An eigenvalue problem for a system of finite difference equations approximating a linear water wave equation

Abstract

We give analytical representations of solutions of an eigenvalue problem for a system of finite difference equations approximating a linear water wave equation in a rectangular region. The eigenvalue problem is derived using the finite element method with the linear basis functions associated with the Friedrichs-Keller type triangulation, and with the generalized mass matrix$\theta M + (1 - \theta ) \bar {\rm M}$ on the water surface at rest, whereM, and$\bar {\rm M}$, are the consistent mass matrix, and the lumped mass matrix, respectively. We also give an analytical representation of relative errors of approximate eigenvalues and consider the limit of the relative errors as the horizontal mesh length,h, tends to 0 under the preservation of the aspect ratiok of the element rectangular, that is, the ratio of the vertical mesh length overh. Finally, our graphical observation is added concerning the dependency of the relative errors and their limits on either θ ork.