Locally Divergence-Free Well-Balanced Path-Conservative Central-Upwind Schemes for Magnetic Thermal Rotating Shallow Water Equations

Abstract

We develop a second-order flux globalization based path-conservative central-upwind (PCCU) scheme for magnetic thermal rotating shallow water (MTRSW) equations, which we have recently introduced in [Y. Cao, A. Kurganov, M. Rostami, C. Wang, and V. Zeitlin; submitted]. The new method is designed to both uphold the divergence-free condition for the magnetic field at the discrete level and satisfy the well-balanced (WB) property by exactly preserving certain physically relevant steady states of the studied system. The locally divergence-free constraint of the magnetic field is enforced by applying the scheme to a Godunov-Powell modified version of the MTRSW system, adding equations for the spatial derivatives of the magnetic fields, and adjusting the reconstruction procedures for the magnetic field variables. The WB property is enforced with the help of a flux globalization technique, which helps to design a method capable of exactly preserving both still- and moving-water equilibria. The performance of the proposed scheme is demonstrated on several numerical examples.

Keywords: Magnetic thermal rotating shallow water equations, divergence-free constraints, path-conservative central-upwind scheme, flux globalization based well-balanced scheme