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

doi:10.12677/aam.2026.151002

期刊菜单

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

DOI: 10.12677/aam.2026.151002, PDF, HTML, XML,
作者: 王胜宇^*, 徐宇航, 刘皓：辽宁师范大学数学学院，辽宁大连
关键词: 随机非线性薛定谔方程；时变函数；中点格式；收敛性分析；Stochastic Nonlinear Schrödinger Equation； Time-Dependent Coefficients； Midpoint Scheme； Convergence Analysis

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.

文章引用：王胜宇, 徐宇航, 刘皓. 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 [J]. 应用数学进展, 2026, 15(1): 6-12. https://doi.org/10.12677/aam.2026.151002

1. 引言

非线性薛定谔方程在物理学的许多分支中有广泛的应用[1] [2]，而在实际观测中，发现由非线性薛定谔方程预测的轨迹受到随机噪声的干扰，因此由Gauss白噪声驱动的随机非线性薛定谔方程引起国内外学者的关注，其中对随机非线性薛定谔方程的数值格式的误差分析是一个重要的研究内容。

近年来，针对于随机非线性薛定谔方程提出了不同的数值分析方法。例如，Liu [3]研究了确定性非线性薛定谔方程的Strang型分裂格式与带乘法噪声的随机非线性薛定谔方程的保质量分裂格式，并证明此格式有1阶概率收敛阶。Cui等[4]指出带白噪声色散的非线性薛定谔方程具有随机辛和多辛结构，据此提出了随机辛与多辛全离散格式，这些格式均能保持离散电荷守恒律，时间收敛阶为依概率1阶，数值实验验证了相关理论结果。Hong等[5]研究了带二次势和加性噪声的随机非线性薛定谔方程，提出了保辛结构的时间中点半离散化格式，在适当正则性条件下证明其依概率一阶收敛，并通过数值实验验证了收敛阶。

受以上文章启发，本文研究具有时变线性损耗/增益的随机非线性薛定谔方程的中点格式。由于非线性项是非全局Lipschitz的，以及方程带有时变函数，本文在非全局Lipschitz条件下，通过截断技术克服了非线性项带来的分析困难，给出了严格的误差估计，以及利用三角不等式性控制时变系数，最终得到该格式的依概率收敛阶为1阶，得到的1阶收敛阶与确定性中点格式的经典收敛阶一致，且在随机框架下已接近最优。与文献[5]中辛中点格式在加性噪声下的结果相比，本文在时变系数情形下保持了相同的收敛阶，说明中点格式在此类问题中具有较强的稳健性。

2. 中点格式

在本文中，我们研究以下具有时变线性损耗/增益的随机非线性薛定谔方程：

${\begin{cases} i d u + (Δ u + λ {| u |}^{α} u + i a (t) u) d t = d W, (t, x) \in [0, \infty) \times ℝ^{d}, \\ u (0, x) = u_{0} (x), x \in ℝ^{d}, \end{cases}$ (2.1)

其中， $λ \in ℝ$ ，非线性项指数满足 $0 < α < \frac{4}{d - 2} (d \geq 3)$ 且当 $d = 1, 2$ 时 $α > 0$ ， $a (t)$ 是定义在区间 $[0, \infty)$ 上的有界实值函数，即 $a (t) \leq C$ ，且 $a (t) > 0$ 或 $a (t) < 0$ 描述损耗或增益的强度。 $W = \sum_{i \in N} β_{i} (t, ω) ϕ e_{i} (x)$ 是复值维纳过程，其中， ${β_{i}}_{i \in N}$ 是一族定义在概率空间 $(Ω, ℱ, P, {(ℱ_{t})}_{t \geq 0})$ 上相互独立的布朗运动， ${e_{i}}_{i \in N}$ 是 $L^{2} (ℝ^{d})$ 上的一组标准正交基， $ϕ \in ℒ_{2}^{0, s}$ 是从 $L^{2} (ℝ^{d})$ 到 $H^{s} (ℝ^{d}) (s \in N)$ 的Hilbert-Schmidt算子，其范数定义是 ${‖ ϕ ‖}_{ℒ_{2}^{s}}^{2} = t r (ϕ^{*} ϕ) = \sum_{i \in N} {‖ ϕ e_{i} ‖}_{H^{s} (ℝ^{d})}^{2}$ 。Miao等[6]已验证，该方程在 $H^{1} (ℝ^{d})$ 上具有全局适定性，类似地，当初值 $u_{0}$ 和 $ϕ$ 满足充分的正则性时，即 $u_{0} \in H^{s}$ ， $ϕ \in ℒ_{2}^{s}$ 时，方程在 $H^{s} (ℝ^{d})$ 中具有全局适定性。现考虑对方程(2.1)的采用时间半离散的中点格式，

