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Accueil > Agenda > Colloques et rencontres > Archives > Agenda 2005 > Atelier du GdR CHANT sur les méthodes numériques pour les équations cinétiques, hyperboliques et de Hamilton-Jacobi

IRMA, 23-25 Novembre 2005

Organisateurs / Organizers

François Castella, IRMAR, Université de Rennes 1.

Michaël Gutnic, IRMA, Université Louis Pasteur, Strasbourg 1

Stéphanie Salmon, IRMA, Université Louis Pasteur, Strasbourg 1

Eric Sonnendrücker, IRMA, Université Louis Pasteur, Strasbourg 1

Conférenciers principaux

Rémi Abgrall (MAB, université de Bordeaux 1)

Francis Filbet (MAPMO, CNRS et université d’Orléans)

Chi-Wang Shu (Brown university, Providence, USA)

Autres conférenciers confirmés / Other confirmed speakers

Emmanuel Audusse (Paris 13)

Nicolas Besse (LPMIA Nancy)

Jean-Pierre Bourgade (MIP Toulouse)

Martin Campos Pinto (RWTH Aachen)

Nicolas Crouseilles (INRIA Lorraine)

Andreas Dedner (AAM Freiburg)

Michael Dumbser (IAG Stuttgart)

Frédéric Lagoutière (LJLL, Paris 7)

Michel Mehrenberger (INRIA Rocquencourt)

Luc Mieussens (MIP Toulouse)

Christian Rohde (Bielefeld)

Présentation / Presentation

Objectifs

L’objectif de cet atelier est à la fois de former les jeunes chercheurs aux méthodes numériques les plus performantes dans les thématiques du GdR CHANT, plus précisément pour les équations cinétiques, hyperboliques et de Hamilton-Jacobi et de permettre à des chercheurs jeunes et confirmés de présenter leurs avancées récentes dans ce domaine. L’atelier consistera en trois cours avancées de deux fois 1h30 et de 11 conférences de 45 minutes.

Objectives

The aim of this workshop is on the one hand to provide a thorough introduction to the numerical methods relevant to the topics of GdR CHANT, more precisely for kinetic, hyperbolic and Hamilton-Jacobi equations. On the other hand young and confirmed researchers will present their recent work in this area. The workshop will consist of 3 advanced lectures of twice 1 hour and 30 minutes and of 11 talks of 45 minutes each.

Résumés/ Abstracts

**Francis Filbet**

*Numerical methods for kinetic equations.*

In these lectures, I will first present different mathematical models which occur in kinetic theory : Vlasov-Maxwell system to describe transport of charged particles and Boltzmann operator to model collision mechanism.

Next I will focus on the numerical approximation of transport equations using particle methods : derivation of the method and some estimates will be also presented. Briefly, I will review Eulerian schemes as semi-Lagrangian and finite volume methods. Finally I will conclude the first lecture by presenting different open problems.

The second lecture will be devoted to stochastic particles method to discretize collision operator as the Boltzmann equation (derivation of Monte-Carlo Method). Next, I will focus on new methods recently developed for the Boltzmann eqaution : the spectral methods. These schemes represent an interesting alternative to DSMC methods to describe transient regimes. Derivation of the methods and numerical results will be presented.

**Rémy Abgrall**

*Numerical approximation of first order Hamilton-Jacobi equation.*

In this series of two lectures, I will first recall, very quickly, the notion of viscosity solution, some results about existence and uniqueness and give some examples. In particular, I will consider some exact formulas that are at the core of several numerical schemes. Then, I will show how to discretizse HJ equations on general meshes, first by using these exact solutions, then by additional approximations.

The approximation of boundary conditions will also be considered. I will end by giving one example of high order scheme.

**Chi-Wang Shu**

*Numerical methods for hyperbolic equations I : finite difference and finite volume WENO schemes.*

In this part we will describe the algorithm formulation of finite difference and finite volume schemes, and the weighted essentially non-oscillatory (WENO) reconstruction technique. A comparison of the finite difference and finite volume methods in one and several space dimensions will be made. A few topics of recent developments in WENO methods, such as well balanced WENO schemes for conservation laws with source terms, anti-diffusive corrections to WENO schemes to sharpen contact discontinuities, and residual distribution WENO methods for steady state solutions, will be briefly discussed if time permits.

*Numerical methods for hyperbolic equations II : discontinuous Galerkin methods.*

In this part we will describe the discontinuous Galerkin (DG) method when applied to hyperbolic equations. We will highlight the algorithm formulation and stability and convergence properties. Limiters for solving solutions with strong discontinuities will be discussed. A few topics of recent developments in DG methods for hyperbolic equations, such as a reinterpretation and simplified implementation of the DG method for Hamilton-Jacobi equations using locally curl-free elements, locally divergence-free DG methods for Maxwell equations and MHD equations, and DG methods based on non-polynomial approximation spaces, will be briefly discussed if time permits.

