A Survey of High Order Schemes for the Shallow Water Equations (Journal of Mathematical Study, 2014)
In this paper, we survey our recent work on designing high order positivity-preservingwell-balanced finite difference and finite volumeWENO (weighted essentially non-oscillatory) schemes, and discontinuous Galerkin finite elementschemes for solving the shallow water equations with a non-flat bottom topography.These schemes are genuinely high order accuratein smooth regions for general solutions, are essentially non-oscillatoryfor general solutions with discontinuities, and at the same timethey 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 twodimensions with triangular meshes, and we prove that the resulting schemes guaranteethe positivity of the water depth.
Keywords: Hyperbolic balance laws; WENO scheme; discontinuous Galerkin method; high order accuracy; source term; conservation laws; shallow water equation.