In the tsunami numerical simulation, the dispersive wave theory is used for simulation of a trans-oceanic tsunami propagation and a near-field tsunami propagation with soliton fission. Many numerical schemes have been proposed to solve the dispersive wave theory, although we do not have a reliable criterion for selecting an adequate scheme at present. In this study, the exact solutions of the finite difference equations of linear dispersive wave theory are derived for some numerical methods on the basis of staggered Leap-Frog scheme. Characteristics of truncation error and numerical stability of schemes are discussed by use of the exact solutions. Leap-Frog implicit scheme appeared to be applicable to the practical problems due to its superior stability as well as high-accuracy of simulation.
A new numerical method of Leap-Frog implicit scheme is developed, in which the dispersive term in the equation of motion is solved as the differential equation of Poisson-type. The computation can be conducted effectively by a high-speed solver. We compare the model results with analytical solution of linear Boussinesq equation to validate the present model. Agreement of the present model is excellent, furthermore the present model requires less computing time with good stability than that of the conventional model. The present model is able to reproduce a solution of the nonlinear dispersive wave theory, and their accuracy is verified by comparing the numerical results with available experiment data.
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