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

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

Fast and Accurate Computation of Time-Domain Acoustic Scattering Problems with Exact Nonreflecting Boundary Conditions

This paper is concerned with fast and accurate computation of exterior wave equations truncated via exact circular or spherical nonreflecting boundary conditions (NRBCs, which are known to be nonlocal in both time and space). We first derive analytic expressions for the underlying convolution kernels, which allow for a rapid and accurate evaluation of the convolution with $O(N_t)$ operations over $N_t$ successive time steps. To handle the onlocality in space, we introduce the notion of boundary perturbation, which enables us to handle general bounded scatters by solving a sequence of wave equations in a regular domain. We propose an efficient spectral-Galerkin solver with Newmark's time integration for the truncated wave equation in the regular domain. We also provide ample numerical results to show high-order accuracy of NRBCs and efficiency of the proposed scheme.Comment: 22 pages with 9 figure

A bootstrap method for sum-of-poles approximations

A bootstrap method is presented for finding efficient sum-of-poles approximations of causal functions. The method is based on a recursive application of the nonlinear least squares optimization scheme developed in (Alpert et al. in SIAM J. Numer. Anal. 37:1138–1164, 2000), followed by the balanced truncation method for model reduction in computational control theory as a final optimization step. The method is expected to be useful for a fairly large class of causal functions encountered in engineering and applied physics. The performance of the method and its application to computational physics are illustrated via several numerical examples

Numerical simulation of quantum waveguides

This chapter is a review of the research of the authors from the last decade and focuses on the mathematical analysis of the Schrödinger model for nano-scale semiconductor devices. We discuss transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation on a two dimensional domain. First we derive the two dimensional discrete TBCs in conjunction with a conservative Crank-Nicolson-type finite difference scheme and a compact nine-point scheme. For this difference equations we derive discrete transparent boundary conditions (DTBCs) in order to get highly accurate solutions for open boundary problems. The presented discrete boundary-valued problem is unconditionally stable and completely reflection-free at the boundary. Then, since the DTBCs for the Schrödinger equation include a convolution w.r.t. time with a weakly decaying kernel, we construct approximate DTBCs with a kernel having the form of a finite sum of exponentials, which can be efficiently evaluated by recursion. In several numerical tests we illustrate the perfect absorption of outgoing waves independent of their impact angle at the boundary, the stability, and efficiency of the proposed method. Finally, we apply inhomogeneous DTBCs to the transient simulation of quantum waveguides with a prescribed electron inflow

Numerical study of ergodicity for the overdamped Generalized Langevin Equation with fractional noise

The Generalized Langevin Equation, in history, arises as a natural fix for the rather traditional Langevin equation when the random force is no longer memoryless. It has been proved that with fractional Gaussian noise (fGn) mostly considered by biologists, the overdamped Generalized Langevin equation satisfying fluctuation-dissipation theorem can be written as a fractional stochastic differential equation (FSDE). While the ergodicity is clear for linear forces, it remains less transparent for nonlinear forces. In this work, we present both a direct and a fast algorithm respectively to this FSDE model. The strong orders of convergence are proved for both schemes, where the role of the memory effects can be clearly observed. We verify the convergence theorems using linear forces, and then present the ergodicity study of the double well potentials in both 1D and 2D setups