Conservation of wave action under multisymplectic discretizations

In this paper we discuss the conservation of wave action under numerical discretization by variational and multisymplectic methods. Both the abstract wave action conservation defined with respect to a smooth, periodic, one-parameter ensemble of flow realizations and the specific wave action based on an approximated and averaged Lagrangian are addressed in the numerical context. It is found that the discrete variational formulation gives rise in a natural way not only to the discrete wave action conservation law, but also to a generalization of the numerical dispersion relation to the case of variable coefficients. Indeed a fully discrete analogue of the modulation equations arises. On the other hand, the multisymplectic framework gives easy access to the conservation law for the general class of multisymplectic Runge-Kutta methods. A numerical experiment confirms conservation of wave action to machine precision and suggests that the solution of the discrete modulation equations approximates the numerical solution to order O(ε) on intervals of O(ε-1)