Jean-Paul Vila

Convergence of an explicit finite volume scheme for first order symmetric systems

Summary. This paper is devoted to the derivation of a O(h1/2) error estimate for the classical upwind, explicit in time, finite volume scheme for linear first order symmetric systems. Such a result already existed for the corresponding implicit in time finite volume scheme, since it can be interpreted as a particular case of the space-time discontinuous Galerkin method but the technique of proof, used in that case, does not extend to explicit schemes. The general framework, recently developed to analyse the convergence rate of finite volume schemes for non linear scalar conservation laws, can not be used either, because it is not adapted for systems, even linear. In this article, we propose a new technique, which takes advantage of the linearity of the problem. The first step consists in controlling the approximation error ∥u−uh∥L2 by an expression of the form −2, where u is the exact solution, g is a particular smooth function, and μh, νh are some linear forms depending on the approximate solution uh. The second step consists in carefully estimating the error terms and , by using uniform stability results for the discrete problem and regularity properties of the continuous solution.

Convergence of SPH Method for Scalar Nonlinear Conservation Laws

This paper is devoted to the study of the convergence of weighted particle approximation of nonlinear multidimensional conservation laws. For Euler equations the method is closely related to the smooth particle hydrodynamics (SPH) method. Extension of the original algorithm is proposed. We use approximate Riemann solvers instead of artificial viscosity in order to stabilize the scheme. The mathematical analysis is performed by connecting this new approach with the finite volume scheme. Convergence of the approximate solution in

Renormalized Meshfree Schemes I: Consistency, Stability, and Hybrid Methods for Conservation Laws

L_{loc}^{p}

A Compressible Model for Separated Two-Phase Flows Computations

(

Shape Optimal Design Problem with Convective and Radiative Heat Transfer: Analysis and Implementation

p<\infty

STABILITY THEORY FOR DIFFERENCE APPROXIMATIONS OF EULER-KORTEWEG EQUATIONS AND APPLICATION TO THIN FILM FLOWS ∗

) towards the unique weak entropy solution of the Cauchy problem is obtained in the scalar nonlinear case by using uniqueness of measure valued solutions.

An accurate low-Mach scheme for a compressible two-fluid model applied to free-surface flows

This paper is devoted to the study of a new kind of meshfree scheme based on a new class of meshfree derivatives: the renormalized meshfree derivatives, which improve the consistency of the original weighted particle methods. The weak renormalized meshfree scheme, built from the weak formulation of general conservation laws, turns out to be

Renormalized Meshfree Schemes II: Convergence for Scalar Conservation Laws

L^2

CONVECTIVE AND RADIATIVE THERMAL TRANSFER WITH MULTIPLE REFLECTIONS. ANALYSIS AND APPROXIMATION BY A FINITE ELEMENT METHOD

stable under some geometrical conditions on the distribution of particles and some regularity conditions of the transport field. A time discretization is then performed by analogy with finite volume methods, and the

Convergence of Meshless Methods for Conservation Laws Applications to Euler equations

L^1