Haut PDF An Asymptotic-Preserving all-speed scheme for the Euler and Navier-Stokes equations

An Asymptotic-Preserving all-speed scheme for the Euler and Navier-Stokes equations

An Asymptotic-Preserving all-speed scheme for the Euler and Navier-Stokes equations

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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On the Eulerian Large Eddy Simulation of disperse phase flows: an asymptotic preserving scheme for small Stokes number flows

On the Eulerian Large Eddy Simulation of disperse phase flows: an asymptotic preserving scheme for small Stokes number flows

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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Study of an asymptotic preserving scheme for the quasi neutral Euler-Boltzmann model in the drift regime

Study of an asymptotic preserving scheme for the quasi neutral Euler-Boltzmann model in the drift regime

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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An Asymptotic Preserving scheme for the shallow-water equations with Manning friction using viscous correction of the HLL scheme

An Asymptotic Preserving scheme for the shallow-water equations with Manning friction using viscous correction of the HLL scheme

The paper is organized as follows. Section 2 is dedicated to the presentation of the considered general framework. It allows to introduce several notations useful in the next sections and to present a generic form for all the schemes presented in this paper. Afterwards, in Section 3 , the scheme given in [ 10 ] for the system under concern ( 1.1 ) is recalled. The failure of this extension of the scheme presented in [ 6 ] is then highlighted. Indeed, this scheme is able to preserve terms of order zero, namely h 0 given by ( 1.8 ), but not of order one. As a consequence, the relation ( 1.7 ) to govern q 1 is not preserved and the resulting scheme is said a priori partially asymptotic preserving. In fact, by studying the asymptotic behavior of the discharge, since the scheme adopts a nonphysical scaling in 1/ε, we exhibit an additional relation satisfied once again by h 0 . This new relation combined with ( 1.8 ) make this scheme not asymptotic preserving.
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STUDY OF A NEW ASYMPTOTIC PRESERVING SCHEME FOR THE EULER SYSTEM IN THE LOW MACH NUMBER LIMIT

STUDY OF A NEW ASYMPTOTIC PRESERVING SCHEME FOR THE EULER SYSTEM IN THE LOW MACH NUMBER LIMIT

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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Splitting up method for the 2D stochastic Navier-Stokes equations

Splitting up method for the 2D stochastic Navier-Stokes equations

NAVIER-STOKES EQUATIONS H. BESSAIH, Z. BRZE´ ZNIAK, AND A. MILLET Abstract. In this paper, we deal with the convergence of an iterative scheme for the 2-D stochastic Navier-Stokes Equations on the torus suggested by the Lie- Trotter product formulas for stochastic differential equations of parabolic type. The stochastic system is split into two problems which are simpler for numerical computations. An estimate of the approximation error is given for periodic bound- ary conditions. In particular, we prove that the strong speed of the convergence in probability is almost 1/2. This is shown by means of an L 2 (Ω, P) convergence
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Slip Boundary Conditions for the Compressible Navier–Stokes Equations

Slip Boundary Conditions for the Compressible Navier–Stokes Equations

For time-independent problems, this problem had been investigated by Y. Sone since 1960’s by systematic asymptotic analysis of the Boltzmann equation, and the complete slip-flow theory, which may be called the generalized slip-flow theory, has been established [53, 54, 59, 55, 60, 61]; the reader is referred to his two books [56, 57]. The generalized slip-flow theory provides the appropriate combinations of the fluid-dynamic-type equations, their boundary conditions of slip or jump type, and the kinetic corrections near the boundary (i.e., inside the so-called Knudsen layer) de- pending on the physical situations. It may be classified as (i) the linear theory for small Reynolds numbers [53, 54, 55], (ii) the weakly nonlinear theory for finite Reynolds numbers [54, 59, 55], (iii) the nonlinear theory for finite Reynolds numbers but large temperature and density variations [60], and (iv) the fully nonlinear theory [61]. The basic fluid-dynamic-type equations are the Stokes equations in (i), the so-called in- compressible NavierStokes type in (ii), and the ghost-effect equations in (iii). In (iv), the overall equations are of the Euler type, but its solution needs to be matched with the solution of the equations of the viscous boundary-layer type with appropriate slip boundary conditions. The two books by Sone [56, 57] give the summary of all these fluid-dynamic-type systems and recipes for applications according to the physical sit- uations under consideration. Therefore, we do not have to go back to the Boltzmann equation and can solve the problems in the framework of macroscopic gas dynamics. It should be noted that the extension of the generalized slip flow theory to time-dependent problems has been discussed in Sect. 3.7 of [57], and the extension of the linear theory has been completed recently [65, 25, 26].
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An asymptotic preserving scheme based on a new formulation for NLS in the semiclassical limit

An asymptotic preserving scheme based on a new formulation for NLS in the semiclassical limit

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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Layer-averaged Euler and Navier-Stokes equations

Layer-averaged Euler and Navier-Stokes equations

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 NavierStokes 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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An asymptotic preserving scheme on staggered grids for the
barotropic Euler system in low Mach regimes

An asymptotic preserving scheme on staggered grids for the barotropic Euler system in low Mach regimes

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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An entropy preserving MOOD scheme for the Euler equations

An entropy preserving MOOD scheme for the Euler equations

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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The incompressible Navier-Stokes equations on non-compact manifolds

The incompressible Navier-Stokes equations on non-compact manifolds

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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A Lagrangian Scheme à la Brenier for the Incompressible Euler Equations

A Lagrangian Scheme à la Brenier for the Incompressible Euler Equations

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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On shape optimization for compressible isothermal Navier-Stokes equations

On shape optimization for compressible isothermal Navier-Stokes equations

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]

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Numerical methods for the Stokes and Navier-Stokes equations driven by threshold slip boundary conditions

Numerical methods for the Stokes and Navier-Stokes equations driven by threshold slip boundary conditions

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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The Euler Scheme for Lévy driven Stochastic Differential Equations

The Euler Scheme for Lévy driven Stochastic Differential Equations

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]

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Convergence study of a DDFV scheme for the Navier-Stokes equations arising in the domain decomposition setting

Convergence study of a DDFV scheme for the Navier-Stokes equations arising in the domain decomposition setting

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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Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations

Remarks on the Cauchy problem for the axisymmetric Navier-Stokes equations

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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A Lagrangian Scheme à la Brenier for the Incompressible Euler Equations

A Lagrangian Scheme à la Brenier for the Incompressible Euler Equations

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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Local exact controllability for the 1-d compressible Navier-Stokes equations

Local exact controllability for the 1-d compressible Navier-Stokes equations

• 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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