Transparent boundary conditions - the pole condition approach

Abstract

A new approach to derive transparent boundary conditions (TBCs) for wave, Schr¨odinger and drift-diffusion equations is presented. It relies on the pole condition approach and distinguishes physical reasonable and unreasonable solutions by the location of the singularities of the spatial Laplace transform U of the exterior solution. By the condition that U is analytic in some region TBCs are established. To realize the pole condition numerically, a Möbius transform is used to map the region of analyticity to the unit disc. There the Laplace transform is expanded in a power series. The equations coupling the coefficients of the power series with the interior provide the TBC. Numerical result for the damped wave equation show that the error introduced by truncating the power series decays exponentially in the number of coefficients.