Multidimensional Riemann problem with self-similar internal structure. Part I – Application to hyperbolic conservation laws on structured meshes

Abstract

Multidimensional Riemann solvers have been formulated recently by the author (Balsara, 2010, 2012) [6,7]. They operate at the vertices of a two-dimensional mesh, taking input from all the neighboring states and yielding the resolved state and fluxes as output. The multidimensional Riemann problem produces a self-similar strongly interacting state which is the result of several one-dimensional Riemann problems interacting with each other. The prior work was restricted to the use of one-dimensional HLLC Riemann solvers as building blocks. In this paper, we formulate the problem in similarity variables. As a result, any self-similar one-dimensional Riemann solver can be used as a building block for the multidimensional Riemann solver. This paper focuses on the structure of the strongly-interacting state. (A video introduction to multidimensional Riemann solvers is available on http://www.nd.edu/~dbalsara/Numerical-PDE-Course.)In this work the strongly-interacting state is expanded in a set of basis functions that depend on the similarity variables. Consequently, the resolved state and the fluxes can be endowed with considerably richer sub-structure compared to prior work. Unlike the multidimensional HLLC Riemann solver, the need to independently specify a direction for the evolution of the contact discontinuity is eliminated. The richer sub-structure in the strongly-interacting state naturally accommodates waves that may be moving in any direction relative to the mesh, thereby minimizing mesh-imprinting. Two formulations are presented. The first formulation does not linearize the problem around a favorable state. Its derivation takes a few cues from the derivation of the multidimensional HLL Riemann solver. The second formulation identifies such a state and carries out a linearization of the fluxes about that state.This paper is the very first time that a series solution of the multidimensional Riemann problem has been presented. Explicit formulae are presented for up to quartic variation in the self-similar variables. While linear variations are sufficient for numerical work, the higher order terms in the series solution could prove useful for analytical studies of the multidimensional Riemann problem.The formulation presented here is general enough to accommodate any hyperbolic conservation law. It can also accommodate any one-dimensional Riemann solver and yields a multidimensional version of the same. It has been incorporated in the author's RIEMANN code. As examples of the different types of hyperbolic conservation laws, we use Euler flow, Magnetohydrodynamics (MHD) and relativistic MHD. As examples of different types of Riemann solvers, we show multidimensional formulations of HLL, HLLC and HLLD Riemann solvers for MHD all working fluently within this formulation. Several stringent test problems are presented.