SIAM Journal on Numerical Analysis · 2003 · 23 citations · 16 references
Natural SciencesMathematicsPhysicsComputational PhysicsDifferential EquationsDiscrete MathematicsApproximation MethodsBoundary ConditionsBoundary-value ProblemsConvergence AnalysisFinite Difference MethodsSchrödinger EquationAbsorbing Boundary ConditionsArtificial Boundary ConditionsFinite Difference Solution Of Partial Differential EquationsNumerical Solution Of The Schrödinger EquationPartial Differential Equation SolversPartial Differential Equation SystemsStability Of Finite Difference SchemesTransparent Boundary Conditions
When a partial differential equation in an unbounded domain is solved numerically, it is necessary to introduce artificial boundary conditions. In this paper, a general class of absorbing boundary conditions is constructed for one-dimensional Schrödinger-type equations discretized in space by finite differences. For this, rational approximations to the transparent boundary conditions are used. We study the simplest case in detail, obtaining an estimate for the full discrete error and showing that the discrete problem is weakly unstable. Moreover, we show numerically that the discrete problems associated to higher order absorbing boundary conditions are more unstable. Several numerical experiments confirm the results previously obtained.
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