Comparison between consistent coupled-mode system and eigenfunction matching method for solving water wave scattering

Abstract

Both the consistent coupled-mode system (CCMS) and theeigenfunction matching method (EMM) are well-known models for simulating the propagation of small-amplitude waterwaves over variable bathymetry. In this study, a thoroughcomparison is performed through numerical experiments. Forthe CCMS, a bottom-sloping mode is coupled in the mildslope equation with evanescent modes, then the CCMS arediscretized by the finite-element method with high-ordershape functions. For the EMM, the bottom profile is approximated in terms of successive flat shelves separated byabrupt steps, and then eigensolutions on the shelves arematched by the conservation of mass and momentum. Toperform error analysis, numerical solutions are compared withRoseau’s analytical solution and the semi-analytical solutionsof the integral equation method. Numerical results indicatethat the CCMS and EMM are accurate up to six and fourdecimal places, respectively. On the other hand, the EMM ismore efficient for short waves because multiple waves can beapproximated by few shelves. In addition, improvements intheir accuracy over the mild-slope system without bottomsloping mode are shown to be significant.