二维非线性Schr\"{o}dinger方程显式格式的最大模误差分析

本文讨论了数值求解二维非线性Schr\"{o}dinger方程周期边值问题的Du Fort-Frankel格式和蛙跳格式. 以解函数的一个广义时间导数作为独立变量, 将非线性方程初边值问题改写成一个混合方程组形式, 应用我们最新提出的离散能量技巧讨论这两个三层显式格式的收敛性. 分析表明, 在必要的网格条件下, 差分解在最大模意义下二阶收敛. 数值算例验证了理论分析结果.

Maximum norm error analysis of explicit schemes for twodimensional nonlinear SchrSdinger equations

Two three-level explicit schemes, including the Du Fort-Frankel and leap-frog schemes, are considered for approximating two-dimensional cubic nonlinear Schrodinger equations with periodic boundary conditions. Based on a mixed formulation of the nonlinear problem, where a generalized time derivative of the solution is introduced as a new unknown function, the recently suggested energy technique is applied to analyze the three- level explicit schemes. It is shown that, under the appropriate mesh conditions, the numerical solutions are convergent in the maximum norm. Numerical experiments are presented to support the theoretical results.