$i \frac{u^{n + 1} - u^{n}}{Δ t} + Δ u^{n + \frac{1}{2}} + λ {| u^{n + \frac{1}{2}} |}^{α} u^{n + \frac{1}{2}} + i a^{n + \frac{1}{2}} u^{n + \frac{1}{2}} = \frac{Δ_{n} W}{Δ t}, n = 0, 1, \dots, N - 1,$ (2.2)

其中， $Δ t = \frac{T}{N}$ 是时间步长， $Δ_{n} W = W (t_{n + 1}) - W (t_{n})$ ， $u^{n + \frac{1}{2}} = \frac{1}{2} (u^{n} + u^{n + 1})$ 。

3. 全局Lipschitz情况强收敛阶

现在我们研究 $α = 2$ 时方程(2.2)的收敛阶。由于非线性项不是全局Lipschitz的，首先要引入阶截断函数 $θ \in C^{\infty} (ℝ^{d})$ 满足 $\sup θ \in [0, 2]$ ，且在 $[0, 1]$ 上 $θ = 1$ 。截断后方程为

$i d u_{R} + (Δ u_{R} + λ F_{R} (u_{R}) + i a (t) u_{R} (t)) d t = d W,$ (3.1)

其对应的中点格式为

$u_{R}^{n + 1} = S_{Δ t} u_{R}^{n} + i Δ t T_{Δ t} F_{R} (u_{R}^{n + \frac{1}{2}}) - Δ t T_{Δ t} a^{n + \frac{1}{2}} u_{R}^{n + \frac{1}{2}} - i \sqrt{Δ t} T_{Δ t} X_{n + 1},$ (3.2)

其中，

$F_{R} (u_{R}) = θ_{R} (u_{R}) {| u_{R} |}^{2} u_{R}, θ_{R} (u) = θ (\frac{{‖ u ‖}_{H^{1}}}{R}),$

$S_{Δ t} = {(1 - \frac{i Δ t}{2} Δ)}^{- 1} (1 + \frac{i Δ t}{2} Δ), T_{Δ t} = {(1 - \frac{i Δ t}{2} Δ)}^{- 1}, X_{n + 1} = \frac{W (t_{n + 1}) - W (t_{n})}{\sqrt{Δ t}} .$

关于算子 $S_{Δ t}$ 和 $T_{Δ t}$ 有以下重要估计[7]，

${‖ S_{Δ t} ‖}_{ℒ (L^{2})} \leq 1, {‖ T_{Δ t} ‖}_{ℒ (L^{2})} \leq 1, {‖ S_{Δ t} - I ‖}_{ℒ (H^{2 + α}, H^{α})} \leq K Δ t,$ (3.3)

${‖ S (t_{n}) - S_{Δ t}^{n} ‖}_{ℒ (H^{3 + α}, H^{α})} \leq K Δ t, {‖ S (- s) - S_{Δ t}^{- n} T_{Δ t} ‖}_{ℒ (H^{3 α}, L^{2})} \leq K Δ t^{α},$ (3.4)

其中， $α \in [0, 1]$ ， $ℒ (U, V)$ 是由从可分Hilbert空间 $U$ 到空间 $V$ 的全体线性有界算子构成的空间，将 $ℒ (U, U)$ 简记为 $ℒ (U)$ ，此时， $F_{R} (u_{R}) = θ_{R} (u_{R}) {| u_{R} |}^{2} u_{R}$ 满足全局Lipschitz条件，其Lipschitz系数为 $L_{F_{R}}$ ，且方程(3.2)具有全局适定性且解 $u_{R}^{n}$ 是由 $u_{0}$ 唯一确定的。在此基础上，需要给出方程(3.1)的解和非线性项 $F_{R} (u_{R})$ 的强正则性，即当 $u_{0} \in H^{4}, ϕ \in ℒ_{2}^{4}$ 时，

