High-order discontinuous element-based schemes for the inviscid shallow water equations: spectral multidomain penalty and discontinuous Galerkin methods

摘要

Two commonly used types of high-order-accuracy element-based schemes, collocationbasedspectral multidomain penalty methods (SMPM) and nodal discontinuousGalerkin methods (DGM), are compared in the framework of the inviscid shallowwater equations. Differences and similarities in formulation are identified,with the primary difference being the dissipative term in the Rusanov form ofthe numerical flux for the DGM that provides additional numerical stability;however, it should be emphasized that to arrive at this equivalence betweenSMPM and DGM requires making specific choices in the construction of bothmethods; these choices are addressed. In general, both methods offer a multitudeof choices in the penalty terms used to introduce boundary conditionsand stabilize the numerical solution. The resulting specialized class of SMPMand DGM are then applied to a suite of six commonly considered geophysicalflow test cases, three linear and three non-linear; we also include results fora classical continuous Galerkin (i.e., spectral element) method for comparison.Both the analysis and numerical experiments show that the SMPM and DGMare essentially identical; both methods can be shown to be equivalent for veryspecial choices of quadrature rules and Riemann solvers in the DGM along withspecial choices in the type of penalty term in the SMPM. Although we onlyfocus our studies on the inviscid shallow water equations the results presentedshould be applicable to other systems of nonlinear hyperbolic equations (suchas the compressible Euler equations) and extendable to the compressible andincompressible Navier-Stokes equations, where viscous terms are included.