Finite Volume Methods for Multi-Symplectic PDEs

We investigate the application of a cell vertex nite volume discretization to multi-symplectic PDEs. The investigated discretization reduces to the Preissmann box scheme when used on a rectangular grid. Concerning arbitrary quadrilateral grids, we show that only methods with parallelogram-like nite volume cells lead to a multi-symplectic discretization, i.e., to a method that preserves a discrete conservation law of symplecticity. One of the advantages of nite volume methods is that they can be easily adjusted to variable meshes. But, although the implementation of moving mesh nite volume methods for multi-symplectic PDEs is rather straightforward, the restriction to parallelogram-like cells implies that only meshes moving with a constant speed are multi-symplectic. To overcome this restriction, we suggest the implementation of reversible moving mesh methods based on a semi-Lagrangian approach. Numerical experiments are presented for a generalized shallow water system with steepening and non-steepening wave solutions and the Korteweg-de Vries (KdV) equation.