A multidimensional HLL‐Riemann solver for non‐linear hyperbolic systems

Abstract

SUMMARYThis article presents a numerical model that enables to solve on unstructured triangular meshes and with a high order of accuracy, Riemann problems that appear when solving hyperbolic systems.For this purpose, we use a MUSCL-like procedure in a 'cell-vertex' finite-volume framework.In the first part of this procedure, we devise a four-state bi-dimensional HLL solver (HLL-2D). This solver is based upon the Riemann problem generated at the barycenter of a triangular cell, from the surrounding cell-averages. A new three-wave model makes it possible to solve this problem, approximately. A first-order version of the bi-dimensional Riemann solver is then generated for discretizing the full compressible Euler equations.In the second part of the MUSCL procedure, we develop a polynomial reconstruction that uses all the surrounding numerical data of a given point, to give at best third-order accuracy. The resulting over determined system is solved by using a least-square methodology. To enforce monotonicity conditions into the polynomial interpolation, we use and adapt the monotonicity-preserving limiter, initially devised by Barth (AIAA Paper 90-0013, 1990).Numerical tests and comparisons with competing numerical methods enable to identify the salient features of the whole model.

Genuinely multidimensional physical-constraints-preserving finite volume schemes for the special relativistic hydrodynamics

“…For the linearized Euler equations, a genuinely multidimensional first-order finite volume scheme was constructed in [1] by computing the exact solution of the Riemann problem for a linear hyperbolic equation obtained by linearizing the Euler equation. Up to now, there have been some further developments on multidimensional Riemann solvers and corresponding numerical schemes, including the multidimensional HLL schemes for solving the Euler equations on unstructured triangular meshes [8,9], the genuinely multidimensional HLL-type scheme with convective pressure flux split Riemann solver [25], the multidimensional HLLE schemes for gas dynamics [41,2], the multidimensional MuSCL solver for magnetohydrodynamics [5], the multidimensional HLLC schemes [3,4] for hydrodynamics and magnetohydrodynamics, the well-balanced two-dimensional HLL scheme for shallow water equations [36], and the genuinely two-dimensional scheme for compressible flows in curvilinear coordinates [33] etc. For the 2D special RHDs, the existing genuinely multidimensional scheme is the finite volume local evolution Galerkin method, developed in [44].…”