An HLLC-Type Approximate Riemann Solver for Ideal Magnetohydrodynamics

Abstract

This paper presents a solver based on the HLLC (Harten-Lax-van Leer contact wave) approximate nonlinear Riemann solver for gas dynamics for the ideal magnetohydrodynamics (MHD) equations written in conservation form. It is shown how this solver also can be considered a modification of Linde's "adequate" solver. This approximation method is intended to resolve slow, Alfvén, and contact waves better than the original HLL (Harten-Lax-van Leer) solver. Compared to exact nonlinear solvers and Roe's solver, this new solver is computationally inexpensive. In addition, the method will exactly resolve isolated contacts and fast shocks. The method also preserves positive density and pressure with two caveats: first, the numerical signal velocities (the eigenvalues of the Roe average matrix) do not underestimate the physical signal velocities, and second, in a very few cases it may be required to change the wavespeeds of the Riemann fan for the underlying HLL method to guarantee positive pressures. These conditions are less restrictive on the definitions of the wavespeeds than the conditions needed to make the HLLC method positively conservative for gas dynamics. While the method is intended for a three-dimensional MHD problem, the simulation results concentrate on one-dimensional test cases.

A new MHD code with adaptive mesh refinement and parallelization for astrophysics

“…The simplest f m flux is Lax-Friedrichs flux but our MAP code also implemented approximate Riemann solvers like HLLC [27], HLLD [28] and Roe solver [29,30]. Since the formulae of these Riemann solvers are too complicated to be given here, thus we only list the Lax-Friedrichs flux as…”

“…(18), we can also use the same Lax-Friedrichs flux Eq. (27) or Riemann solvers like HLLC, HLLD and Roe solvers in…”

E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

“…For example, Roe's Riemann solvers are developed by Brio and Wu [1], Dai and Woodward [6], Zachary and Collelaz [7], Roe and Balsara [2] and Cargo and Gallice [8]. HLL (Harten-Lax-van Leer)-type schemes are developed by Janhunen [9], Honkkila and Janhunen [10], Gurski [11], Li [12], Miyoshi and Kusano [13] and Balsara et al [14]. Flux vector splitting methods are developed by MacCormack [15] and Jiang and Wu [4].…”

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

“…Other popular numerical fluxes used for the solution of MHD equations include linearized Roe solvers [9,36,37] and HLL type solvers [7,17,21,26,32]. Detailed comparisons of different solvers are performed in [18,31].…”