Relaxation-projection schemes, the ultimate approximate Riemann solvers

Christian Klingenberg Dept. of Mathematics, W ̈urzburg University, Germany

Speaker	Christian Klingenberg Dept. of Mathematics, W ̈urzburg University, Germany
When	Sep 17, 2019 from 02:00 PM to 03:00 PM
Where	LH 006
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Abstract:A classical method to numerically solve hyperbolic conservation laws is the finite volume method, which involves as its essential ingredient the numerical solution of a Riemann problem. Phil Roe noticed in 1981 that an approximation of the Riemann solver suffices. This led to a quest for finding particularly useful approximate Riemann solvers.

Such an approximate Riemann solver was developed in Paris by Frederic Coquel and co-workers around the turn of the century, see e.g. [3], [1]. It is inspired by an idea of Shi Jin and Zhou-ping Xin, where the solutions to a system of conservation laws are approximated by a particularly straightforward relaxation system. The ensuing French idea was two-fold:

find a particularly clever relaxation system that approximates a given system of conservation laws

translate this into a numerical scheme by first solving the left hand side of the relaxation system (a linear transport and thus numerically easy to do) and then projecting the thus found solution to the equilibrium variables (again easy).

We shall show how this leads to approximate Riemann solvers with good properties, like stability and entropy consistency, which implies positivity of density and temperature for the shallow water (see [7]), Euler (see [4], [5], [6], [8]) and the equations of ideal magnetohydrodynamics (see [2]).