MODELING WITH FINITE DIFFERENCES

Many geophysical numerical algorithms assume linearity. Thus they will work only on equations that have a linear behavior. For modeling nonlinear phenomena, the choice of algorithms becomes rather restrictive. In most cases an analytical solution is not possible or would involve some sort of linearization. That in turn would defeat the original purpose of investigating nonlinear effects. For modeling nonlinearities finite difference algorithms have the advantage that no linearity assumptions have to be made in order to model nonlinear wave propagation. One problem however remains with using finite difference methods: It is now much harder to calculate stability criteria properly. Many rigorous stability criteria are based on linearity. The usual methods use linear transforms to come up with an easy to calculate estimate of stability and dispersion of the algorithm. In a recent paper Kosik 1993 makes use of a Crank-Nicholson FD scheme which is implicit and thus guaranteed to exhibit stability for a certain range of parameters.