Addressing the issue of mass conservation error and the connected problem of Carbuncle formation

Abstract: We study mass conservation errors (momentum density spike) and the related phenomenon of post shock oscillations in numerical solutions of compressible Euler equations. These phenomena and their causes have been reported in literature [34, 1]. In this paper, first, we compare the mass conservation and post shock oscillation errors obtained using combinations of different numerical methods (Finite Volume, Finite Difference with WENO and DG with simple WENO limiter) and upwind flux functions (ROE, AUSM+-up, and others) for moving shocks, modelled using one-dimensional Euler equations. Next, the mass conservation error is quantified for stationary shocks modelled using one-dimensional, quasi-one-dimensional and two-dimensional Euler equations. It is shown that using a fine mesh or refining mesh near shocks using multiple over set meshes lead to mitigation of the mass conservation error. This is demonstrated using the problem of flow through a variable area duct modelled using quasi-one-dimensional Euler equations. The link between mass conservation error and carbuncle formation is shown and preliminary results indicating that the carbuncle can be cured using multiple overset meshes are also shown.