Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions
Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions
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DOI：
10.1016/j.jcp.2013.03.007
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发表时间：
2013-06
期刊：
影响因子：
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通讯作者：
Ting-chun Wang;B. Guo;Qiubin Xu
中科院分区：
文献类型：
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作者：
Ting-chun Wang;B. Guo;Qiubin Xu
本文对二维非线性薛定谔方程提出了一种能量守恒的四阶紧致差分格式，并在不对网格比作任何限制的情况下，得到了离散L2范数下，时间步长为τ，网格尺寸为h时的最优收敛速度为O（h4+τ2）阶.除了能量法的标准技巧外，还提出了一种新的技巧和一些重要的引理来证明高阶收敛性。为了避免外迭代，推导了一种线性化的能量守恒紧致差分格式。最后给出了数值例子来支持理论分析。
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.