This paper studies the numerical solution of the semiclassical nonlinear Schrödinger equation on the d-dimensional torus \(\mathbb {T}^d\) , with highly oscillatory initial data depending on a small parameter \(\varepsilon \in (0,1]\) . We first show that a WKB-type approximation attains an \(\mathcal {O}(\varepsilon )\) error in the \(L^2\) norm for \(H^2\) initial data theoretically, although its accuracy deteriorates as \(\varepsilon \) 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 \(H^2\) regularity, which is weaker than the regularity assumptions in the literature. Furthermore, our analysis shows that the CFL-type conditions linking h, \(\tau \) , and \(\varepsilon \) —typically imposed in the semiclassical regime in the literature—are not required in our scheme to obtain second-order convergence with respect to \(\tau \) and h, uniformly with respect to \(\varepsilon \) , under the weaker regularity condition. Numerical experiments support the theoretical results and demonstrate the robustness of the method across a wide range of \(\varepsilon \) .
