Positivity-preserving new low-dissipation central-upwind schemes for compressible Euler equations

Abstract

One major challenge in developing accurate and robust numerical schemes for compressible Euler equations arises due to the potential emergence of discontinuous structures in the solution. Recently proposed low-dissipation central-upwind (LDCU) schemes achieve sharp resolution of these structures without introducing spurious oscillations. However, unlike many other Godunov-type methods, the LDCU schemes cannot be written as a convex combination of first-order positivity-preserving (PP) schemes. Therefore, the PP property of the LDCU schemes cannot be analyzed by standard techniques. In this paper, we overcome this difficulty by first decomposing the studied schemes into a convex combination of several intermediate solution states, and then analyzing their PP properties. The performed analysis helps us to construct PPLDCU schemes for Euler equations of compressible gas dynamics, guaranteeing the positivity of computed density and pressure. To achieve the PP property, the built-in anti-diffusion terms in the two-dimensional case and the piecewise linear reconstruction procedure in both the one- and two-dimensional cases are redesigned. The effectiveness and robustness of the proposed PPLDCU schemes are demonstrated in several challenging numerical examples.