**Nicolas Crouseilles**

*Numerical approximation of collisional plasmas by high order methods.*

In this talk, we investigate the approximation of the solution to the Vlasov equation coupled with the Fokker-Planck-Landau collision operator using a phase space grid. On the one hand, the algorithm is based on the conservation of the flux of particles and the distribution function is reconstructed allowing to control spurious oscillations and preserving positivity and energy. On the other hand, the method preserves the main properties of the collision operators in order to reach the correct stationary state. Several numerical results will be presented in one dimension in space and three dimensions in velocity.

**Andreas Dedner**

*A-posteriori error estimates for the discontinuous Galerkin method applied to conservation laws.*

Starting with the work from Cockburn and Shu in the 90s, the Discontinuous Galerkin (DG) method has become very popular for solving non-linear conservation laws. Although extensively studied over the last years there are still problems with the efficiency of the scheme on unstructured grids in higher space dimensions --- especially for problems with strong shocks waves. Constructing suitable limiters is therefore still an important issue.

We propose an a—posteriori error estimate for the semi-implicit DG method for arbitrary order in arbitrary space dimensions. The estimate is derived following the ideas of Kruzkov’s a-priori error estimates. We then apply the estimate to the fully discrete Runge-Kutta DG method using it for local adaptivity of the underlying grid but more importantly also for detecting non-smooth regions and for the gradient limiting. Numerical experiments --- at the moment in 1d --- demonstrate the stability of the scheme and the gain in efficiency in comparison with computations on uniform grids.

**Emmanuel Audusse**

*A well-balanced positivity preserving scheme for shallow water flows. Extension to a multilayer model.*

We consider the 2D shallow water system with topographic source term. This hyperbolic system of balance laws is commonly used to describe various geophysical flows. Its numerical treatment raises two major questions : the first one is related to the preservation of the nonnegativity of the discrete water height ; the second one is related to the preservation of the steady states and thus to the numerical discretization of the source term. We first describe a positive homogeneous solver that is based on a kinetic formulation of the shallow water system. Then we present a new hydrostatic reconstruction procedure that allows to preserve the so-called still water steady state and the positivity properties of the original solver. Some academic and geophysical test cases illustrate the method.

In a second step we extend our approach to a new multilayer Saint-Venant system. The idea is to preserve the 2D shallow water approach but to allow a 3D velocity profile. We first derive the new system from the three dimensional free surface incompressible Navier-Stokes equations through a classical asymptotic analysis. Then we study some questions related to the conservativity and hyperbolicity of the model that we obtained. Finally we extend the numerical techniques of the first part of the talk to this new system (that can be seen as a set of coupled modified 2D shallow water systems). Some comparisons with hydrostatic and non-hydrostatic Navier-Stokes solutions are presented for academic and geophysical problems.

**Jean-Pierre Bourgade**

*Numerical study of a SHE type model for semiconductor physics.*

The goal of this talk is to give a precise account of the reliability of two diffusion models, the SHE and coupled SHE models, by accurate numerical comparisons with a Boltzmann like equation. These models describe the transport of particles subject to collisions with a surrounding medium. Comparisons are given at three levels of description : at small times (transient regime), at the diffusion time (diffusion regime) and for long times (stationary regime). The three regimes are well described by both SHE models. We discuss the main benefits and drawbacks of each SHE model.

**Luc Mieussens**

*Coupling methods for kinetic and fluid equations.*

I will present several new methods that provide a smooth transition between a kinetic and a fluid domain. The idea is to use a buffer zone, in which both fluid and kinetic equations will be solved.

In the first method, the solution of the original kinetic equation will be recovered as the sum of the solutions of these two equations. We use an artificial connecting function which makes the equation on each domain degenerate at the end of the buffer zone, thus no boundary condition is needed at the transition point. Consequently this model avoids the delicate issue of finding the interface condition in a typical domain decomposition method that couples a kinetic equation with fluid equations.

In the second method, the same approach is used, but on the non-equilibrium part of the solution only. The new important features of this method is a general property of preservation of uniform flows, and its applicability to a large class of kinetic models.

In this talk, I will give the ideas of these methods, as well as several applications to different models and various numerical tests.

**Michael Dumbser**

*Application of arbitrary high order discontinuous Galerkin schemes.
Schwartzkopff et al. successfully constructed a finite volume scheme of arbitrary high order of accuracy in space and time for linear two-dimensional hyperbolic systems using Toro’s and Titarev’s ADER approach for time discretization.*

In our presentation we will show the application of this strategy to Discontinuous Galerkin finite element schemes allowing the construction of arbitrarily accurate schemes in space and time even on unstructured grids. This should be useful for solving evolution equations in the time domain in complex geometries. Furthermore, high order schemes permit good resolution of physical phenomena even on very coarse grids. We will show numerical convergence results of our method for the linearized and the nonlinear Euler equations up to 10th order of accuracy in space and time on triangular grids.

