  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 an
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.
