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