Time-relaxation structure-preserving explicit low-regularity integrators for the nonlinear Schrödinger equation

TL;DR

The paper addresses numerical approximation of the nonlinear Schrödinger equation on $\mathbb{T}^d$ with low-regularity data, where preserving invariants and long-time stability are challenging for explicit schemes. It introduces a time-relaxation framework built on a resonance-based, explicit update in a twisted variable $v=e^{-it\Delta}u$, with an adaptive relaxation parameter $\gamma_n$ that enforces exact mass conservation and preserves computational efficiency. The authors prove mass conservation, establish a convergence theory under low regularity, and show that the proposed RLRIs achieve second-order accuracy without order reduction, unlike relaxing the original variable $u$. Numerical experiments in 1D validate second-order convergence, excellent long-time $L^2$-norm preservation (near machine precision), and superior efficiency relative to existing methods. The approach is general enough to extend to other dispersive semilinear PDEs governed by contraction semigroups, offering a practical, structure-preserving tool for rough-data simulations.

Abstract

We propose and rigorously analyze a novel family of explicit low-regularity exponential integrators for the nonlinear Schrödinger (NLS) equation, based on a time-relaxation framework. The methods combine a resonance-based scheme for the twisted variable with a dynamically adjusted relaxation parameter that guarantees exact mass conservation. Unlike existing symmetric or structure-preserving low-regularity integrators, which are typically implicit and computationally expensive, the proposed methods are fully explicit, mass-conserving, and well-suited for solutions with low regularity. Furthermore, the schemes can be naturally extended to a broad class of evolution equations exhibiting the structure of strongly continuous contraction semigroups. Numerical results demonstrate the accuracy, robustness, and excellent long-time behavior of the methods under low-regularity conditions.

Figures (4)

• Figure 1: $H^1$-error at $T=1$ as a function of $\tau$ for low-regularity and smooth initial data.
• Figure 2: $H^1$-error at $T=1$ versus CPU time for low-regularity and smooth initial data.
• Figure 3: Evolutions of the relaxation coefficients $\gamma_n$ with $\tau=0.01$ and $d(\gamma_n)$ at $T=1$ as a function of $\tau$ for different initial data.
• Figure 4: Relative error in the $L^2$-norm for $\tau=0.02$ and different initial data.

Theorems & Definitions (16)

• Remark 1.1
• Lemma 2.1
• Lemma 2.2
• Proof 1
• Remark 2.4
• Remark 2.4
• Remark 2.5
• Theorem 3.1: Conservation property of general RLRIs-v
• Proof 2
• Proposition 3.3: Stability estimate