${F_{R} (u_{R})}_{t \in [0, T]} \in L^{2 p} (Ω; L^{\infty} (0, T; H^{4}))$ ,

$E \sup_{0 \leq t \leq T} {‖ u_{R} (t) ‖}_{H^{4}}^{2 p} \leq C (p, T, R, {‖ u_{0} ‖}_{H^{4}}, {‖ ϕ ‖}_{ℒ_{2}^{4}})$ .

为了简化符号，将 $u_{R}^{n}$ 和 $u_{R}$ 分别写为 $u^{n}$ 和 $u$ 。假设 $Δ t < 2 / (C (M) + C (| λ |, L_{F_{R}}))$ ，那么对于任意的 $n$ ，方程(3.2)存在如下数值解，

$u^{n} = S_{Δ t}^{n} u_{0} + i Δ t λ \sum_{l = 1}^{n} S_{Δ t}^{n - l} T_{Δ t} F_{R} (u^{l - \frac{1}{2}}) - Δ t \sum_{l = 1}^{n} S_{Δ t}^{n - l} T_{Δ t} a^{l - \frac{1}{2}} u^{l - \frac{1}{2}} - i \sqrt{Δ t} \sum_{l = 1}^{n} S_{Δ t}^{n - l} T_{Δ t} X_{l} .$

接下来，我们给出方程(3.1)和(3.2)解之间的误差估计。

定理3.1假设 $α = 2, u_{0} \in H^{4}, ϕ \in ℒ_{2}^{4}$ ，对于任意的 $T > 0$ 和 $p \geq 1$ ，都存在常数 $K_{1}$ 使得

$E \max_{n = 0, \dots, N} {‖ u_{R} (t_{n}) - u_{R}^{n} ‖}_{H^{1}}^{2 p} \leq K_{1} Δ t^{2 p} .$

证明：由于 $a (t)$ 是有界时变函数，不妨假设上界为M。定义 $g_{n}$ 是将解 $u^{0}, \dots, u^{n - 1}$ 映射到 $u^{n}$ 的映射，即 $g_{n} (u^{0}, \dots, u^{n - 1}) = u^{n}$ ，那么对于任意满足 $u_{0} = v_{0}$ 的序列 ${u_{k}}$ 和 ${v_{k}}$ ，有

$\begin{array}{l} {‖ g_{n} (u^{0}, \dots, u^{n - 1}) - g_{n} (v^{0}, \dots, v^{n - 1}) ‖}_{H^{1}} \\ \leq Δ t {‖ λ \sum_{l = 1}^{n} S_{Δ t}^{n - l} T_{Δ t} (F_{R} (u^{l - \frac{1}{2}}) - F_{R} (v^{l - \frac{1}{2}})) ‖}_{H^{1}} + Δ t {‖ \sum_{l = 1}^{n} S_{Δ t}^{n - l} T_{Δ t} a^{l - \frac{1}{2}} (u^{l - \frac{1}{2}} - v^{l - \frac{1}{2}}) ‖}_{H^{1}} \\ \leq Δ t C (| λ |, L_{F_{R}}) \sum_{l = 1}^{n - 1} {‖ u^{l} - v^{l} ‖}_{H^{1}} + \frac{Δ t C (| λ |, L_{F_{R}})}{2} {‖ g_{n} (u^{0}, \dots, u^{n - 1}) - g_{n} (v^{0}, \dots, v^{n - 1}) ‖}_{H^{1}} \\ + C (M) \sum_{l = 1}^{n - 1} {‖ u^{l} - v^{l} ‖}_{H^{1}} + \frac{Δ t C (M)}{2} {‖ g_{n} (u^{0}, \dots, u^{n - 1}) - g_{n} (v^{0}, \dots, v^{n - 1}) ‖}_{H^{1}} \end{array}$

