An Embedded Boundary Method for the Wave Equation with Discontinuous Coefficients

Abstract

A second order accurate embedded boundary method for the two-dimensional wave equation with discontinuous wave propagation speed is described. The wave equation is discretized on a Cartesian grid with constant grid size and the interface (across which the wave speed is discontinuous) is allowed to intersect the mesh in an arbitrary fashion. By using ghost points on either side of the interface, previous embedded boundary techniques for the Neumann and Dirichlet problems are generalized to satisfy the jump conditions across the interface to second order accuracy. The resulting discretization of the jump conditions has the desirable property that each ghost point can be updated independently of all other ghost points, resulting in a fully explicit time-integration method. We prove that the one-dimensional restriction of the method is stable without damping for arbitrary locations of the interface relative to the grid. For the two-dimensional case, the previously developed fourth order $A^T A$-dissipation is generalized to handle jump conditions. We demonstrate that this operator provides sufficient stabilization to enable long-time simulations while being weak enough to preserve the accuracy of the solution. Numerical examples are given where the method is used to study electromagnetic scattering of a plane wave by a dielectric cylinder. The numerical solutions are evaluated against the analytical solution due to Mie, and pointwise second order accuracy is confirmed.