Finite difference methods for the reduced water wave equation

Abstract A comparative study of various finite difference schemes for numerical solution of the reduced water wave equation is carried out. The nine-point optimal scheme is newly derived, and the derivation involves a tactical combination of the Taylor expansions of wave function at the relevant nodes of a representative grid element. It is demonstrated that there is a proper range of the relative grid size for each scheme in practical computations, beyond which a numerical solution is either inaccurate, due to the fact that approximations adopted by the scheme are valid only when the ratio of the grid size to the local wave length is small, or unobtainable, due to the ill-condition of the difference equation system. A high-order scheme is shown to be advantageous in terms of not only the numerical accuracy but also the validity when relatively large grid elements are employed.