Mathematical modeling of the absorption of weaker shock wave by relatively stronger one using the semi-Lagrangian Godunov-type method without numerical viscosity for shocks

This paper describes a modification of the semi-Lagrangian Godunov-type method without numerical viscosity for shocks, which was proposed by the author in a previously published paper, and its application to the problems of absorption of a weaker shock wave by stronger one. Numerous high-resolution Euler-type methods have been proposed to resolve smooth flow scales accurately and to capture the discontinuities simultaneously. One of the disadvantages of these methods is the numerical viscosity for 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 few mesh cells. In previous works, the author proposed a semi-Lagrangian Godunov-type method without numerical viscosity for shocks. In this method, the one-dimensional Euler equations are solved, but the equations are divided into two parts that describe the convection and acoustic processes separately, with corresponding different time steps. The iterative Godunov exact solver is additionally used because the Riemann invariants are non-conserved for moderate and strong shocks in an ideal gas. The iterative Godunov exact solver determines the accuracy of the proposed method for flow discontinuities. The calculations used the condition of iteration termination when the pressure difference between the current and previous iterations was less than 10-5. The proposed method is a particle-in-cell (PIC) method. To the best of the author’s knowledge, no similar PIC numerical schemes using the Riemann invariants or an iterative Godunov exact solver existed at the time of his first paper on this method in 2011. In shock waves for the proposed method, the flow properties change instantaneously (with accuracy depending on the mesh cell size). In a previous paper published in 2024, a unified formula for the density distribution of rarefaction waves was proposed. In the 2022 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 old nonlinear law derived from the Riemann invariants was used for the remaining test problems. This article proposes a further advancement of the numerical scheme of the proposed method described in the author’s previous article of 2024, namely, another modification of the method for modeling strong shock waves and its application to the problem of the absorption of weaker shock wave by stronger one. The obtained numerical analysis results for the two test cases were compared with both the results of the method presented in the 2022 paper and data of the Total Variation Deminishing (TVD) scheme proposed by Harten. Unfortunately, it is difficult to obtain exact solutions to the considered problems.