Godunov-type schemes for the pressureless gas dynamics and related models

Abstract

We study and develop new first-order Godunov-type schemes for the weakly hyperbolic pressureless gas dynamics equations and augmented Burgers’ equations. Each of these systems carries the information of propagation of waves with the same fluid velocity. The goal is achieved by first obtaining an Engquist-Osher (EO) type scheme for the pressureless system and then by enhancing the upwinding information present in the EO-type scheme to construct more accurate Godunov-type schemes. The resulting schemes present a lesser amount of numerical dissipation than existing Jordan decomposition-based Flux Difference Splitting (FDS) schemes recently proposed in [N. K. GARG, Numer. Algorithms, 83 (2020) 1091–1121] and are compact and robust. These schemes are tested on a number of numerical examples for one- and two-dimensional pressureless equations of gas dynamics and then to augmented Burgers’ equations.