因此，我们推断

$‖ g_{n} (u^{0}, \dots, u^{n - 1}) - g_{n} (v^{0}, \dots, v^{n - 1}) ‖ \leq \frac{2 Δ t (C (M) + C (| λ |, L_{F_{R}}))}{2 - Δ t (C (M) + C (| λ |, L_{F_{R}}))} \sum_{l = 1}^{n - 1} {‖ u^{l} - v^{l} ‖}_{H^{1}} .$ (3.5)

由于

$\begin{array}{l} u (t_{n}) = S (t_{n}) u_{0} + i λ \int_{0}^{t_{n}} S (t_{n} - r) F_{R} (u (r)) d r \\ - \int_{0}^{t_{n}} S (t_{n} - r) a (r) u (r) d r - i \int_{0}^{t_{n}} S (t_{n} - r) d W (r) . \end{array}$

那么

$\begin{array}{l} u (t_{n}) - g_{n} (u^{0}, \dots, u (t_{n - 1})) \\ = (S (t_{n}) - S_{Δ t}^{n}) u_{0} + i λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} (S (t_{n} - r) F_{R} (u (r)) - S_{Δ t}^{n - l} T_{Δ t} F_{R} (u (t_{l - \frac{1}{2}}))) d r \\ - \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} (S (t_{n} - r) a (r) u (r) - S_{Δ t}^{n - l} T_{Δ t} a (t_{l - \frac{1}{2}}) u (t_{l - \frac{1}{2}})) d r \\ - i \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} (S (t_{n} - r) - S_{Δ t}^{n - l} T_{Δ t}) d W (r) \\ : = A_{n} - i B_{n} - C_{n} - i D_{n}, \end{array}$

首先用(3.3)式处理 $A_{n}$ ，用(3.4)式处理 $D_{n}$ ，则有

$E \max_{n = 0, \dots, N} {‖ A_{n} + D_{n} ‖}_{H^{1}}^{2 p} \leq C Δ t^{2 p} (E ({‖ u_{0} ‖}_{H^{4}}^{2 p}) + {‖ ϕ ‖}_{ℒ_{2}^{4}}^{2 p}) .$

接下来我们处理 $B_{n}$ ，将 $B_{n}$ 分裂成如下形式，

$\begin{array}{l} λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} (S (t_{n} - r) F_{R} (u (r)) - S_{Δ t}^{n - l} T_{Δ t} F_{R} (u (t_{l - \frac{1}{2}}))) d r \\ = λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} (S (t_{n} - r) - S_{Δ t}^{n - l} T_{Δ t}) F_{R} (u (r)) d r + λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} S_{Δ t}^{n - l} T_{Δ t} F_{R} (u (r) - u (t_{l - 1})) d r \\ + λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} S_{Δ t}^{n - l} T_{Δ t} F_{R} (u (t_{l - 1}) - u (t_{l - \frac{1}{2}})) d r \\ : = B_{n}^{1} + B_{n}^{2} + B_{n}^{3} . \end{array}$

下面我们分别来估计这三项。对于 $B_{n}^{1}$ ，

$\begin{array}{l} {‖ λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} (S (t_{n} - r) - S_{Δ t}^{n - l} T_{Δ t}) F_{R} (u (r)) d r ‖}_{H^{1}} \\ \leq {‖ λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} (S (t_{n} - r) - S (t_{n}) S_{Δ t}^{- l} T_{Δ t}) F_{R} (u (r)) d r ‖}_{H^{1}} + {‖ λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} (S (t_{n}) - S_{Δ t}^{n}) S_{Δ t}^{- l} T_{Δ t} F_{R} (u (r)) d r ‖}_{H^{1}} \\ \leq {‖ λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} (S (- r) - S_{Δ t}^{- l} T_{Δ t}) F_{R} (u (r)) d r ‖}_{H^{1}} + {‖ S (t_{n}) - S_{Δ t}^{n} ‖}_{ℒ (H^{3}, L^{2})} {‖ λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} S_{Δ t}^{- l} T_{Δ t} F_{R} (u (r)) d r ‖}_{H^{4}} \\ \leq C T Δ t \sup_{0 \leq t \leq T} {‖ F_{R} (u (t)) ‖}_{H^{4}}, \end{array}$

