Numerical multistep methods for the efficient solution of quantum mechanics and related problems

作者：Anastassi Z A; Simos T E^*

来源：Physics Reports-Review Section of Physics Letters, 2009, 482: 1-240.

DOI：10.1016/j.physrep.2009.07.005

• Ordinary differential equations
• Initial value problems
• Oscillatory
• Trigonometric fitting
• Exponential fitting
• Phase-lag
• Phase fitting
• Schrodinger equation
• N-body problem
• Orbital problems
• Symmetric
• Multistep
• Explicit
• Implicit
• Predictor-corrector
• Hybrid
• Linear
• Exponential order
• Phase order
• Energy
• Local truncation error
• Stability analysis
• Interval of periodicity
• Region of periodicity

In this paper we present the recent development in the numerical integration of the Schrodinger equation and related systems of ordinary differential equations with oscillatory solutions, such as the N-body problem. We examine several types of multistep methods (explicit, implicit, predictor-corrector, hybrid) and several properties (P-stability, trigonometric fitting of various orders, phase fitting, high phase-lag order, algebraic order). We analyze the local truncation error and the stability of the methods. The error for the Schrodinger equation is also presented, which reveals the relation of the error to the energy. The efficiency of the methods is evaluated through the integration of five problems. Figures are presented and analyzed and some general conclusions are made. Code written in Maple is given for the development of all methods analyzed in this paper. Also the subroutines written in Matlab, that concern the integration of the methods, are presented.

• 出版日期2009-10