The ADER approach also provides a unified framework in which source terms can be included quite easily. They are automatically taken into account with the full accuracy of the scheme in space and time, both inside the numerical flux and in the time integration. We will show a simple example of a steady-state solution of the two-dimensional Euler equations with a gravitational source term and compare the results of our scheme to non-balanced and well-balanced second order finite volume schemes.

To handle complex geometries with curved boundaries we use superparametric elements on curved triangles. Numerical examples are presented which show that this is necessary when computing e.g. the inviscid subsonic flow around a cylinder or the flow around a NACA 0012 profile discretized with only very few high order elements. We show via numerical evidence that if the boundary of the domain is not discretized carefully, one obtains wrong results.

Recent applications of ADER-DG schemes on three-dimensional unstructured grids concern aeroacoustics and linear elasticity, for which several results are presented.

**Nicolas Besse**

*Analysis of a semi-Lagrangian method for the spherically symmetric Vlasov-Einstein system.*

We consider the spherically symmetric Vlasov-Einstein system in the case of asymptotically flat spacetime. From the physical point of view this system of equations can modelize the formation of a spherical black hole by the gravitational collapse of an isolated star. We present high-order numerical schemes based on semi-Lagrangian technique and multiresolution analysis in $H^s$. The convergence of the solution of the discretized problem to the exact solution is proven and high-order error estimates are supplied. More precisely the metric coefficients converge in $L^∞$ and the statistical distribution function of the matter and its moments converge in $L²$ with a rate of $O(Δt^2 + h^m/Δt)$, when the exact solution belongs to $H^m$.

**Christian Rohde**

*Numerics for conservation laws modelling phase transitions.*

Many phase transition phenomena can be modelled by systems of conservation laws that typically are not purely hyperbolic anymore but of mixed elliptic-hyperbolic type.

As a consequence almost all standard numerical methods fail, not to mention the even more fundamental analytical problems. These difficulties will be discussed and we propose two approaches that might help to overcome them. One approach relies on a sharp interface theory, i.e., the base is the (approximative) solution of the Riemann problem with a special treatment of the phase boundary. The other starts from a regularized model (phase field equ-ations). In this case interfaces have to be fully resolved which requires special numerical treatment.

As applications we present computations for liquid-vapour phase transitions and the dynamics of different phases in solids.

**Frédéric Lagoutière**

*Non-dissipative reconstruction schemes for transport equations.*

We develop and analyze non-dissipative reconstruction schemes for conservation laws. The aim is to ensure the convergence of the numerical solutions toward the entropy solutions without introducing numerical dissipation, especially at discontinuities. We first focus on deriving new stability and convergence conditions on reconstruction algorithms. This then allows to consider a wider set of algorithms, among which discontinuous reconstruction schemes. They are first applied to scalar conservation laws in dimension 1. A link with classical algorithms is established. We then extend the ideas to transport equations in dimension 2 and more. Some numerical results will be reported.

**Martin Campos Pinto**

*A fully adaptive and accurate semi-Lagrangian scheme for the Vlasov-Poisson equation.*

In this talk, we shall propose a new adaptive scheme for solving the one-dimensional Vlasov-Poisson equation and give an a priori error estimate. In order to save computational costs while approximating the complex and thin structures that may appear in the solutions, several adaptive schemes have been proposed in the past few years, inspired from the semi-Lagrangian method of Cheng and Knorr, later revisited by Sonnendrücker, Roche, Bertrand and Ghizzo. Yet no one has been proven to converge.

A key feature of our new scheme relies on the simple structure of the graded dyadic quadrangulations which are used to build hierarchical finite element approximations of the phase space distribution functions $f(t_n,·,·)$. Based on the regularity analysis of the numerical solution and how it gets transported by the numerical flow, the method performs an accurate evolution of the adaptive mesh from one time step to the next one, in the sense that the accuracy of the scheme is monitored by a prescribed tolerance parameter ε which represents the local interpolation error at each time step. As a consequence, the numerical solutions can be proved to converge in $L^∞$ towards the exact ones as ε and Δt tend to zero, provided the initial data is in $W^*1,∞* ∩ W^*2,1*$. The rate of convergence is in $O(Δt^2 + ε/Δt)$, and several numerical tests illustrate the effectiveness of our approach for generating the optimal adaptive meshes.

**Michel Mehrenberger**

*High order semi-Lagrangian schemes.*

We present some results concerning high order semi-Lagrangian schemes in the 1Dx1D phase space. In the uniform case, we consider the stability of several reconstructions : B-splines, Lagrange interpolation and interpolets. In the adaptive case, we give first numerical results concerning a Hermite reconstruction.

Dernière mise à jour le 17-10-2006