Difference approximations of the Neumann problem for the second order wave equation

Stability theory and numerical experiments are presented for a finite difference method that directly discretizes the Neumann problem for the second order wave equation. Complex geometries are discretized using a Cartesian embedded boundary technique. Both second and third order accurate approximations of the boundary conditions are presented. Away from the boundary, the basic second order method can be corrected to achieve fourth order spatial accuracy. To integrate in time, we present both a second order and a fourth order accurate explicit method. The stability of the method is ensured by adding a small fourth order dissipation operator, locally modified near the boundary to allow its application at all grid points inside the computational domain. Numerical experiments demonstrate the accuracy and long-time stability of the proposed method.