因此，

$E \max_{n = 0, \dots, N} {‖ B_{n}^{1} ‖}_{H^{1}}^{2 p} \leq C Δ t^{2 p} .$

对于 $B_{n}^{2}$ 使用Taylor公式，我们有

$\begin{array}{l} B_{n}^{2} = λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} S_{Δ t}^{n - l} T_{Δ t} {F^{'}}_{R} (u (t_{l - 1})) (S (r - t_{l - 1}) - I d) u (t_{l - 1}) d r \\ + λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} S_{Δ t}^{n - l} T_{Δ t} {F^{'}}_{R} (u (t_{l - 1})) (i λ \int_{t_{l - 1}}^{r} S (r - s) F_{R} (u (s)) d s) d r \\ - λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} S_{Δ t}^{n - l} T_{Δ t} {F^{'}}_{R} (u (t_{l - 1})) (\int_{t_{l - 1}}^{r} S (r - s) a (s) u (s) d s) d r \\ - i λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} S_{Δ t}^{n - l} T_{Δ t} {F^{'}}_{R} (\int_{t_{l - 1}}^{r} S (r - s) d W (s)) d r \\ + \frac{1}{2} \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} S_{Δ t}^{n - l} T_{Δ t} (\int_{0}^{1} F^{″} (ρ u (r) + (1 - ρ) u (t_{l - 1})) {(u (r) - u (t_{l - 1}))}^{2} d ρ) d r \\ = B_{n}^{21} + i B_{n}^{22} - B_{n}^{23} - i B_{n}^{24} + B_{n}^{25}, \end{array}$

其中，对于任意 $ρ \in ℝ$ ， $S (ρ)$ 是等距映射，满足 ${‖ S (ρ) - I ‖}_{ℒ (H^{4}, H^{2})} \leq C ρ,$ 其中I是单位算子，则

$E \max_{n = 0, \dots, N} {‖ B_{n}^{21} ‖}_{H^{1}}^{2 p} \leq C (T Δ t \sup_{0 \leq t \leq T} {‖ {F^{'}}_{R} ‖}_{ℒ (H^{4})} \sup_{0 \leq t \leq T} {‖ u ‖}_{H^{4}}) \leq C Δ t^{2 p} .$

对于 $B_{n}^{22}$ 和 $B_{n}^{23}$ ，

${‖ B_{n}^{22} ‖}_{H^{1}} \leq C T Δ t \sup_{0 \leq t \leq T} {‖ F_{R} (u (t)) ‖}_{H^{4}}, {‖ B_{n}^{23} ‖}_{H^{1}} \leq C T Δ t \sup_{0 \leq t \leq T} {‖ u (t) ‖}_{H^{4}},$

因此，

$E \max_{n = 0, \dots, N} {‖ B_{n}^{22} ‖}_{H^{1}}^{2 p} \leq C Δ t^{2 p}, E \max_{n = 0, \dots, N} {‖ B_{n}^{23} ‖}_{H^{1}}^{2 p} \leq C Δ t^{2 p} .$

接下来利用Fubini定理和鞅不等式处理 $B_{n}^{24}$ ，我们得到

$\begin{matrix} E \max_{n = 0, \dots, N} {‖ B_{n}^{24} ‖}_{H^{1}}^{2 p} = E \max_{n = 0, \dots, N} {‖ λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} S_{Δ t}^{n - l} T_{Δ t} {F^{'}}_{R} (\int_{t_{l - 1}}^{r} S (r - s) d W (s)) d r ‖}_{H^{1}}^{2 p} \\ \leq C_{ρ} E {(\sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} {‖ S_{Δ t}^{n - l} T_{Δ t} {F^{'}}_{R} (\int_{t_{l - 1}}^{r} S (r - s) ϕ e_{k} d r) ‖}_{ℒ_{2}^{1}}^{2} d s)}^{p} \leq C Δ t^{2 p} . \end{matrix}$

