  The finite element method (FEM) is applied to obtain numerical solutions to a
recently derived nonlinear equation for the shallow water wave problem. A weak
formulation and the Petrov-Galerkin method are used. It is shown that the FEM
gives a reasonable description of the wave dynamics of soliton waves governed
by extended KdV equations. Some new results for several cases of bottom shapes
are presented. The numerical scheme presented here is suitable for taking into
account stochastic effects, which will be discussed in a subsequent paper.
