A consistent analysis of Boussinesq-type water wave equations in continuous and discrete form

Abstract

Previous derivations of the continuum Boussinesq-type equations are investigated and compared through a nondimensional order-of-approximation to establish a unique equation set. The resulting continuum equation set is transformed into a highly accurate discrete equation set whereby numerical falsification of the third order dispersive term found in the continuum equation of motion is eliminated or minimized. The resulting discrete form is compared to each representative continuum derivative counterpart by a nondimensional order of approximation, insuring a consistency between the continuum and discrete form. A computer algorithm is developed employing an implicit, three-level-in-time numerical scheme that solves the resulting equations for constant water depth and constant sloping bottom conditions. The numerical results are compared to laboratory data, to verify the accuracy of the algorithm and also all methods that are employed in the study.