Volume 1, Issue 5
Symplectic Euler Method for Nonlinear High Order Schrödinger Equation with a Trapped Term

Adv. Appl. Math. Mech., 1 (2009), pp. 699-710.

Published online: 2009-01

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• Abstract

In this paper, we establish a family of symplectic integrators for a class of high order Schrödinger equations with trapped terms. First, we find its symplectic structure and reduce it to a finite dimensional Hamilton system via spatial discretization. Then we apply the symplectic Euler method to the Hamiltonian system. It is demonstrated that the scheme not only preserves symplectic geometry structure of the original system, but also does not require to resolve coupled nonlinear algebraic equations which is different from the general implicit symplectic schemes. The linear stability of the symplectic Euler scheme and the errors of the numerical solutions are investigated. It shows that the semi-explicit scheme is conditionally stable, first order accurate in time and $2l^{th}$ order accuracy in space. Numerical tests suggest that the symplectic integrators are more effective than non-symplectic ones, such as backward Euler integrators.

• Keywords

Symplectic Euler integrator, high order Schrödinger equation, stability, trapped term.

65M06, 65M12, 65Z05, 70H15

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@Article{AAMM-1-699, author = {Fangfang and Fu and and 20565 and and Fangfang Fu and Linghua and Kong and and 20566 and and Linghua Kong and Lan and Wang and and 20567 and and Lan Wang}, title = {Symplectic Euler Method for Nonlinear High Order Schrödinger Equation with a Trapped Term}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2009}, volume = {1}, number = {5}, pages = {699--710}, abstract = {

In this paper, we establish a family of symplectic integrators for a class of high order Schrödinger equations with trapped terms. First, we find its symplectic structure and reduce it to a finite dimensional Hamilton system via spatial discretization. Then we apply the symplectic Euler method to the Hamiltonian system. It is demonstrated that the scheme not only preserves symplectic geometry structure of the original system, but also does not require to resolve coupled nonlinear algebraic equations which is different from the general implicit symplectic schemes. The linear stability of the symplectic Euler scheme and the errors of the numerical solutions are investigated. It shows that the semi-explicit scheme is conditionally stable, first order accurate in time and $2l^{th}$ order accuracy in space. Numerical tests suggest that the symplectic integrators are more effective than non-symplectic ones, such as backward Euler integrators.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.09-m0929}, url = {http://global-sci.org/intro/article_detail/aamm/8392.html} }
TY - JOUR T1 - Symplectic Euler Method for Nonlinear High Order Schrödinger Equation with a Trapped Term AU - Fu , Fangfang AU - Kong , Linghua AU - Wang , Lan JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 699 EP - 710 PY - 2009 DA - 2009/01 SN - 1 DO - http://doi.org/10.4208/aamm.09-m0929 UR - https://global-sci.org/intro/article_detail/aamm/8392.html KW - Symplectic Euler integrator, high order Schrödinger equation, stability, trapped term. AB -

In this paper, we establish a family of symplectic integrators for a class of high order Schrödinger equations with trapped terms. First, we find its symplectic structure and reduce it to a finite dimensional Hamilton system via spatial discretization. Then we apply the symplectic Euler method to the Hamiltonian system. It is demonstrated that the scheme not only preserves symplectic geometry structure of the original system, but also does not require to resolve coupled nonlinear algebraic equations which is different from the general implicit symplectic schemes. The linear stability of the symplectic Euler scheme and the errors of the numerical solutions are investigated. It shows that the semi-explicit scheme is conditionally stable, first order accurate in time and $2l^{th}$ order accuracy in space. Numerical tests suggest that the symplectic integrators are more effective than non-symplectic ones, such as backward Euler integrators.

Fangfang Fu, Linghua Kong & Lan Wang. (1970). Symplectic Euler Method for Nonlinear High Order Schrödinger Equation with a Trapped Term. Advances in Applied Mathematics and Mechanics. 1 (5). 699-710. doi:10.4208/aamm.09-m0929
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