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

摘要

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.