Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions

Antoine, Xavier; Besse, Christophe; Mouysset, Vincent

doi:10.1090/S0025-5718-04-01631-X

Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions
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by Xavier Antoine, Christophe Besse and Vincent Mouysset PDF

Math. Comp. 73 (2004), 1779-1799 Request permission

Abstract:

This paper adresses the construction and study of a Crank-Nicolson-type discretization of the two-dimensional linear Schrödinger equation in a bounded domain $\Omega$ with artificial boundary conditions set on the arbitrarily shaped boundary of $\Omega$. These conditions present the features of being differential in space and nonlocal in time since their definition involves some time fractional operators. After having proved the well-posedness of the continuous truncated initial boundary value problem, a semi-discrete Crank-Nicolson-type scheme for the bounded problem is introduced and its stability is provided. Next, the full discretization is realized by way of a standard finite-element method to preserve the stability of the scheme. Some numerical simulations are given to illustrate the effectiveness and flexibility of the method.

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