Exact boundary condition for semi-discretized Schrödinger equation and heat equation in a rectangular domain. (English) Zbl 1371.65099
Summary: 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.
MSC:
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35K05 | Heat equation |
Keywords:
exact boundary condition; Schrödinger equation; heat equation; kernel function; corner effect; semidiscretization; numerical examples; Crank-Nicolson schemeReferences:
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