| Abstract: | In this paper, we survey our recent work on designing high order positivity-preserving
well-balanced finite difference and finite volume
WENO (weighted essentially non-oscillatory) schemes, and discontinuous Galerkin finite element
schemes for solving the shallow water equations with a non-flat bottom topography.
These schemes are genuinely high order accurate
in smooth regions for general solutions, are essentially non-oscillatory
for general solutions with discontinuities, and at the same time
they preserve exactly the water at rest or the more general moving water steady state solutions.
A simple positivity-preserving limiter, valid under suitable CFL condition,
has been introduced in one dimension and reformulated to two
dimensions with triangular meshes, and we prove that the resulting schemes guarantee
the positivity of the water depth. |
