Mathematical modeling of compressible one-dimensional flows using the semi-Lagrangian numerical Godunov method without computational viscosity for shock waves

One of the most important and complex effects in compressible fluid flow simulation is a shock-capturing mechanism. 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 a 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 grid. Due to the numerical viscosity, the aforementioned Euler-type methods stretch the parameter change in the shock over few grid cells. We introduce a semi-Lagrangian Godunov-type method without numerical viscosity for shocks. Another well-known approach is a method of characteristics that has no numerical viscosity and uses the Riemann invariants or solvers for water hammer phenomenon modeling, but in its formulation the convective terms are typically neglected. We use a similar approach to solve the one-dimensional adiabatic gas dynamics equations, but we split the equations into parts describing convection and acoustic processes separately, with corresponding different time steps. When we are looking for the solution to the one-dimensional problem of the scalar hyperbolic conservation law by the proposed method, we additionally use the iterative Godunov exact solver, because the Riemann invariants are non-conserved for moderate and strong shocks in an ideal gas. In the shock for the proposed method, the flow properties change instantaneously (with an accuracy dependent on the grid cell size). The proposed method belongs to a group of particle-in-cell (PIC) methods. To the best of the author’s knowledge, there were no similar PIC numerical schemes utilizing the Riemann invariants or the iterative Godunov exact solver prior to 2011 (as described in the first author’s publication detailing the numerical method). 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 as outlined in the author’s second article from 2022. In the 2022 article, a linear law of distribution of flow parameters was employed for a rarefaction wave when modeling the Shu-Osher problem, aimed at reducing parasitic oscillations. Additionally, a nonlinear law derived from Riemann invariants was utilized for the remaining test problems. The obtained results of numerical analysis for these cases, including the standard shock-tube problem of Sod, the Riemann problem of Lax, and the Shu-Osher shock-tube problem, are compared with both the exact solution and the data presented in the 2022 paper. The iterative Godunov exact solver determines the accuracy of the proposed method for flow discontinuities. In calculations, we use the iteration termination condition less than 10-5 to find the pressure difference between the current and previous iterations.