Osc Methods for Schr Odinger Equation
نویسنده
چکیده
Two discrete-time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time-dependent Schrr odinger equation in two space variables. These are Crank-Nicolson and alternating direction implicit (ADI) schemes employing C 1 piecewise polynomial spaces of arbitrary degree 3 in each space variable. The stability of the schemes is examined and optimal order a priori L 2-and H 1-error estimates at each time step are derived. Parallel implementation of the ADI scheme is discussed.
منابع مشابه
A New High Order Closed Newton-Cotes Trigonometrically-fitted Formulae for the Numerical Solution of the Schrodinger Equation
In this paper, we investigate the connection between closed Newton-Cotes formulae, trigonometrically-fitted methods, symplectic integrators and efficient integration of the Schr¨odinger equation. The study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry (see the relevant lit...
متن کاملOdinger-type Equations in One Space Variable
We examine the use of orthogonal spline collocation for the semi-discretization of the cubic Schr odinger equation and the two-dimensional parabolic equation of Tappert. In each case, an optimal order L 2 estimate of the error in the semidiscrete approximation is derived. For the cubic Schr odinger equation, we present the results of numerical experiments in which the integration in time is p...
متن کاملA Modulation Method for Self-focusing in the Perturbed Critical Nonlinear Schr Odinger Equation
In this Letter we introduce a systematic perturbation method for analyzing the e ect of small perturbations on critical self focusing by reducing the perturbed critical nonlinear Schrodinger equation (PNLS) to a simpler system of modulation equations that do not depend on the transverse variables. The modulation equations can be further simpli ed, depending on whether PNLS is power conserving ...
متن کاملOn the Use of Schr odinger ' s Equation in the Analytic Determination of Horn Re ectance
The ared horn is modeled assuming that Webster's horn equation is satis ed. Any shape can be assumed for the wavefront within the horn. This paper presents a technique for solving Webster's horn equation as follows: The equation is converted to the form of the Schrodinger wave equation used for one-dimensional particle scattering. The horn characteristics can then be obtained directly in terms...
متن کاملFactorization of Scattering Matrices Due to Partitioning of Potentials in One-dimensional Schr Odinger-type Equations
The one-dimensional Schrodinger equation and two of its generalizations are considered, as they arise in quantum mechanics, wave propagation in a nonhomogeneous medium, and wave propagation in a nonconservative medium where energy may be absorbed or generated. Generically, the zero-energy transmission coe cient vanishes when the potential is nontrivial, but in the exceptional case this coe cie...
متن کامل