Numerical multistep methods for the efficient solution of quantummechanics and related problems

摘要

In this paper we present the recent development in the numerical integration ofthe Schrodinger equation and related systems of ordinary differential equations withoscillatory solutions, such as the N-body problem. We examine several types of multistepmethods (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 theSchrodinger 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. Figuresare presented and analyzed and some general conclusions are made. Code written in Mapleis given for the development of all methods analyzed in this paper. Also the subroutineswritten in Matlab, that concern the integration of the methods, are presented.