High-Order Exponential Operator Splitting Methods for Time-Dependent Schrödinger Equations

2008 ◽  
Vol 46 (4) ◽  
pp. 2022-2038 ◽  
Author(s):  
Mechthild Thalhammer
Keyword(s):  
Operator Splitting ◽  
Splitting Methods ◽  
High Order ◽  
Time Dependent ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Operator Splitting Methods ◽  
Exponential Operator

scholarly journals Optimized Operator-Splitting Methods in Numerical Integration of Maxwell's Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Z. X. Huang ◽  
X. L. Wu ◽  
W. E. I. Sha ◽  
B. Wu
Keyword(s):  
Numerical Integration ◽  
Time Domain ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Operator Splitting ◽  
Splitting Methods ◽  
High Order ◽  
Numerical Results ◽  
Operator Splitting Methods ◽  
The Time Domain

Optimized operator splitting methods for numerical integration of the time domain Maxwell's equations in computational electromagnetics (CEM) are proposed for the first time. The methods are based on splitting the time domain evolution operator of Maxwell's equations into suboperators, and corresponding time coefficients are obtained by reducing the norm of truncation terms to a minimum. The general high-order staggered finite difference is introduced for discretizing the three-dimensional curl operator in the spatial domain. The detail of the schemes and explicit iterated formulas are also included. Furthermore, new high-order Padé approximations are adopted to improve the efficiency of the proposed methods. Theoretical proof of the stability is also included. Numerical results are presented to demonstrate the effectiveness and efficiency of the schemes. It is found that the optimized schemes with coarse discretized grid and large Courant-Friedrichs-Lewy (CFL) number can obtain satisfactory numerical results, which in turn proves to be a promising method, with advantages of high accuracy, low computational resources and facility of large domain and long-time simulation. In addition, due to the generality, our optimized schemes can be extended to other science and engineering areas directly.

scholarly journals Convergence analysis of high-order commutator-free quasi-Magnus exponential integrators for nonautonomous linear Schrödinger equations

2020 ◽  
Author(s):  
Sergio Blanes ◽  
Fernando Casas ◽  
Cesáreo González ◽  
Mechthild Thalhammer
Keyword(s):  
Evolution Equations ◽  
High Order ◽  
Time Dependent ◽  
Schrodinger Equations ◽  
Local Error ◽  
Operator Family ◽  
Schrödinger Equations ◽  
Unbounded Operators ◽  
Exponential Integrators ◽  
Variation Of Constants Formula

Abstract This work is devoted to the derivation of a convergence result for high-order commutator-free quasi-Magnus (CFQM) exponential integrators applied to nonautonomous linear Schrödinger equations; a detailed stability and local error analysis is provided for the relevant special case where the Hamilton operator comprises the Laplacian and a regular space-time-dependent potential. In the context of nonautonomous linear ordinary differential equations, CFQM exponential integrators are composed of exponentials involving linear combinations of certain values of the associated time-dependent matrix; this approach extends to nonautonomous linear evolution equations given by unbounded operators. An inherent advantage of CFQM exponential integrators over other time integration methods such as Runge–Kutta methods or Magnus integrators is that structural properties of the underlying operator family are well preserved; this characteristic is confirmed by a theoretical analysis ensuring unconditional stability in the underlying Hilbert space and the full order of convergence under low regularity requirements on the initial state. Due to the fact that convenient tools for products of matrix exponentials such as the Baker–Campbell–Hausdorff formula involve infinite series and thus cannot be applied in connection with unbounded operators, a certain complexity in the investigation of higher-order CFQM exponential integrators for Schrödinger equations is related to an appropriate treatment of compositions of evolution operators; an effective concept for the derivation of a local error expansion relies on suitable linearisations of the evolution equations for the exact and numerical solutions, representations by the variation-of-constants formula and Taylor series expansions of parts of the integrands, where the arising iterated commutators determine the regularity requirements on the problem data.

scholarly journals Adaptive Time Propagation for Time-dependent Schrödinger equations

2020 ◽  
Vol 7 (1) ◽  
Author(s):  
Winfried Auzinger ◽  
Harald Hofstätter ◽  
Othmar Koch ◽  
Michael Quell
Keyword(s):  
Splitting Methods ◽  
Time Variable ◽  
Time Dependent ◽  
Schrodinger Equations ◽  
Local Error ◽  
Schrödinger Equations ◽  
Time Integrators ◽  
Potential Part ◽  
Adaptive Time ◽  
Time Propagation

AbstractWe compare adaptive time integrators for the numerical solution of linear Schrödinger equations where the Hamiltonian explicitly depends on time. The approximation methods considered are splitting methods, where the time variable is split off and advanced separately, and commutator-free Magnus-type methods. The time-steps are chosen adaptively based on asymptotically correct estimators of the local error in both cases. It is found that splitting methods are more efficient when the Hamiltonian naturally suggests a separation into kinetic and potential part, whereas Magnus-type integrators excel when the structure of the problem only allows to advance the time variable separately.

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