Invariant Region Preserving Reconstruction and Enhanced Stability of the Central Scheme in Two Dimensions

Abstract

In this paper, our focus is on examining the robustness of the central scheme in two dimensions. Although stability analyses are available in the literature for the scheme’s solution of scalar conservation laws, the associated Courant-Friedrichs-Lewy (CFL) number is often notably small, occasionally degenerating to zero. This challenge is traced back to the initial data reconstruction. The interface value limiter used in the reconstruction proves insufficient to maintain the invariant region of the updated solutions. To overcome this limitation, we introduce the vertex value limiter, resulting in a more suitable CFL number that is half of the one-dimensional value. We present a unified analysis of stability applicable to both types of limiters. This enhanced stability condition enables the utilization of larger time steps, offering improved resolution to the solution and ensuring faster simulations. Our analysis extends to general conservation laws, encompassing scalar problems and nonlinear systems. We support our findings with numerical examples, validating our claims and showcasing the robustness of the enhanced scheme.