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Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions

Authors
Journal
Journal of Computational Physics
0021-9991
Publisher
Elsevier
Publication Date
Volume
243
Identifiers
DOI: 10.1016/j.jcp.2013.03.007
Keywords
  • Nonlinear Schrödinger Equation
  • Compact And Conservative Difference Scheme
  • A Prioriestimate
  • Unconditional Convergence

Abstract

Abstract In this paper, a fourth-order compact and energy conservative difference scheme is proposed for solving the two-dimensional nonlinear Schrödinger equation with periodic boundary condition and initial condition, and the optimal convergent rate, without any restriction on the grid ratio, at the order of O(h4+τ2) in the discrete L2-norm with time step τ and mesh size h is obtained. Besides the standard techniques of the energy method, a new technique and some important lemmas are proposed to prove the high order convergence. In order to avoid the outer iteration in implementation, a linearized compact and energy conservative difference scheme is derived. Numerical examples are given to support the theoretical analysis.

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