2006
DOI: 10.4310/cms.2006.v4.n4.a4
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On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I

Abstract: Abstract. We consider initial-boundary value problems for a generalized time-dependent Schrödinger equation in 1D on the semi-axis and in 2D on a semi-bounded strip. For Crank-Nicolson finite-difference schemes, we suggest an alternative coupling to approximate transparent boundary conditions and present a condition ensuring unconditional stability. In the case of discrete transparent boundary conditions, we revisit the statement and the proof of stability together with the derivation of the conditions.

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Cited by 23 publications
(41 citation statements)
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“…The derivation of transparent numerical boundary conditions for (67) was performed in [EA01] so we shall not reproduce it here. Approximate (namely, absorbing) numerical boundary conditions for (67) are proposed and studied in [EA01, AES03, AAB + 08, DZ06]. We also refer to [Sze04, AAB + 08] and references therein for the construction of absorbing boundary conditions for the nonlinear Schrödinger equation.…”
Section: Numerical Schemes For Dispersive Equationsmentioning
confidence: 99%
“…The derivation of transparent numerical boundary conditions for (67) was performed in [EA01] so we shall not reproduce it here. Approximate (namely, absorbing) numerical boundary conditions for (67) are proposed and studied in [EA01, AES03, AAB + 08, DZ06]. We also refer to [Sze04, AAB + 08] and references therein for the construction of absorbing boundary conditions for the nonlinear Schrödinger equation.…”
Section: Numerical Schemes For Dispersive Equationsmentioning
confidence: 99%
“…Since the formulation of the boundary condition is independent of the cross section's shape, a cylindrical waveguide is depicted just for illustration. (42) where denotes at . The time is discretized in length of several picoseconds (as in a normal gyrotron cavity simulation).…”
Section: Numerical Validationmentioning
confidence: 99%
“…The reasons for the observed instability of high-order polynomials should be numerical, most probably due to the used finite-difference process and improper space-time discretization [40]- [42]. Exact reasons are still under investigation.…”
Section: Numerical Validationmentioning
confidence: 99%
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“…In Part I of this study (see [10]), a new form of the approximate TBCs has been suggested, with a non-local operator S governing properties of the schemes. The uniform-in-time stability bounds in L 2 have been proved under suitable condition (inequality) on S, for non-uniform meshes in space and time.…”
Section: Introductionmentioning
confidence: 99%