Finite difference method for the Fornberg-Whitham equation

Abstract

In this thesis, we design a family of finite difference (θ)-schemes to approximate solutions of the viscous Fornberg-Whithem equation which is a shallow-water wave model describing waves breaking. The model admits multiple invariants including momentum and energy. Although the model is nonlinear, our present schemes are linear and involve information from three time steps, i.e. three-level scheme. For (θ) = 1/3, it can be shown that the conservation of some invariants is still maintained. The methods have second order of accuracy in time and space. Moreover, we can prove that the convergence of the numerical solutions is uniform. Finally, numerical examples are used to demonstrate effectiveness as well as to verify our theoretical results.