Galerkin methods applied to some model equations for non-linear dispersive waves

The application of a dissipative Galerkin scheme to the numerical solution of the Korteweg de Vries (KdV) and Regularised Long Wave (RLW) equations, is investigated. The accuracy and stability of the proposed schemes is derived using a localised Fourier analysis. With cubic splines as basis functions, the errors in the numerical solutions of the KdV equation for different mesh-sizes and different amounts of dissipation is determined. It is shown that the Galerkin scheme for the RLW equation gives rise to much smaller errors (for a given mesh-size), and allows larger steps to be taken for the integrations in time (for a specified error tolerance). Also, the interaction of two solitons is compared for the KdV and RLW equations, and several differences in their behaviour are found.