Finite volume evolution Galerkin methods for Euler equations of gas dynamics

Abstract

The aim of this paper is a derivation of a new multidimensional high-resolution ÿnite volume evolution Galerkin method for system of the Euler equations of gas dynamics. Instead of solving one-dimensional Riemann problems in directions normal to cell interfaces the ÿnite volume evolution Galerkin schemes are based on a genuinely multidimensional approach. The approximate solution at cell interfaces is computed by means of an approximate evolution operator taking all of the inÿnitely many bicharacteristics explicitly into account. Integrals along the Mach cones are evaluated exactly or by means of numerical quadratures. Second-order resolution is obtained with a conservative piecewise bilinear recovery and the second-order midpoint rule for the time integration. A numerical experiment which illustrates the good multidimensional approximation as well as higher-order resolution is presented.

On the Stability of Evolution Galerkin Schemes Applied to a Two‐Dimensional Wave Equation System

“…For more details on higher order finite volume evolution Galerkin (FVEG) schemes, see [10], [15], [17], where the error analysis as well as numerical experiments are presented. Using the L 2 -projection (4.4), the approximate evolution operator E τ , and (4.7), (4.8) the EG and the FVEG schemes can be written in the finite difference form…”

Finite Volume Evolution Galerkin Methods for Nonlinear Hyperbolic Systems

“…Higher order finite volume evolution Galerkin (FVEG) schemes were studied for linear wave equation systems in [18,19]. The aim of this contribution is to derive and analyze new multidimensional high-resolution FVEG methods for systems of nonlinear hyperbolic conservation laws (see also [21] for a related paper).…”

The finite volume local evolution Galerkin method for solving the hyperbolic conservation laws

“…In this section, we recall the work of Lukáčová-Medvid'ová et al [8,9,11,12] to present the basic idea of the FVEG schemes and to introduce the notations that will be used in deriving the FVLEG scheme. For this purpose, it is convenient to start with the Euler equations in primitive variable form,…”

“…In fact, it turned out that in certain cases, e.g. when there are strong shocks or waves are propagating in directions that are oblique with respect to the mesh, this approach leads to structural deficiencies and large errors in the solutions [8,16].…”