2004
DOI: 10.1090/s0025-5718-04-01631-x
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Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions

Abstract: 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 Ω with artificial boundary conditions set on the arbitrarily shaped boundary of Ω. 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-disc… Show more

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Cited by 89 publications
(67 citation statements)
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References 24 publications
(47 reference statements)
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“…In this goal, we first have to define the E-quasi hyperbolic, elliptic and glancing zones [7], with E = (1, 2). Definition 3.1.…”
Section: Remark About the Convergence Of Cswr Methods In Real Timementioning
confidence: 99%
“…In this goal, we first have to define the E-quasi hyperbolic, elliptic and glancing zones [7], with E = (1, 2). Definition 3.1.…”
Section: Remark About the Convergence Of Cswr Methods In Real Timementioning
confidence: 99%
“…[41,42,43]. Let us finally remark that applications to generalized Schrödinger equations could also be developed by adapting the methods developed in [2,44]. …”
Section: Resultsmentioning
confidence: 99%
“…The natural and mathematically correct types of artificial boundary conditions are approximate TBCs (either non-local or local). It is well-known that their construction is not trivial since such approximate TBCs should ensure both stability of the resulting method in the bounded domain and small numerical reflections from the artificial boundary, see [1]- [4], [7,9,10,13], [15]- [17], [19,20].…”
Section: Introductionmentioning
confidence: 99%