Local artificial boundary conditions for Schrödinger and heat equations by using high-order azimuth derivatives on circular artificial boundary

The aim of the paper is to design high-order artificial boundary conditions for the Schrödinger equation on unbounded domains in parallel with a treatment of the heat equation. We first introduce a circular artificial boundary to divide the unbounded definition domain into a bounded computational domain and an unbounded exterior domain. On the exterior domain, the Laplace transformation in time and Fourier series in space are applied to achieve the relation of special functions. Then the rational functions are used to approximate the relation of the special functions. Applying the inverse Laplace transformation to a series of simple rational function, we finally obtain the corresponding high-order artificial boundary conditions, where a sequence of auxiliary variables are utilized to avoid the high-order derivatives in respect to time and space. Furthermore, the finite difference method is formulated to discretize the reduced initial-boundary value problem with high-order artificial boundary conditions on a bounded computational domain. Numerical experiments are presented to illustrate the performance of our method.