High order CG schemes for KdV and Saint-Venant flows

Description

Hyperbolic systems and dispersive equations remain challenging for the FEM community. Onthe basis of an arbitrarily high order FEM, namely the spectral element method (SEM), here weaddress:- The Korteweg-de Vries equation, to explain how high order derivative terms can be efficientlyhandled with a C 0 continuous Galerkin approximation. Two strategies are proposed, both of themallowing the SEM approximation of the high order derivative term to remain in the usual H1space. The conservation of the invariants is also focused on, especially by using in time embeddedimplicit-explicit Runge Kutta schemes [1].- The 2D shallow water equations, to show how a stabilized SEM can solve problems involvingshocks. Moreover, we especially focus on flows involving dry-wet transitions and propose to thisend an efficient variant of the entropy viscosity method [2, 3].Results obtained for well known benchmark problems are provided to illustrate the capabilities ofthe proposed high order algorithms.

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