Solution of non-linear dispersive wave problems using a moving finite element method

The solution of the fully non-linear time-dependent two-dimensional shallow water equations is considered. Dispersive effects due to the Coriolis forces are taken into account. Such effects are of major importance in geophysical fluid dynamics applications. The recently proposed string gradient weighted moving finite element method is extended for this class of problems. This method simultaneously determines, at each time step, the solution of the governing partial differential equations and an optimal location of the finite element nodes. It has previously been applied to non-dispersive wave problems; here its performance under the demanding conditions of large Coriolis forces, inducing large mesh and field rotation, is studied. Optimal rates of convergence are obtained. Results for some example problems of water hump release are presented. Non-linear and linearized solutions are compared.