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A rezoning-type adaptive moving mesh discontinuous Galerkin method isproposed for the numerical solution of the shallow water equations withnon-flat bottom topography. The well-balance property is crucial to thesimulation of perturbation waves over the lake-at-rest steady state such aswaves on a lake or tsunami waves in the deep ocean. To ensure the well-balanceand positivity-preserving properties, strategies are discussed in the use ofslope limiting, positivity-preservation limiting, and data transferring betweenmeshes. Particularly, it is suggested that a DG-interpolation scheme be usedfor the interpolation of both the flow variables and bottom topography from theold mesh to the new one and after each application of thepositivity-preservation limiting on the water depth, a high-order correction bemade to the approximation of the bottom topography according to themodifications in the water depth. Moreover, mesh adaptation based on theequilibrium variable and water depth is shown to give more desirable resultsthan that based on the commonly used entropy function. Numerical examples inone and two spatial dimensions are presented to demonstrate the well-balanceand positivity-preserving properties of the method and its ability to capturesmall perturbations of the lake-at-rest steady state.

A high-order well-balanced positivity-preserving moving mesh DG method for the shallow water equations with non-flat bottom topography