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 T d , with highly oscillatory initial data depending on a small parameter ε ∈ (0,1] ε ∈ ( 0 , 1 ] . We first show that a WKB-type approximation attains an O(ε ) O ( ε ) error in the L² L 2 norm for H² H 2 initial data theoretically, although its accuracy deteriorates as ε 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 τ , and ε 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 τ and h , uniformly with respect to ε 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 ε .