Modified Semi-Lagrangian Godunov-Type Method Without Numerical Viscosity for Shocks

Numerous high-resolution Euler-type methods have been proposed to resolve smooth flow scales accurately and to simultaneously capture the discontinuities. A disadvantage of these methods is the numerical viscosity of the shocks. In the shock, the flow parameters change abruptly at a distance equal to the mean free path of a gas molecule, which is much smaller than the cell size of the computational mesh. Due to the numerical viscosity, Euler-type methods stretch the parameter change in the shock over a few mesh cells. This paper describes a modification of the semi-Lagrangian Godunov-type method without numerical viscosity for shocks, which was proposed by the author in the previously published paper. In the previous article, a linear law for the distribution of flow parameters was employed for a rarefaction wave when modeling the Shu-Osher problem with the aim of reducing parasitic oscillations. Additionally, the nonlinear law derived from the Riemann invariants was used for the remaining test problems. This article proposes an advanced method, namely, a unified formula for the density distribution of rarefaction waves and modification of the scheme for modeling moderately strong shock waves. The obtained results of numerical analysis including the standard shock-tube problem of Sod, the Riemann problem of Lax, the Shu–Osher shock-tube problem and a few author’s test cases are compared with the exact solution, the data of the previous method and the Total Variation Deminishing (TVD) scheme results. This article delineates the further advancement of the numerical scheme of the proposed method, specifically presenting a unified mathematical formulation for an expanded set of test problems.