FINITE ELEMENT MODELING OF ONE-DIMENSIONAL BOUSSINESQ EQUATIONS

Abstract

Finite element modeling of one-dimensional Beji and Nadaoka Boussinesq equation is presented. The continuous equations are spatially discretized using standard Galerkin method. Since the extended Boussinesq equations contain high-order derivatives, two different numerical techniques are proposed in this paper in order to simplify the discretization task of the third-order terms. In the first technique, an auxiliary equation is introduced to eliminate the third-order derivatives of the momentum equation while non-overlapping elements with linear interpolating functions are employed to account for the dependent variables. However, in the second method, overlapping elements with quadratic interpolating functions are applied for discretizing the governing equations. Time integration is performed using the Adams–Bashforth–Moulton predictor–corrector method. By considering the truncation error and theoretical analysis for both of the numerical techniques, accuracy and stability of the adopted finite element schemes have been studied. Finally, a computer code is developed based on the proposed schemes. To show the validity as well as the practicality of the developed code, five different test cases are presented, and the results are compared with some analytical solutions and experimental data. Favorable agreements have been achieved in all cases.