible **and** inviscid flu- ids as geodesics in **the** space of measure-**preserving** diffeomorphisms, **and** **an** extrinsic approximation of **the** **equations** of geodesics due to Brenier. Using recently developed semi-discrete optimal transport solvers, this approach yields a numerical **scheme** which is able to handle problems of realistic size in 2D. Our purpose in this article is to estab- lish **the** convergence of this **scheme** towards regular solutions of **the** incompressible **Euler** **equations**, **and** to provide numerical experiments on a few simple test cases in 2D.

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Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex France Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vi[r]

It should be noted that one of **the** first breakthrough **for** a systematic mathematical analysis of problems formulated as variational inequalities was due to G. Duvaut **and** J.L. Lions [8].
**The** main goal of our work is to discuss **the** numerical solution of some fluid flows driven by nonlinear slip boundary conditions of friction type. These include **the** **Stokes**, **Navier** **Stokes** in two **and** three dimensions. **The** point of departure of this study is **the** work of J.K. Djoko **and** M. Mbehou [7], where **the** resulting formulation had been solved by making use of **the** “Lagrange multiplier” **and** application of Uzawa’s algorithm. In this work, we solve **the** problem associated with **the** **Stokes** **equations** by exploiting **the** minimization structure of **the** variational formulation, **and** apply to it **an** alternating direction method reminiscent to those used in [6, 22, 20, 18, 38, 37]. Next, we solve **the** stationary **Navier** **Stokes** **equations** in two steps. Firstly, we associate with **the** stationary problem a time dependent problem in which only **the** long time behavior is considered. **The** time dependent problem is solved using **an** operator splitting **scheme**.

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Unite´ de recherche INRIA Lorraine, Technopoˆle de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LE` S NANCY Unite´ de recherche INRIA Rennes, Ir[r]

4 Numerical results
In this Section, **the** **scheme** ( f P) is validated by some numerical experiments. **The** computational domain is Ω = [−1,1]×[0,1] **and** **the** interface Γ is placed at x = 0. **For** **the** tests, we give **the** expression of **the** exact solution (u,p), from which we de- duce **the** source term f. We compare **the** L 2 -norm of **the** error (difference between a centered projection of **the** exact solution **and** **the** approximated solution obtained with DDFV **scheme**) **for** **the** velocity (denoted Ervel), **the** velocity gradient (Ergrad- vel) **and** **the** pressure (Erpre). **The** error estimates are discussed by working with a family of meshes (see Fig. 2), obtained by refining successively **and** uniformly **the** original mesh. **The** sub-index in **the** name of **the** mesh denotes **the** level of refine- ment, i.e. Mesh k

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which is bounded by Cξ −1 in case a) **and** by Cξ −3 in case b). This concludes **the** proof of (27). Now, since **the** integral kernel G = (G r , G z ) in (22) satisfies **the** same bound as **the** two-
dimensional Biot-Savart kernel in R 2 , properties i) **and** ii) in Proposition 2.3 can be established
exactly as in **the** 2D case. Estimate (25) thus follows from **the** Hardy-Littlewood-Sobolev in- equality, **and** **the** bound (26) can be proved by splitting **the** integration domain **and** applying H¨older’s inequality, see e.g. [8, Lemma 2.1]. **The** proof of Proposition 2.3 shows that **the** axisymmetric Biot-Savart law (22) has **the** same properties as **the** usual Biot-Savart law in **the** whole plane R 2 . In fact, it is possible to obtain in **the** axisymmetric situation weighted inequalities (involving powers of **the** distance r to **the** vertical axis) which have no analogue in **the** 2D case. As **an** example, we state here **an** interesting extension of estimate (25).

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distance can be computed efficiently thanks to recently developed numerical solvers **for** op- timal transport problems between probability densities **and** finitely-supported probability measures [3, 20, 13, 18]. This alternative discretization has already been used successfully to compute minimizing geodesics between measure-**preserving** maps in [21], allowing **the** recovery of non-deterministic solutions to Euler’s **equations** predicted by Shnirelman **and** Brenier in dimension two. **The** object of this article is to study whether this strategy can be used to construct Lagrangian schemes **for** **the** more classical Cauchy problem **for** **the** Euler’s **equations** ( 1.1 ), able to cope with problems of realistic size in dimension two. Discretization in space: approximate geodesics. **The** construction of approximate geodesics presented here is strongly inspired by a particle **scheme** introduced by Brenier [8]. We first approximate **the** Hilbert space M = L 2 (Ω, R d ) by finite dimensional subspaces.

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• Getting a local exact controllability result around smooth target trajectory should be possible under a suitable geometric condition. A possible way to do that is to include **the** target flow in **the** Carleman weight itself. This issue is currently under investigation.
• Our approach is based on **the** linearized compressible **Navier**-**Stokes** equation **and** consider **the** non- linear effects as a perturbation. Of course, another way to proceed would be to think **all** **the** way around by using **the** non-linear effects to control **the** fluid. This is **the** idea beyond **the** return method of J.-M. Coron [3, 4], which has already been used several times in **the** control theory of fluid flows. Whether or not these ideas can be applied in **the** context of compressible **Navier** **Stokes** equation is **an** interesting open problem.

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