High order CG schemes for KdV and Saint-Venant flows
Résumé
Hyperbolic systems and dispersive equations remain challenging for the FEM community. On
the basis of an arbitrarily high order FEM, namely the spectral element method (SEM), here we
address:
- The Korteweg-de Vries equation, to explain how high order derivative terms can be efficiently
handled with a C 0 continuous Galerkin approximation. Two strategies are proposed, both of them
allowing the SEM approximation of the high order derivative term to remain in the usual H1
space. The conservation of the invariants is also focused on, especially by using in time embedded
implicit-explicit Runge Kutta schemes [1].
- The 2D shallow water equations, to show how a stabilized SEM can solve problems involving
shocks. Moreover, we especially focus on flows involving dry-wet transitions and propose to this
end an efficient variant of the entropy viscosity method [2, 3].
Results obtained for well known benchmark problems are provided to illustrate the capabilities of
the proposed high order algorithms.