Abstract
This paper is focused on the residual distribution (RD) interpretation of the Dobrev, Kolev, and Rieben scheme [SIAM J. Sci. Comput., 34 (2012), pp. B606--B641] for the numerical solution of the Euler equations in Lagrangian form. The first ingredient of the original scheme is the staggered grid formulation which uses continuous node-based finite element approximations for the kinematic variables and cell-centered discontinuous finite elements for the thermodynamic parameters. The second ingredient of the Dobrev et al. scheme is an artificial viscosity technique applied in order to make possible the computation of strong discontinuities. The aim of this paper is to provide an efficient mass matrix diagonalization method in order to avoid the inversion of the global sparse mass matrix while keeping all the accuracy properties and to construct a parameter-free stabilization of the scheme to get rid of the artificial viscosity. In addition, we study the conservation and entropy properties of the constructed RD scheme. To demonstrate the robustness of the proposed RD scheme, we solve several one-dimensional shock tube problems from rather mild to very strong ones. This paper also illustrates a general technique that enables, from a nonconservative formulation of a system that has a conservative formulation, design of a numerical approximation that will provably give sequences of solution converging to a weak solution of the problem. This enables us to use directly variables that are more pertinent, from an engineering point of view, than the standard conserved variables: here the specific internal energy.
Abgrall, Rémi