对于 $B_{n}^{25}$ ，已知 $F^{″}$ 有界，那么通过Minkowski不等式与Burkholder-Davis-Gundy不等式，我们有

$\begin{matrix} E \max_{n = 0, \dots, N} {‖ B_{n}^{25} ‖}_{H^{1}}^{2 p} \leq C E (\sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{^{l}}} {‖ u (r) - u (t_{l - 1}) ‖}_{H^{1}}^{4 p} d r) \\ \leq C E (\max_{n = 0, \dots, N} \sup_{t_{l - 1} \leq r \leq t_{l}} {‖ u (r) - u (t_{l - 1}) ‖}_{H^{1}}^{4 p}) \\ = C E (\max_{n = 0, \dots, N} \sup_{t_{l - 1} \leq r \leq t_{l}} {‖ (S (r - t_{l - 1}) - I d) u (t_{l - 1}) + i λ \int_{t_{l - 1}}^{r} S (r - s) F_{R} (u (s)) d s ‖}_{H^{1}}^{4 p}) \\ + C E (\max_{n = 0, \dots, N} \sup_{t_{l - 1} \leq r \leq t_{l}} {‖ - \int_{t_{l - 1}}^{r} S (r - s) a (s) u (s) d s - i \int_{t_{l - 1}}^{r} S (r - s) d W (s) ‖}_{H^{1}}^{4 p}) \\ \leq C Δ t^{2 p} . \end{matrix}$

利用上述类似的不等式处理 $B_{n}^{3}$ 项，得到

$\begin{array}{l} E \max_{n = 0, \dots, N} {‖ B_{n}^{3} ‖}_{H^{1}}^{2 p} = E \max_{n = 0, \dots, N} {‖ λ \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} S_{Δ t}^{n - l} T_{Δ t} F_{R} (u (t_{l - 1}) - u (t_{l - \frac{1}{2}})) d r ‖}_{H^{1}}^{2 p} \\ \leq C T {‖ u (t_{l - 1}) - u (t_{l - \frac{1}{2}}) ‖}_{H^{1}} \\ \leq C Δ t^{2 p} . \end{array}$

对于 $C_{n}$ ，仍利用三角不等式将其拆分得到，

$\begin{array}{l} C_{n} = \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} (S (t_{n} - r) a (r) u (r) - S_{Δ t}^{n - l} T_{Δ t} a (t_{l - \frac{1}{2}}) u (t_{l - \frac{1}{2}})) d r \\ \leq \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} (S (t_{n} - r) a (r) u (r) - S_{Δ t}^{n - l} T_{Δ t} a (r) u (r)) d r \\ + \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} (S_{Δ t}^{n - l} T_{Δ t} a (r) u (r) - S_{Δ t}^{n - l} T_{Δ t} a (t_{l - \frac{1}{2}}) u (t_{l - \frac{1}{2}})) d r \\ + \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} (S_{Δ t}^{n - l} T_{Δ t} a (t_{l - \frac{1}{2}}) u (t_{l - \frac{1}{2}}) - S_{Δ t}^{n - l} T_{Δ t} a (t_{l - \frac{1}{2}}) u (t_{l - \frac{1}{2}})) d r \\ : = C_{n}^{1} + C_{n}^{2} + C_{n}^{3}, \end{array}$

其中，

$\begin{array}{l} C_{n}^{2} \leq \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} (S_{Δ t}^{n - l} T_{Δ t} a (r) u (r) - S_{Δ t}^{n - l} T_{Δ t} a (r) u (t_{l - \frac{1}{2}})) d r \\ + \sum_{l = 1}^{n} \int_{t_{l - 1}}^{t_{l}} (S_{Δ t}^{n - l} T_{Δ t} a (r) u (t_{l - \frac{1}{2}}) - S_{Δ t}^{n - l} T_{Δ t} a (t_{l - \frac{1}{2}}) u (t_{l - \frac{1}{2}})) d r \\ : = C_{n}^{21} + C_{n}^{22} . \end{array}$

此时，由于 $a (t)$ 是有界的，项 $C_{n}^{1}, C_{n}^{21}, C_{n}^{22}, C_{n}^{3}$ 可以用上述处理 $B_{n}$ 的方法进行类似的处理，最终得到

