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This paper proposes high-order accurate well-balanced (WB) energy stable (ES)adaptive moving mesh finite difference schemes for the shallow water equations(SWEs) with non-flat bottom topography. To enable the construction of the ESschemes on moving meshes, a reformulation of the SWEs is introduced, with thebottom topography as an additional conservative variable that evolves in time.The corresponding energy inequality is derived based on a modified energyfunction, then the reformulated SWEs and energy inequality are transformed intocurvilinear coordinates. A two-point energy conservative (EC) flux isconstructed, and high-order EC schemes based on such a flux are proved to be WBthat they preserve the lake at rest. Then high-order ES schemes are derived byadding suitable dissipation terms to the EC schemes, which are newly designedto maintain the WB and ES properties simultaneously. The adaptive moving meshstrategy is performed by iteratively solving the Euler-Lagrangian equations ofa mesh adaptation functional. The fully-discrete schemes are obtained by usingthe explicit strong-stability preserving third-order Runge-Kutta method.Several numerical tests are conducted to validate the accuracy, WB and ESproperties, shock-capturing ability, and high efficiency of the schemes.

journal of computational physics

High-order accurate well-balanced energy stable adaptive moving mesh  finite difference schemes for the shallow water equations with non-flat  bottom topography