Bruno Després

Application of an Ultra Weak Variational Formulation of Elliptic PDEs to the Two-Dimensional Helmholtz Problem

A new technique to solve elliptic linear PDEs, called ultra weak variational formulation (UWVF) in this paper, is introduced in [B. Despres, C. R. Acad. Sci. Paris, 318 (1994), pp. 939--944]. This paper is devoted to an evaluation of the potentialities of this technique. It is applied to a model wave problem, the two-dimensional Helmholtz problem. The new method is presented in three parts following the same style of presentation as the classical one of the finite elements method, even though they are definitely conceptually different methods. The first part is committed to the variational formulation and to the continuous problem. The second part defines the discretization process using a Galerkin procedure. The third part actually studies the efficiency of the technique from the order of convergence point of view. This is achieved using theoretical proofs and a series of numerical experiments. In particular, it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom. An application to scattering problems is presented in a fourth part.

A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension

We describe a cell-centered Godunov scheme for Lagrangian gas dynamics on general unstructured meshes in arbitrary dimension. The construction of the scheme is based upon the definition of some geometric vectors which are defined on a moving mesh. The finite volume solver is node based and compatible with the mesh displacement. We also discuss boundary conditions. Numerical results on basic 3D tests problems show the efficiency of this approach. We also consider a quasi-incompressible test problem for which our nodal solver gives very good results if compared with other Godunov solvers. We briefly discuss the compatibility with ALE and/or AMR techniques at the end of this work. We detail the coefficients of the isoparametric element in the appendix.

Using Plane Waves as Base Functions for Solving Time Harmonic Equations with the Ultra Weak Variational Formulation

This article deals with the use of the Ultra Weak Variational Formulation to solve Helmholtz equation and time harmonic Maxwell equations. The method, issued from domain decomposition techniques, lies in partitioning the domain into subdomains with the use of adapted interface conditions. Going further than in domain decomposition, we make so that the problem degenerates into an interface problem only. The new formulation is equivalent to the weak formulation. The discretization process is a Galerkin one. A possible advantage of the UWVF applied to wave equations is that we use the physical approach that consists in approximating the solution with plane waves. The formulation allows to use a very large mesh as compared to the frequency, on the contrary to the Finite Element Method when applied to time harmonic equations. Furthermore, the convergence analysis shows the method is a high order one: the order evolves as the square root of the number of degrees of freedom.

Contact Discontinuity Capturing Schemes for Linear Advection and Compressible Gas Dynamics

We present a non-diffusive and contact discontinuity capturing scheme for linear advection and compressible Euler system. In the case of advection, this scheme is equivalent to the Ultra-Bee limiter of [24], [29]. We prove for the Ultra-Bee scheme a property of exact advection for a large set of piecewise constant functions. We prove that the numerical error is uniformly bounded in time for such prepared (i.e., piecewise constant) initial data, and state a conjecture of non-diffusion at infinite time, based on some local over-compressivity of the scheme, for general initial data. We generalize the scheme to compressible gas dynamics and present some numerical results.

Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics

Abstract This work addresses some asymptotic regimes for the coupling of radiation and hydrodynamics, and is inspired by the still non-answered need of high resolution and robust schemes for the numerical solutions of these problems. Using a simple characterization of the isotropy of the scattering in the comobile reference frame, we derive various asymptotic regimes. Among them is the non-equilibrium regime. Then we prove that the method of moments is compatible with the non-equilibrium regime. We also study the Rankine–Hugoniot relations.

Asymptotic preserving and positive schemes for radiation hydrodynamics

In view of radiation hydrodynamics computations, we propose an implicit and positive numerical scheme that captures the diffusion limit of the two-moments approximate model for the radiative transfer even on coarses grids. The positivity of the scheme is equivalent to say that the scheme preserves the limited flux property. Various test cases show the accuracy and robustness of the scheme.

Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme

We present a new cell-centered Lagrangian scheme on unstructured mesh for hyperelasticity. It is based on the recently proposed Glace scheme [11] for compressible gas dynamics. We show how to use the multiplicative decomposition of the gradient of deformation and the entropy property to derive the new scheme. We also prove the compatibility of this discretization with usual calculations of mass. Our motivation is to use hyperelasticity models for the study of finite plasticity, which is an extension of hypoelasticity to finite deformations. Hyperelasticity is a natural choice for extended models in solid mechanics, because of its mathematical structure which is a system of conservation laws with full rotational invariance. We study these properties for the Lagrangian system, and detail the various Eulerian formulations. We present several test problems, in 1D and 2D planar cases, which shows the capability of the scheme to capture complex shock-waves and to simulate solid-fluid problems. In this article, we use a special equation of state [21]. Its interest is twofold: we can calculate multi-dimensional plastic phenomenon (such as split shock in 1D uniaxial cases or particular shapes in Taylor test-case), and it gives interesting multi-dimensional test cases for hyperelastic planar schemes.

Lax theorem and finite volume schemes

This work addresses a theory of convergence for finite volume methods applied to linear equations. A non-consistent model problem posed in an abstract Banach space is proved to be convergent. Then various examples show that the functional framework is non-empty. Convergence with a rate h 1/2 of all TVD schemes for linear advection in 1D is an application of the general result. Using duality techniques and assuming enough regularity of the solution, convergence of the upwind finite volume scheme for linear advection on a 2D triangular mesh is proved in L α , 2 < a < +∞: provided the solution is in W 1, ∞, it proves a rate of convergence h 1/4-e in L∞.

Lagrangian systems of conservation laws

Summary. We study the mathematical structure of 1D systems of conservation laws written in the Lagrange variable. Modifying the symmetrization proof of systems of conservation laws with three hypothesis, we prove that these models have a canonical formalism. These hypothesis are i) the entropy flux is zero, ii) Galilean invariance, iii) reversibility for smooth solutions. Then we study a family of numerical schemes for the solution of these systems. We prove that they are entropy consistent. We also prove from general considerations the symmetry of the spectrum of the Jacobian matrix.

Symmetrization of Lagrangian gas dynamic in dimension two and multidimensional solvers

Abstract We propose a new formulation of multidimensional Euler equations in Lagrange coordinates as a system of conservation laws associated with free-divergent constraints. This formulation leads to a natural class of entropic Lagrangian schemes, based on a multidimensional node solver. For the sake of simplicity the study is done in 2D, but most of the ideas can be generalized in 3D. To cite this article: B. Despres, C. Mazeran, C. R. Mecanique 331 (2003).