[Paper Review] Error estimates of a Fourier integrator for the cubic Schr\\"odinger equation at low regularity

Alexander Ostermann, Frédéric Rousset+1 more|arXiv (Cornell University)|Feb 18, 2019

Advanced Mathematical Physics Problems34 references52 citations

TL;DR

The paper introduces a filtered low-regularity Fourier integrator for the cubic nonlinear Schrödinger equation and provides global L2 error estimates from H1 data in dimensions 1–3, breaking the natural 1/2 order barrier. It combines a filtered discretization with discrete Strichartz-type estimates to achieve improved convergence rates at low regularity.

ABSTRACT

We present a new filtered low-regularity Fourier integrator for the cubic nonlinear Schr\\"odinger equation based on recent time discretization and filtering techniques. For this new scheme, we perform a rigorous error analysis and establish better convergence rates at low regularity than known for classical schemes in the literature so far. In our error estimates, we combine the better local error properties of the new scheme with a stability analysis based on general discrete Strichartz-type estimates. The latter allow us to handle a much rougher class of solutions as the error analysis can be carried out directly at the level of $L^2$ compared to classical results \\black in dimension $d$, \\black which are limited to higher-order (sufficiently smooth) Sobolev spaces $H^s$ with $s>d/2$. In particular, we are able to establish a global error estimate in $L^2$ for $H^1$ solutions which is roughly of order $\ au^{ {1\\over 2} + { 5-d \\over 12} }$ in dimension $d \\leq 3$ ($\ au$ denoting the time discretization parameter). This breaks the "natural order barrier" of $\ au^{1/2}$ for $H^1$ solutions which holds for classical numerical schemes (even in combination with suitable filter functions).

Motivation & Objective

• Motivate and address the loss of regularity in numerical approximation of the cubic NLS.
• Develop a Fourier integrator with filtering that requires only H1 regularity of the solution.
• Establish global L2 error bounds in low regularity spaces using discrete Strichartz-type estimates.
• Optimize time-step convergence rates across dimensions d=1,2,3 under low-regularity assumptions.

Proposed method

• Define a one-step Fourier integrator with a frequency projection operator Pi_K and a filter function phi_1 to manage nonlinear interactions.
• Analyze the local error structure to exploit the favorable form tau^{1+gamma} |nabla|^{gamma} u(t) and avoid the full-derivative growth typical of classical schemes.
• Introduce discrete Strichartz-type estimates for the filtered linear propagator S_K(t) = e^{itΔ} Pi_K, including controlled loss of derivatives.
• Balance the filter parameter K = tau^{-alpha/2} to optimize the total global error by trading off local error improvements against stability/Strichartz losses.
• Prove a global L2 error bound for solutions in H1 in dimensions d ≤ 3: tau^{5/6} (d=1), tau^{3/4} (d=2), and tau^{2/3} (d=3) up to logarithmic factors.
• Employ frequency-truncated continuous problems u^K and show that the scheme is stable and convergent by combining Strichartz estimates with bounds on the difference u^K - u.

Experimental results

Research questions

• RQ1 Can a low-regularity Fourier integrator achieve improved global convergence rates for the cubic NLS with initial data in H1?
• RQ2 How do discrete Strichartz-type estimates for a filtered, frequency-truncated evolution influence stability and error bounds in low-regularity regimes?
• RQ3 What is the optimal choice of the filter parameter K (in terms of tau) to balance local error and stability for d ≤ 3?
• RQ4 To what extent can the method break the natural order barrier of tau^{1/2} for H1 solutions in low dimensions?
• RQ5 Do the results extend to both defocusing and focusing cubic NLS when the exact solution remains in H1?

Key findings

• The proposed filtered Fourier integrator achieves global L2 error bounds for H1 data in dimensions 1–3 with rates tau^{5/6}, tau^{3/4}, and tau^{2/3} (up to log factors).
• The error analysis combines improved local error structure with discrete Strichartz-type estimates to handle low regularity directly at L2, avoiding higher Sobolev requirements.
• Discrete Strichartz estimates for the filtered propagator S_K(t) exhibit controlled loss of derivatives that can be converted into uniform derivative-loss bounds.
• Choosing K = tau^{-alpha/2} with alpha ≥ 1 balances the two error contributions and yields the sub-half-order convergence in L2 that surpasses classical methods for H1 data.
• The results hold for the defocusing cubic NLS and extend to the focusing case as long as the exact solution remains in H1 (no global blow-up within the time interval).
• A global bound on the accumulated L2 error is established via a stability argument around the filtered solution u^K and the true solution u, with uniform-in-time control of relevant norms.