Evolution Galerkin methods for hyperbolic systems in two space dimensions

Abstract

Abstract. The subject of the paper is the analysis of three new evolution Galerkin schemes for a system of hyperbolic equations, and particularly for the wave equation system. The aim is to construct methods which take into account all of the infinitely many directions of propagation of bicharacteristics. The main idea of the evolution Galerkin methods is the following: the initial function is evolved using the characteristic cone and then projected onto a finite element space. A numerical comparison is given of the new methods with already existing methods, both those based on the use of bicharacteristics as well as commonly used finite difference and finite volume methods. We discuss the stability properties of the schemes and derive error estimates.

On the connection between some Riemann-solver free approaches to the approximation of multi-dimensional systems of hyperbolic conservation laws

“…In this section, we will show that the integral representation derived in Lemma 4.3 is closely related to the one developed by Ostkamp [24,25] and later used by Lukáčová, Morton and Warnecke [20] for the derivation of their EG 3 scheme. Following these authors, we will call their integral representation the evolution Galerkin, or EG operator.…”

“…• the evolution Galerkin (EG) approach of Butler [3], Morton et al [19] (exploiting the transport collapse operator of Brenier [2]), Ostkamp [24,25], Lukáčová, Morton, Warnecke [20] as well as (based on this) the finite volume evolution Galerkin (FVEG) approach of Lukáčová, Morton, Saibertová and Warnecke [21,22]; and • the kinetic approach of Deshpande [6] and Perthame [26,27].…”

“…The already mentioned EG and FVEG schemes by Ostkamp [24,25] and Lukáčová, Morton, Saibertová, Warnecke [20][21][22] are also derived from an exact integral representation using quadrature rules. This integral representation (which we will call the EG operator) is based on the classical characteristic theory, see for example Courant and Hilbert [5].…”

“…We are now able to lift this connection up from the level of schemes to the level of exact integral representations. More precisely, we can find a canonical continuous state decomposition (derived from the classical characteristic theory) for which our exact integral representation becomes identical to the EG integral representation, called evolution operator in [20] for linear, constant coefficient systems.…”

Constraint preserving schemes using potential-based fluxes. III. Genuinely multi-dimensional schemes for MHD equations

“…Considerable effort has been devoted for devising numerical methods which address the multi-dimensional character of nonlinear system such as (1.1). These methods include dimensional splitting [25], wave propagation algorithms [24,25], method of transport [15,16,34], bi-characteristics based evolution Galerkin methods [27,28] and fluctuation splitting schemes [12].…”