Second-order uniformly accurate method for the semiclassical nonlinear Schrödinger equation with initial data in $${{\text {H}}^2}$$
Year: 2026
Type: article
Abstract: This paper studies the numerical solution of the semiclassical nonlinear Schrödinger equation on the d -dimensional torus $$\mathbb {T}^d$$ T d , with highly oscillatory initial data depending on a small parameter $$\varepsilon \in (0,1]$$ ε ∈ ( 0 , 1 ] . We first show that a WKB-type approximation attains an $$\mathcal {O}(\varepsilon )$$ O ( ε ) error in the $$L^2$$ L 2 norm for $$H^2$$ H 2 initial data theoretically, although its accuracy deteriorates as $$\varepsilon $$ 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 $$H^2$$ H 2 regularity, which is weaker than the regularity assumptions in the literature. Furthermore, our analysis shows that the CFL-type conditions linking h , $$\tau $$ τ , and $$\varepsilon $$ ε —typically imposed in the semiclassical regime in the literature—are not required in our scheme to obtain second-order convergence with respect to $$\tau $$ τ and h , uniformly with respect to $$\varepsilon $$ ε , under the weaker regularity condition. Numerical experiments support the theoretical results and demonstrate the robustness of the method across a wide range of $$\varepsilon $$ ε .
Source: Numerische Mathematik
Authors: Buyang Li, Yifei Wu, Fangyan Yao
Institutions: Hong Kong Polytechnic University, Nanjing Normal University
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Funder: Hong Kong Polytechnic University
OpenAlex ID: W7150844068