High-order accurate well-balanced energy stable finite difference schemes for multi-layer shallow water equations on fixed and adaptive moving meshes
Zhihao Zhang, Huazhong Tang, Junming Duan
TL;DR
This work tackles the stable and accurate simulation of multi-layer shallow water flows with $M\ge 2$ layers over complex bottom topography. It develops high-order well-balanced energy-stable finite-difference schemes on fixed and adaptive moving meshes by proving convexity of the ML-SWEs energy $\eta(\mathbf{U})$, constructing a two-point energy-conservative flux, and enforcing compatible source-term discretizations; high-order accuracy is achieved with WENO dissipation and SSP-RK3 time stepping, preserving the lake at rest. The moving-mesh extension relies on a reformulated energy $\widehat{\eta}$ including the bottom as a conservative variable and a geometric conservation law framework. Numerical tests for two- and three-layer cases demonstrate fifth-order accuracy, robust well-balancing, energy stability, and efficiency gains on moving meshes, despite the lack of a closed-form eigenstructure, highlighting the practical viability for stratified flow simulations in oceanographic contexts.
Abstract
This paper develops high-order well-balanced (WB) energy stable (ES) finite difference schemes for multi-layer (the number of layers $M\geqslant 2$) shallow water equations (SWEs) on both fixed and adaptive moving meshes, extending our previous works [20,51]. To obtain an energy inequality, the convexity of an energy function for an arbitrary $M$ is proved by finding recurrence relations of the leading principal minors or the quadratic forms of the Hessian matrix of the energy function with respect to the conservative variables, which is more involved than the single-layer case due to the coupling between the layers in the energy function. An important ingredient in developing high-order semi-discrete ES schemes is the construction of a two-point energy conservative (EC) numerical flux. In pursuit of the WB property, a sufficient condition for such EC fluxes is given with compatible discretizations of the source terms similar to the single-layer case. It can be decoupled into $M$ identities individually for each layer, making it convenient to construct a two-point EC flux for the multi-layer system. To suppress possible oscillations near discontinuities, WENO-based dissipation terms are added to the high-order WB EC fluxes, which gives semi-discrete high-order WB ES schemes. Fully-discrete schemes are obtained by employing high-order explicit SSP-RK methods and proved to preserve the lake at rest. The schemes are further extended to moving meshes based on a modified energy function for a reformulated system, relying on the techniques proposed in [51]. Numerical experiments are conducted for some two- and three-layer cases to validate the high-order accuracy, WB and ES properties, and high efficiency of the schemes, with a suitable amount of dissipation chosen by estimating the maximal wave speed due to the lack of an analytical expression for the eigenstructure of the multi-layer system.