Transparent boundary conditions based on the Pole Condition for time-dependent, two-dimensional problems
Provides a new method for transparent boundary conditions in 2D time-dependent simulations, though instabilities limit its applicability to certain equation types.
The pole condition approach for transparent boundary conditions is extended to 2D time-dependent problems, achieving super-algebraic error decay with few coefficients. It works well for Schrödinger and drift-diffusion equations but shows instabilities for wave and Klein-Gordon equations.
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 Schrödinger 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.