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| 摘要: |
| 本文研究了带五次项的非线性Schrödinger方程初边值问题.利用有限差分法构造了一个四阶紧致差分格式，证明格式在离散意义下保持原问题的两个守恒性质，即质量守恒和能量守恒.引入“抬升”技巧，运用标准的能量方法和数学归纳法建立了误差的最优估计，证明数值解在空间和时间两个方向分别具有四阶和二阶精度.数值实验对理论结果进行了验证，并与已有结果进行了对比，结果表明本文格式在保持精度相当的前提下具有更高的计算效率. |
| 关键词: 五次非线性Schrödinger方程 紧致有限差分格式 离散守恒律 最优误差估计 计算效率 |
| DOI： |
| 分类号:O241.82 |
| 基金项目:国家自然科学基金资助（11571181）；江苏省自然科学基金资助（BK20171454）；江苏省高校“青蓝工程”. |
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| A NEW COMPACT FINITE DIFFERENCE SCHEME FOR THE QUANTIC NONLINEAR SCHRÖDINGER EQUATION |
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XUE Xiang,WANG Ting-chun
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| Abstract: |
| In this paper, we study the nonlinear Schrödinger equation with a quintic terma of the initial boundary value problem. By using the finite difference method to construct a fourth-order compact finite difference scheme, we prove that the scheme preserves the total mass and energy, respectively. By introducing the lifting technique, the optimal error estimate of the proposed scheme is established by using the standard energy method and the mathematical induction. It is proved that the numerical solution has accuracy of fourth-order and second-order in space and time, respectively. Numerical experiments are given to verify the theoretical results and compared with the existing results, which show that the proposed scheme has higher computational efficiency under the condition of maintaining high accuracy. |
| Key words: quintic nonlinear Schrödinger equation compact finite difference scheme discrete conservation laws optimal error estimate computational efficiency |
