具有时变系数的随机非线性薛定谔方程中点格式的收敛性分析
Convergence Analysis of the Midpoint Scheme for Stochastic Nonlinear Schrödinger Equations with Time-Dependent Coefficients
摘要: 对于由加性噪声驱动的且具有时变函数项的随机非线性薛定谔方程，其解析解的构造具有本质困难，这表明数值方法是研究该方程性质的必要手段，其核心问题就是分析数值格式的收敛阶。本文利用中点格式的时间半离散方法，在空间具有充分正则性的条件下，结合截断方法和Gronwall不等式证明其在非全局Lipschitz情况下的概率收敛阶为1阶。
Abstract: For the stochastic nonlinear Schrödinger equation driven by additive noise and with time-dependent coefficients, constructing its analytical solution is intrinsically difficult. This indicates that numerical methods are a necessary means to study the properties of this equation, and the core issue is to analyze the convergence order of the numerical scheme. In this paper, by employing the time semi-discrete method of the midpoint scheme, under the condition that the space has sufficient regularity, and combining the truncation method with Gronwall’s inequality, it is proved that its probability convergence order is 1st order in the case of non-global Lipschitz.
参考文献
|
[1]
|
Kagan, Y., Muryshev, A.E. and Shlyapnikov, G.V. (1998) Collapse and Bose-Einstein Condensation in a Trapped Bose Gas with Negative Scattering Length. Physical Review Letters, 81, 933-937. [Google Scholar] [CrossRef]
|
|
[2]
|
Cazenave, T. (2003) Semilinear Schrödinger Equations. American Mathematical Society. [Google Scholar] [CrossRef]
|
|
[3]
|
Liu, J. (2013) Order of Convergence of Splitting Schemes for Both Deterministic and Stochastic Nonlinear Schrödinger Equations. SIAM Journal on Numerical Analysis, 51, 1911-1932. [Google Scholar] [CrossRef]
|
|
[4]
|
Cui, J., Hong, J., Liu, Z. and Zhou, W. (2017) Stochastic Symplectic and Multi-Symplectic Methods for Nonlinear Schrödinger Equation with White Noise Dispersion. Journal of Computational Physics, 342, 267-285. [Google Scholar] [CrossRef]
|
|
[5]
|
Hong, J., Miao, L. and Zhang, L. (2019) Convergence Analysis of a Symplectic Semi-Discretization for Stochastic NLS Equation with Quadratic Potential. Discrete & Continuous Dynamical Systems B, 24, 4295-4315. [Google Scholar] [CrossRef]
|
|
[6]
|
Miao, L., Yu, J. and Qiu, L. (2025) On Global Existence for the Stochastic Nonlinear Schrödinger Equation with Time-Dependent Linear Loss/Gain. Electronic Research Archive, 33, 3571-3583. [Google Scholar] [CrossRef]
|
|
[7]
|
de Bouard, A. and Debussche, A. (2006) Weak and Strong Order of Convergence of a Semidiscrete Scheme for the Stochastic Nonlinear Schrodinger Equation. Applied Mathematics and Optimization, 54, 369-399. [Google Scholar] [CrossRef]
|