Error Estimate of a Quasi-Monte Carlo Time-Splitting Pseudospectral Method for Nonlinear Schrödinger Equation with Random Potentials

SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 1, Page 1-29, March 2024.
Abstract. In this paper, we consider the numerical solution of a nonlinear Schrödinger 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.