Efficient elliptic solvers for the mild-slope equation using the multigrid technique

Abstract

The multigrid method is used to solve very efficiently the elliptic form of the mild-slope equation for water wave propagation over large areas in the presence of currents, taking into account the combined effects of shoaling, refraction, diffraction and wave breaking. Wave reflections may also be taken into account but in this case additional computational resources are required. The present scheme offers significant advantages over other existing elliptic and hyperbolic solution techniques for the mild-slope equation with regard to computational efficiency and speed. The original equation as well as its variant which includes the effects of wave-current interaction are first recast into forms which can be readily handled by the multigrid method. Solutions of the governing equations are successively obtained for a number of increasingly coarser grid meshes, using the Gauss-Seidel Iterative Method for all grid meshes apart from the coarsest for which the Gauss Elimination Method may also be used. The main advantage of this approach is that, while traditional elliptic and hyperbolic solution schemes for the mild-slope equation require a large number of grid points per wavelength, the present scheme requires no more than two to three points, thus reducing overall computational effort. Moreover, the solution procedure is as efficiently computationally as the parabolic approximations for the mild-slope equation, without imposing any of the constraints of those schemes. Verification of the model for a number of test cases confirms that it is stable, highly accurate and economical to use.