A finite element method for the one-dimensional extended Boussinesq equations

A new finite element method for Nwogu's (O. Nwogu, ASCE J. Waterw., Port, Coast., Ocean Eng., 119, 618-638 (1993)) one-dimensional extended Boussinesq equations is presented using a linear element spatial discretisation method coupled with a sophisticated adaptive time integration package. The accuracy of the scheme is compared to that of an existing finite difference method (G. Wei and J.T. Kirby, ASCE J. Waterw., Port, Coast., Ocean Eng., 121, 251-261 (1995)) by considering the truncation error at a node. Numerical tests with solitary and regular waves propagating in variable depth environments are compared with theoretical and experimental data. The accuracy of the results confirms the analytical prediction and shows that the new approach competes well with existing finite difference methods. The finite element formulation is shown to enable the method to be extended to irregular meshes in one dimension and has the potential to allow for extension to the important practical case of unstructured triangular meshes in two dimensions. This latter case is discussed. Copyright (C) 1999 John Wiley & Sons, Ltd.