A staggered semi-implicit spectral discontinuous Galerkin scheme for the shallow water equations

Abstract

A spatially arbitrary high order, semi-implicit spectral discontinuous Galerkin (DG) scheme for the numerical solution of the shallow water equations on staggered control volumes is derived and discussed. The free surface elevation and the momentum are expressed in terms of the same polynomials that are used as basis, and as test functions. Each unknown is, however, defined on a different set of control volumes that are spatially staggered with respect to each other. A semi-implicit time integration yields a stable and efficient mass conservative algorithm. The use of a staggered mesh has the advantage that after substitution of the momentum equations into the mass conservation equation only one single block-penta-diagonal system must be solved for the new free surface location, which is a scalar quantity. Subsequently the new momentum components are directly obtained. The staggered semi-implicit approach makes the present DG scheme different from other published DG schemes. The sparse linear system of the resulting discrete wave equation for the free surface can be conveniently solved by a matrix-free GMRES algorithm. Furthermore, for the shallow water equations the proposed scheme can be written as a quadrature-free method, where all surface and volume integrals can be precomputed once and for all on the reference element and assembled into universal matrices and tensors. For the special case N=0 the proposed method reduces to a classical semi-implicit finite difference scheme. The proposed semi-implicit scheme is particularly well suited for low Froude number flows. The method is validated on some typical academic benchmark problems, using polynomial degrees of up to N=20.