Flux globalization-based well-balanced path-conservative central-upwind scheme for two-dimensional two-layer thermal rotating shallow water equations. (English) Zbl 07922770
Summary: We develop a flux globalization-based well-balanced path-conservative central-upwind scheme on Cartesian meshes for the two-dimensional (2-D) two-layer 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 2-D 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 2-D equilibria. Numerical experiments demonstrate the performance of the proposed scheme in computationally non-trivial situations: in the presence of shocks, dry areas, non-trivial topographies, including discontinuous ones, and in the case of hyperbolicity loss. The scheme works equally well in both the \(f\)-plane and beta-plane frameworks.
MSC:
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 86-08 | Computational methods for problems pertaining to geophysics |
| 86-10 | Mathematical modeling or simulation for problems pertaining to geophysics |
| 86A10 | Meteorology and atmospheric physics |
| 35L67 | Shocks and singularities for hyperbolic equations |
Keywords:
two-layer thermal rotating shallow water equations; well-balanced schemes; flux globalization; path-conservative central-upwind schemesReferences:
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