The applicability of a time-stepping approximate finite difference method is tested for the response of a plane incoming tsunami of small amplitude meeting an idealized island (see Fig 1). The resulting amplitudes are compared with the exact solution, which comes out of solving the linear shallow water wave equation for the area in question. Since this wave equation excludes dissipation (bottom friction) and the Coriolis force, these terms are omitted in the Boussinesq equations, formulated as mass and momentum conservation, which are the bases of the finite difference scheme. Grid size is 1 x 1 km. The incoming wave is time-harmonic with a period of T = 480 s; the (test) solution to the wave equation is thus a truly steady-state solution. The finite difference scheme, however, has a so-called "cold start" and so it is transient in principle. During a time corresponding to three periods, in which disturbances from the open boundaries still have only a small effect on the wave field near the island, the time-series of signals in selected points can define a steady response, though. Considering the inevitable shortcomings of a provisional study like the present, satisfactory agreement with the exact solution is met over the shoal in Fig 1. We have thus a promising starting point for more elaborate studies, comprising new filtering algorithms for the boundaries, tests with real transient input signals, and including non-linearity, bottom friction, and the Coriolis force. The numerical scheme used is the so-called System 21, developed at the Danish Hydraulic Institute and placed at our disposal for the present study.
Solution of the Dirichlet problem by a finite difference analog of the boundary integral equation
