Multi-symplectic Runge-Kutta-Nyström methods for nonlinear Schrödinger equations with variable coefficients

  • Authors:

  • Jialin Hong;Xiao-yan Liu;Chun Li

  • Affiliations:

  • State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Acade ...;State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Acade ...;State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Acade ...

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2007

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Abstract

In this paper, we consider Runge-Kutta-Nystrom (RKN) methods applied to nonlinear Schrodinger equations with variable coefficients (NLSEvc). Concatenating symplectic Nystrom methods in spatial direction and symplectic Runge-Kutta methods in temporal direction for NLSEvc leads to multi-symplectic integrators, i.e. to numerical methods which preserve the multi-symplectic conservation law (MSCL), we present the corresponding discrete version of MSCL. It is shown that the multi-symplectic RKN methods preserve not only the global symplectic structure in time, but also local and global discrete charge conservation laws under periodic boundary conditions. We present a (4-order) multi-symplectic RKN method and use it in numerical simulation of quasi-periodically solitary waves for NLSEvc, and we compare the multi-symplectic RKN method with a non-multi-symplectic RKN method on the errors of numerical solutions, the numerical errors of discrete energy, discrete momentum and discrete charge. The precise conservation of discrete charge under the multi-symplectic RKN discretizations is attested numerically. Some numerical superiorities of the multi-symplectic RKN methods are revealed.