Transparent boundary conditions based on the Pole Condition for time-dependent, two-dimensional problems

TL;DR

This paper extends the pole condition method for transparent boundary conditions to two-dimensional, time-dependent problems, effectively suppressing non-physical modes with super-algebraic error decay, but faces stability issues for certain wave equations.

Contribution

It introduces a novel extension of the pole condition approach to 2D time-dependent problems, enabling efficient boundary condition approximation with super-algebraic error decay.

Findings

Effective for Schrödinger and drift-diffusion equations

Super-algebraic error decay with few coefficients

Instabilities observed for wave and Klein-Gordon equations

Abstract

The pole condition approach for deriving transparent boundary conditions is extended to the time-dependent, two-dimensional case. Non-physical modes of the solution are identified by the position of poles of the solution's spatial Laplace transform in the complex plane. By requiring the Laplace transform to be analytic on some problem dependent complex half-plane, these modes can be suppressed. The resulting algorithm computes a finite number of coefficients of a series expansion of the Laplace transform, thereby providing an approximation to the exact boundary condition. The resulting error decays super-algebraically with the number of coefficients, so relatively few additional degrees of freedom are sufficient to reduce the error to the level of the discretization error in the interior of the computational domain. The approach shows good results for the Schr\"odinger and the drift-diffusion equation but, in contrast to the one-dimensional case, exhibits instabilities for the wave and Klein-Gordon equation. Numerical examples are shown that demonstrate the good performance in the former and the instabilities in the latter case.