High Order CG Schemes for KdV and Saint-Venant Flows

Strategies we have recently proposed to efficiently address dispersive equations and hyperbolic systems with high order continuous Galerkin schemes are first recalled. Using the Spectral Element Method (SEM), we especially consider the Korteweg-De Vries equation to explain how to handle the third order derivative term with an only C0-continuous approximation. Moreover, we focus on the preservation of two invariants, namely the mass and momentum invariants. With a stabilized SEM, we then address the Saint-Venant system to show how a strongly non linear viscous stabilization, namely the entropy viscosity method (EVM), can allow to support the presence of dry-wet transitions and shocks. The new contribution of the paper is a sensitivity study to the EVM parameters, for a shallow water problem involving many interactions and shocks. A comparison with a computation carried out with a second order Finite Volume scheme that implements a shock capturing technique is also presented.