Abstract This paper studies the numerical solution of the semiclassical nonlinear Schrödinger equation on the d -dimensional torus Tᵈ T d, with highly oscillatory initial data depending on a small parameter (0, 1] ε ∈ (0, 1 ]. We first show that a WKB-type approximation attains an O () O (ε) error in the L² L 2 norm for H² H 2 initial data theoretically, although its accuracy deteriorates as ε increases. To address this limitation, we propose a numerical scheme that (i) applies a Galilean transform to remove the oscillations in the initial data, (ii) establishes sharp space–time estimates for the transformed equation, and (iii) employs a new low-regularity integrator to achieve second-order accuracy under the minimal H² H 2 regularity, which is weaker than the regularity assumptions in the literature. Furthermore, our analysis shows that the CFL-type conditions linking h, τ, and ε —typically imposed in the semiclassical regime in the literature—are not required in our scheme to obtain second-order convergence with respect to τ and h, uniformly with respect to ε, under the weaker regularity condition. Numerical experiments support the theoretical results and demonstrate the robustness of the method across a wide range of ε.
Li et al. (Mon,) studied this question.