Explicit finite difference schemes for solving the one - Dimensional tsunami wave propagation equations

Abstract

Because the tsunami wavelength is much longer than the sea depth, the propagation of this powerful destructive wave can be modelled by the non-linear shallow water equations (SWEs). To solve accurately these hyperbolic partial differential equations, one can resort to a variety of numerical techniques, including finite difference method (FDM) or finite volume method (FVM). In this paper, we have selected FDM, as a well verified instrument on other fluid flow problems, to estimate the tsunami wave propagation in one dimension. The ability of four explicit scheme (Forward Euler, Lax Friedrichs, Mac Cormack and Richtmyer) to solve SWEs was tested on three different test problems. Starting from the idea that a tsunami may begin as a sudden rise of a column of water, the first test represents a square or a sinusoidal pulse which breaks up into two identical waves moving in opposing directions on a constant depth seabed. In the other two test problems we simulate a tsunami wave approaching the shore on a variable seabed depth having either o steady or a highly variable slope. The numerical results allow us to assert that, in general, the used explicit schemes lead to a correct description of a tsunami wave propagation, both offshore and near the coastline. The exception is represented by the Forward Euler scheme that, due to its instability, was only used in the first test.