Numerical analysis of a multi-symplectic scheme for a strongly coupled Schrödinger system

Abstract

In this paper, we analyze a multi-symplectic scheme for a strongly coupled nonlinear Schrödinger equations. We derive a general box scheme which is equivalent to the multi-symplectic scheme by reduction method. Based on the general box scheme, we prove that the multi-symplectic scheme preserves not only the multi-symplectic structure of the equation but also conservation law of mass. In general, the multi-symplectic schemes are not conservative to energy in the nonlinear case, so it is difficult to obtain the estimates of numerical solutions in ‖ · ‖ ∞ norm. Hence proofs of convergence and stability are difficult for multi-symplectic schemes of nonlinear equations. A deduction argument and the energy analysis method are used to prove that the numerical solution is stable for initial values, and second order convergent to the exact solutions in ‖ · ‖ 2 norm. A fixed point theorem is introduced and used to prove the unique solvability of the numerical solutions.