Second-order Uniformly Accurate Method for The Semiclassical Nonlinear Schrödinger Equation with Initial Data in H2

This paper studies the numerical solution of the semiclassical nonlinear Schrödinger equation on the d-dimensional torus Td, with highly oscillatory initial data depending on a small parameter ε∈(0,1]. We first show that a WKB-type approximation attains an O(ε) error in the L2 norm for H2 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 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.