Optimal convergence and long-time conservation of exponential integration for Schrödinger equations in a normal or highly oscillatory regime

2020/07/01 by Wang, Bin, Jiang, Yaolin
#65M70 #65P10 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2007.00592

Abstract

In this paper, we formulate and analyse exponential integrations when applied to nonlinear Schrödinger equations in a normal or highly oscillatory regime. A kind of exponential integrators with energy preservation, optimal convergence and long time near conservations of actions, momentum and density will be formulated and analysed. To this end, we derive continuous-stage exponential integrators and show that the integrators can exactly preserve the energy of Hamiltonian systems. Three practical energy-preserving integrators are presented. It is shown that these integrators exhibit optimal convergence and have near conservations of actions, momentum and density over long times. A numerical experiment is carried out to support all the theoretical results presented in this paper. Some applications of the integrators to other kinds of ordinary/partial differential equations are also presented.

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