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Volume 15, Issue 4
Stochastic Symplectic Exponential Runge-Kutta Integrators for Semilinear SDEs and Applications to Stochastic Nonlinear Schrödinger Equation

Feng Wang, Qiang Ma & Xiaohua Ding

East Asian J. Appl. Math., 15 (2025), pp. 716-740.

Published online: 2025-06

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  • Abstract

Symplecticity is a significant property of stochastic Hamiltonian systems, and the symplectic methods are very attractive. Compared with the classical non-exponential Runge-Kutta methods, the exponential Runge-Kutta methods are more suitable for stiff problems. Therefore, the focus of this paper is on constructing stochastic symplectic exponential Runge-Kutta (SSERK) integrators for semilinear stochastic differential equations (SDEs) driven by multiplicative noise. The first is to establish the symplectic conditions of stochastic exponential Runge-Kutta (SERK) methods. It can be found that when the stiffness matrix is 0, these conditions will degenerate into the symplectic conditions of classical stochastic Runge-Kutta methods. Based on this idea, we construct a class of SSERK integrators with remarkable properties of structure-preservation. In addition, we verify the existence of the quadratic first integral of the stochastic Hamiltonian system and investigate the connection between preserving both the quadratic first integral and the symplectic structure by the SERK methods. Numerical experiments demonstrate a better structure-preserving ability and a higher accuracy of the SSERK integrators in solving the considered semilinear SDEs than the corresponding stochastic symplectic Runge-Kutta integrators. Excitingly, the SSERK integrators perform well when applied to the temporal discretization of stochastic nonlinear Schrödinger equation.

  • AMS Subject Headings

37N30, 65C20, 65C30, 78A48

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-15-716, author = {Wang , FengMa , Qiang and Ding , Xiaohua}, title = {Stochastic Symplectic Exponential Runge-Kutta Integrators for Semilinear SDEs and Applications to Stochastic Nonlinear Schrödinger Equation}, journal = {East Asian Journal on Applied Mathematics}, year = {2025}, volume = {15}, number = {4}, pages = {716--740}, abstract = {

Symplecticity is a significant property of stochastic Hamiltonian systems, and the symplectic methods are very attractive. Compared with the classical non-exponential Runge-Kutta methods, the exponential Runge-Kutta methods are more suitable for stiff problems. Therefore, the focus of this paper is on constructing stochastic symplectic exponential Runge-Kutta (SSERK) integrators for semilinear stochastic differential equations (SDEs) driven by multiplicative noise. The first is to establish the symplectic conditions of stochastic exponential Runge-Kutta (SERK) methods. It can be found that when the stiffness matrix is 0, these conditions will degenerate into the symplectic conditions of classical stochastic Runge-Kutta methods. Based on this idea, we construct a class of SSERK integrators with remarkable properties of structure-preservation. In addition, we verify the existence of the quadratic first integral of the stochastic Hamiltonian system and investigate the connection between preserving both the quadratic first integral and the symplectic structure by the SERK methods. Numerical experiments demonstrate a better structure-preserving ability and a higher accuracy of the SSERK integrators in solving the considered semilinear SDEs than the corresponding stochastic symplectic Runge-Kutta integrators. Excitingly, the SSERK integrators perform well when applied to the temporal discretization of stochastic nonlinear Schrödinger equation.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.2023-265.230624}, url = {http://global-sci.org/intro/article_detail/eajam/24194.html} }
TY - JOUR T1 - Stochastic Symplectic Exponential Runge-Kutta Integrators for Semilinear SDEs and Applications to Stochastic Nonlinear Schrödinger Equation AU - Wang , Feng AU - Ma , Qiang AU - Ding , Xiaohua JO - East Asian Journal on Applied Mathematics VL - 4 SP - 716 EP - 740 PY - 2025 DA - 2025/06 SN - 15 DO - http://doi.org/10.4208/eajam.2023-265.230624 UR - https://global-sci.org/intro/article_detail/eajam/24194.html KW - Semilinear stochastic differential equations, stochastic symplectic exponential Runge-Kutta integrators, symplecticity, quadratic first integral, stochastic nonlinear Schrödinger equation. AB -

Symplecticity is a significant property of stochastic Hamiltonian systems, and the symplectic methods are very attractive. Compared with the classical non-exponential Runge-Kutta methods, the exponential Runge-Kutta methods are more suitable for stiff problems. Therefore, the focus of this paper is on constructing stochastic symplectic exponential Runge-Kutta (SSERK) integrators for semilinear stochastic differential equations (SDEs) driven by multiplicative noise. The first is to establish the symplectic conditions of stochastic exponential Runge-Kutta (SERK) methods. It can be found that when the stiffness matrix is 0, these conditions will degenerate into the symplectic conditions of classical stochastic Runge-Kutta methods. Based on this idea, we construct a class of SSERK integrators with remarkable properties of structure-preservation. In addition, we verify the existence of the quadratic first integral of the stochastic Hamiltonian system and investigate the connection between preserving both the quadratic first integral and the symplectic structure by the SERK methods. Numerical experiments demonstrate a better structure-preserving ability and a higher accuracy of the SSERK integrators in solving the considered semilinear SDEs than the corresponding stochastic symplectic Runge-Kutta integrators. Excitingly, the SSERK integrators perform well when applied to the temporal discretization of stochastic nonlinear Schrödinger equation.

Wang , FengMa , Qiang and Ding , Xiaohua. (2025). Stochastic Symplectic Exponential Runge-Kutta Integrators for Semilinear SDEs and Applications to Stochastic Nonlinear Schrödinger Equation. East Asian Journal on Applied Mathematics. 15 (4). 716-740. doi:10.4208/eajam.2023-265.230624
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