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This paper develops high-order well-balanced (WB) energy stable (ES) finitedifference 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, theconvexity of an energy function for an arbitrary $M$ is proved by findingrecurrence relations of the leading principal minors or the quadratic forms ofthe Hessian matrix of the energy function with respect to the conservativevariables, which is more involved than the single-layer case due to thecoupling between the layers in the energy function. An important ingredient indeveloping high-order semi-discrete ES schemes is the construction of atwo-point energy conservative (EC) numerical flux. In pursuit of the WBproperty, a sufficient condition for such EC fluxes is given with compatiblediscretizations of the source terms similar to the single-layer case. It can bedecoupled into $M$ identities individually for each layer, making it convenientto construct a two-point EC flux for the multi-layer system. To suppresspossible oscillations near discontinuities, WENO-based dissipation terms areadded to the high-order WB EC fluxes, which gives semi-discrete high-order WBES schemes. Fully-discrete schemes are obtained by employing high-orderexplicit SSP-RK methods and proved to preserve the lake at rest. The schemesare further extended to moving meshes based on a modified energy function for areformulated system, relying on the techniques proposed in [51]. Numericalexperiments are conducted for some two- and three-layer cases to validate thehigh-order accuracy, WB and ES properties, and high efficiency of the schemes,with a suitable amount of dissipation chosen by estimating the maximal wavespeed due to the lack of an analytical expression for the eigenstructure of themulti-layer system.

High-order accurate well-balanced energy stable finite difference  schemes for multi-layer shallow water equations on fixed and adaptive moving  meshes