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In this paper, the idea of nonstandard finite difference discretization is employed to develop two variational integrators for the nonlinear Schrödinger equation with variable coefficients. These integrators are naturally multi-symplectic, and their multi-symplectic structures are presented by the multi-symplectic form formulas. Local truncation errors and convergences of the integrators are briefly discussed. The effectiveness and efficiency of the proposed schemes, such as the convergence order, numerical stability, and the capability in preserving the norm conservation, are verified in the numerical experiments. © 2013 Liao and Ding; licensee Springer.

Nonstandard finite difference variational integrators for nonlinear Schrödinger equation with variable coefficients