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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
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Abstract:
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。
For the stochastic nonlinear Schr？dinger 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.
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