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

Abstract

Abstract.In this paper, we present some interesting connections between a number of Riemannsolver free approaches to the numerical solution of multi-dimensional systems of conservation laws. As a main part, we present a new and elementary derivation of Fey's Method of Transport (MoT) (respectively the second author's ICE version of the scheme) and the state decompositions which form the basis of it. The only tools that we use are quadrature rules applied to the moment integral used in the gas kinetic derivation of the Euler equations from the Boltzmann equation, to the integration in time along characteristics and to space integrals occurring in the finite volume formulation. Thus, we establish a connection between the MoT approach and the kinetic approach. Furthermore, Ostkamp's equivalence result between her evolution Galerkin scheme and the method of transport is lifted up from the level of discretizations to the level of exact evolution operators, introducing a new connection between the MoT and the evolution Galerkin approach. At the same time, we clarify some important differences between these two approaches.Mathematics Subject Classification. 35C15, 35L65, 65D32, 65M25, 76M12, 76N15, 76P05.

Well-balanced finite volume evolution Galerkin methods for the shallow water equations

“…Extensive numerical experiments confirm robustness, good multidimensional behaviour, high accuracy, stability, and efficiency of the FVEG schemes, see Section 5 and also references [26,27]. We refer the reader to [1,3,7,8,9,16,19,30] for other recent multidimensional schemes.…”

Finite Volume Evolution Galerkin Methods for Hyperbolic Systems

“…where b k is the x k component of the th characteristic velocity; the so-called ray velocity; see [3] or [5]. We integrate the th equation of the system (2.3) from the point P down to the point Q (n), where the bicharacteristic hits the plane through P .…”

An evolution Galerkin scheme for the shallow water magnetohydrodynamic equations in two space dimensions

“…On the other hand, Krö ger et al [15,14] developed a framework of the so-called state decompositions, in which they examined the connection between the EG approach and the Method of Transport more deeply and managed to clarify an important difference between these approaches. At the same time, this framework offers the possibility to consider the EG approach from a different point of view.…”

“…Next, in Section 3, we state the integral equations in an abstract, but compact, form which is also used in the works of Krö ger et al [15,14]. A brief description of the FVEG schemes follows in Section 4.…”