Spectral element schemes for the Korteweg-de Vries and Saint-Venant equations

Richard Pasquetti (1, 2)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 CASTOR - Control, Analysis and Simulations for TOkamak Research

Hyperbolic systems and dispersive equations remain challenging for finite element methods (FEMs). On the basis of an arbitrarily high order FEM, namely the spectral element method (SEM), we address : -The Korteweg-de Vries equation, to explain how high order derivative terms can be efficiently handled with a C0-continuous Galerkin approximation. The conservation of the invariants is also focused on, especially by using in time embedded implicit-explicit Runge Kutta schemes. -The 2D shallow water equations, to show how a stabilized SEM can solve problems involving shocks. We especially focus on flows involving dry-wet transitions and propose to this end an efficient variant of the entropy viscosity method.

Mots clés: Spectral element method; dispersive equations; hyperbolic systems; Korteweg-de Vries equation; shallow water equations; conservation of invariants; entropy viscosity method

Domaines: Mécanique des fluides [physics.class-ph]

Richard Pasquetti. Spectral element schemes for the Korteweg-de Vries and Saint-Venant equations. 23ème Congrès Français de Mécanique (Mini-symposium Rencontres Mathématiques-Mécanique), Aug 2017, Lille, France. ⟨hal-01582605⟩