A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations. (English) Zbl 1364.65178
Summary: In this review article we discuss different techniques to solve numerically the time-dependent Schrödinger equation on unbounded domains. We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.
MSC:
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65R20 | Numerical methods for integral equations |
| 65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |
| 35Q40 | PDEs in connection with quantum mechanics |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 45K05 | Integro-partial differential equations |
