In this paper, we consider the numerical solution of a nonlinear Schrodinger equation with spatial random potential. The randomly shifted quasi-Monte Carlo (QMC) lattice rule combined with the time-splitting pseudospectral discretization is applied and analyzed. The nonlinearity in the equation induces difficulties in estimating the regularity of the solution in random space. By the technique of weighted Sobolev space, we identify the possible weights and show the existence of QMC that converges optimally at the almost-linear rate without dependence on dimensions. The full error estimate of the scheme is established. We present numerical results to verify the accuracy and investigate the wave propagation.

翻译：本文研究具有空间随机势的非线性薛定谔方程的数值求解方法。我们采用随机移位拟蒙特卡罗(QMC)网格规则结合时间分裂拟谱离散化方法，并对其进行分析。方程中的非线性项给随机空间解的正则性估计带来困难。通过加权Sobolev空间技术，我们确定了可能的权重，并证明了QMC能以几乎线性速率实现最优收敛，且不依赖于维度。建立了该格式的完整误差估计。我们通过数值结果验证了精度，并研究了波传播特性。