A perfectly matched layer formulation for the nonlinear shallow water equations models: The split equation approach

Perfectly matched layer (PML) equations for the treatment of boundary conditions are constructed for the two dimensional linearized shallow-water equations. The method uses the splitting technique, i.e. the absorbing layer equations are obtained by splitting the governing equations in the coordinate directions and absorbing coefficients are introduced in each split equation. The shallow water equations are discretized using the explicit Miller Pierce finite difference approach. The method was tested and the transparency of the boundaries was demonstrated on the nonlinear shallow-water equations including the Coriolis factor on a limited-area rectangular domain for a Gaussian pulse with convective mean flow. Different scenarios involving various values of the mean flow speeds, depth of the PML layer and different PML absorbtion coefficients are tested. The numerical results indicate that outgoing waves are leaving the domain without perturbing the flow in the physical domain. No filters were used in the numerical experiments which included both a stationary as well as a convected Gaussian. The exterior domain was ended using well posed boundary conditions for the shallow water equations. The L2 error in the height field between the PML layer domain and a reference solution along a line located inside the interior domain was computed also for the case using only characteristic boundary conditions as well as the second order Engquist and Majda boundary conditions for comparison purposes. The efficacy of the PML scheme over the other two aforementioned schemes for the nonlinear shallow water equations was confirmed . PML layers of increasing thickness yielded better results than either the characteristic treatment or the second order Majda Engquist absorbing boundary conditions.