Large time step HLL and HLLC schemes
Marin Prebeg, Tore Flåtten and Bernhard Müller
Abstract: We present Large Time Step (LTS) extensions of the Harten–Lax–van Leer (HLL) and Harten–Lax–van Leer Contact (HLLC) schemes. Herein, LTS denotes a class of explicit methods stable for Courant numbers greater than one. The original LTS method [★] was constructed as an extension of the Godunov scheme, and successive versions have been developed in the framework of Roe's approximate Riemann solver.
We first formulate the LTS extension of the original HLL scheme in conservation form. Next, we provide explicit expressions for the flux-difference splitting coefficients and the numerical viscosity coefficients. We then formulate the LTS extension of the HLLC scheme in conservation form.
We apply the new schemes to the one dimensional Euler equations and compare them to their non-LTS counterparts. As test cases, we consider the classical Sod shock tube problem and the Woodward–Colella blast-wave problem. It is shown that the LTS–HLL scheme smears out the contact discontinuity, while the LTS–HLLC scheme improves the resolution of both shocks and contact discontinuities. In addition, we numerically demonstrate that for the right choice of wave velocity estimates both schemes calculate entropy satisfying solutions.