Flux globalization based well-balanced central-upwind schemes for the Euler equations with gravitation

Abstract

We develop a second-order flux globalization based well-balanced (WB) central-upwind (CU) scheme for compressible Euler equations with gravitation. Within the flux globalization framework, the gravitational term is incorporated into the global flux, and the resulting quasi-conservative system is numerically solved using the Riemann-problem-solver-free CU scheme. The constructed scheme is WB in the sense that it is capable of exactly preserving both hydrostatic and non-hydrostatic equilibria at the discrete level. This is achieved by performing piecewise linear reconstruction for the nonlocal global flux variables rather than the conservative variables, and by switching off a part of numerical viscosity when the flow is near steady state regime. The performance of the proposed flux globalization based WB CU schemes is tested on several one- and two-dimensional numerical examples. In these examples, we clearly demonstrate the advantage of the proposed scheme over its non-WB counterparts.