Symplectic Euler Method for Nonlinear High Order Schr ¨ odinger Equation with a Trapped Term
Symplectic Euler Method for Nonlinear High Order Schr ¨ odinger Equation with a Trapped Term
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DOI：
10.4208/aamm.09-m0929
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发表时间：
2009-06
影响因子：
1.4
通讯作者：
Fangfang Fu;L. Kong;Lan Wang
中科院分区：
文献类型：
--
作者：
Fangfang Fu;L. Kong;Lan Wang
抽象的。在本文中，我们为一类带有捕获项的高阶薛定谔方程建立了一系列辛积分器。首先，我们找到它的辛结构，并通过空间离散化将其简化为有限维哈密顿系统。然后将辛欧拉方法应用到哈密顿系统中，证明该格式不仅保留了原系统的辛几何结构，而且与一般隐式辛格式不同，不需要求解耦合非线性代数方程。研究了辛欧拉格式的线性稳定性和数值解的误差。它表明半显式方案是条件稳定的，时间上一阶精度，空间上二阶精度。数值试验表明辛积分器比非辛积分器（例如后向欧拉积分器）更有效。 AMS 主题分类：65M06、65M12、65Z05、70H15
Abstract. In this paper, we establish a family of symplectic integrators for a classof high order Schrodinger equations with trapped terms. First, we find its symplec-¨tic structure and reduce it to a finite dimensional Hamilton system via spatial dis-cretization. Then we apply the symplectic Euler method to the Hamiltonian system.It is demonstrated that the scheme not only preserves symplectic geometry struc-ture of the original system, but also does not require to resolve coupled nonlinearalgebraic equations which is different with the general implicit symplectic schemes.The linear stability of the symplectic Euler scheme and the errors of the numericalsolutions are investigated. It shows that the semi-explicit scheme is conditionallystable, first order accurate in time and 2 l th order accuracy in space. Numerical testssuggest that the symplectic integrators are more effective than non-symplectic ones,such as backward Euler integrators. AMS subject classifications : 65M06, 65M12, 65Z05, 70H15