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A high-order, well-balanced, positivity-preserving quasi-Lagrange moving meshDG method is presented for the shallow water equations with non-flat bottomtopography. The well-balance property is crucial to the ability of a scheme tosimulate perturbation waves over the lake-at-rest steady state such as waves ona lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrangemoving mesh DG method, a hydrostatic reconstruction technique, and a change ofunknown variables. The strategies in the use of slope limiting,positivity-preservation limiting, and change of variables to ensure thewell-balance and positivity-preserving properties are discussed. Compared torezoning-type methods, the current method treats mesh movement continuously intime and has the advantages that it does not need to interpolate flow variablesfrom the old mesh to the new one and places no constraint for the choice of anupdate scheme for the bottom topography on the new mesh. A selection of one-and two-dimensional examples are presented to demonstrate the well-balanceproperty, positivity preservation, and high-order accuracy of the method andits ability to adapt the mesh according to features in the flow and bottomtopography.

A well-balanced positivity-preserving quasi-Lagrange moving mesh DG method for the shallow water equations