Flux Globalization Based Well-Balanced Path-Conservative Central-Upwind Scheme for the Thermal Rotating Shallow Water Equations

Abstract

We present an extension of the flux globalization based well-balanced path-conservative central-upwind scheme to the one- and two-dimensional thermal rotating shallow water equations. The scheme is well-balanced in the sense that it can exactly preserve a variety of physically relevant steady states. In the one-dimensional case, it can preserve different “lake-at-rest” equilibria, thermo-geostrophic equilibria, as well as general moving-water steady states. In the two-dimensional case, preserving general moving-water steady states is difficult, and to the best of our knowledge, none of existing schemes can achieve this ultimate goal. The proposed scheme can exactly preserve the $x$- and $y$-directional jets in the rotational frame as well as certain genuinely two-dimensional equilibria. Furthermore, our approach employs a path-conservative technique for discretizing nonconservative product terms, which are incorporated into the global fluxes. This allows the developed scheme to exactly preserve some of the discontinuous steady states as well. We provide a number of numerical examples to demonstrate the advantages of the proposed scheme over some alternative finite-volume methods.