On uniformly accurate high-order Boussinesq difference equations for water waves

A new accurate finite-difference (AFD) numerical method is developed specifically for solving high-order Boussinesq (HOB) equations. The method solves the water-wave flow with much higher accuracy compared to the standard finite-difference (SFD) method for the same computer resources. It is first developed for linear water waves and then for the nonlinear problem. It is presented for a horizontal bottom, but can be used for variable depth as well. The method can be developed for other equations as long as they use Padé approximation, for example extensions of the parabolic equation for acoustic wave problems. Finally, the results of the new method and the SFD method are compared with the accurate solution for nonlinear progressive waves over a horizontal bottom that is found using the stream function theory. The agreement of the AFD to the accurate solution is found to be excellent compared to the SFD solution.