Exact Boundary Condition for Semi-discretized Schrödinger Equation and Heat Equation in a Rectangular Domain

A convolution type exact/transparent boundary condition is proposed for simulating a semi-discretized linear Schrödinger equation on a rectangular computational domain. We calculate the kernel functions for a single source problem, and subsequently those over the rectangular domain. Approximate kernel functions are pre-computed numerically from discrete convolutionary equations. With a Crank–Nicolson scheme for time integration, the resulting approximate boundary conditions effectively suppress boundary reflections, and resolve the corner effect. The proposed boundary treatment, with a parameter modified, applies readily to a semi-discretized heat equation.