Flux Globalization Based Well-Balanced Path-Conservative Central-Upwind Scheme for the Thermal Rotating Shallow Water Equations (Communications in Computational Physics, 2023)
We present an extension of the flux globalization based well-balanced path-conservative central-upwind scheme to the one- and two-dimensional thermal rotatingshallow water equations. The scheme is well-balanced in the sense that it can exactlypreserve a variety of physically relevant steady states. In the one-dimensional case, itcan preserve different “lake-at-rest” equilibria, thermo-geostrophic equilibria, as wellas 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 ofexisting 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 genuinelytwo-dimensional equilibria. Furthermore, our approach employs a path-conservativetechnique for discretizing nonconservative product terms, which are incorporated intothe global fluxes. This allows the developed scheme to exactly preserve some of thediscontinuous steady states as well. We provide a number of numerical examplesto demonstrate the advantages of the proposed scheme over some alternative finite-volume methods.
Keywords: Thermal rotating shallow water equations; well-balanced schemes; flux globalization; path-conservative central-upwind schemes.