$E \max_{n = 0, \dots, N} {‖ C_{n} ‖}_{H^{1}}^{2 p} \leq C Δ t^{2 p} .$

综合以上不等式，我们有

$E \max_{n = 0, \dots, N} {‖ u (t_{n}) - g_{n} (u^{0}, \dots, u (t_{n - 1})) ‖}_{H^{1}}^{2 p} \leq C Δ t^{2 p} .$

又由(3.5)和Minkowski不等式可知

${(E \max_{n = 0, \dots, N} {‖ u^{\tilde{n}} - u (t_{\tilde{n}}) ‖}_{H^{1}}^{2 p})}^{\frac{1}{2 p}} \leq C \frac{2 (C (M) + C (| λ |, L_{F_{R}}))}{2 - Δ t (C (M) + C (| λ |, L_{F_{R}}))} Δ t \sum_{l = 1}^{n - 1} {(E (\max_{\tilde{l} = 0, \dots, l} {‖ u^{\tilde{l}} - u (t_{\tilde{l}}) ‖}^{2 p}))}^{\frac{1}{2 p}} + C τ .$

再结合Gronwall引理[5]，证得引理3.1。

4. 数值解收敛性

通过上述的探讨，可以得到方程(2.2)的数值解的收敛性，也是本文的中心定理。

定理4.1令 $α = 2, u_{0} \in H^{4}, ϕ \in ℒ_{2}^{4}$ ，则对于任意的 $0 \leq n \leq N$ ，有

$\lim_{C \to \infty} P (\max_{n = 0, \dots, N} {‖ u (t_{n}) - u^{n} ‖}_{H^{1}} \geq C Δ t) = 0.$

证定义停时 $τ_{R} = \inf_{n = 0, \dots, N} {t_{n} : {‖ u_{R}^{n - 1} ‖}_{H^{1}} \geq R 或者 {‖ u_{R}^{n} ‖}_{H^{1}} \geq R}$ ，且当 $t_{n} < τ_{R}$ 时有离散解 $u^{n} = u_{R}^{n}$ ，由定理3.1可知，对于任意的 $R > 0$ ，有

$E \max_{n = 0, \dots, N} {‖ u_{R} (t_{n}) - u_{R}^{n} ‖}_{H^{1}}^{2 p} \leq C Δ t^{2 p} .$

这表明当 $τ \to 0$ 时， $\max_{n = 0, \dots, N} {‖ u_{R} (t_{n}) - u_{R}^{n} ‖}_{H^{1}}$ 依概率收敛于0。由Chebyshev不等式可知

$\begin{array}{l} P (\max_{n = 0, \dots, N} {‖ u (t_{n}) - u^{n} ‖}_{H^{1}} \geq C Δ t) \\ \leq P (\max_{n = 0, \dots, N} {‖ u (t_{n}) ‖}_{H^{1}} \geq R) + P (\max_{n = 0, \dots, N} {‖ u^{n} ‖}_{H^{1}} \geq R) + P (\max_{n = 0, \dots, N} {‖ u_{R} (t_{n}) - u_{R}^{n} ‖}_{H^{1}} \geq C Δ t) \\ \leq P (\max_{n = 0, \dots, N} {‖ u (t_{n}) ‖}_{H^{1}} \geq R) + P (\max_{n = 0, \dots, N} {‖ u^{n} ‖}_{H^{1}} \geq R) + \frac{E \max_{n = 0, \dots, N} {‖ u_{R} (t_{n}) - u_{R}^{n} ‖}_{H^{1}}^{2}}{C^{2} Δ t^{2}} . \end{array}$

结合 $u^{n}, u (t_{n})$ 的一致有界性和定理3.1，我们可以得到当 $C \to 0$ 时有

$P (\max_{n = 0, \dots, N} {‖ u (t_{n}) - u^{n} ‖}_{H^{1}} \geq C Δ t) \to 0.$

因此该格式的依概率收敛阶为1阶。

本文仅研究了时间半离散格式，后续可结合谱方法或有限元方法进行空间离散，建立全离散格式的误差估计，并探讨时空步长的耦合影响。

NOTES

^*通讯作者。

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]

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