Optimal error bounds on time-splitting methods for the nonlinear Schrödinger equation with low regularity potential and nonlinearity

TL;DR

The paper advances numerical analysis for the nonlinear Schrödinger equation with low-regularity potential and nonlinearity by establishing optimal $L^2$ and $H^1$ error bounds for Lie-Trotter and Strang time-splitting Fourier spectral methods. Central to the approach is the regularity compensation oscillation (RCO) technique, which replaces higher-order spatial derivatives with a lower-order temporal derivative to capture error cancellations under a CFL-type condition $\tau\lesssim h^2$. The results significantly relax regularity requirements on $V$ and the exponent $\sigma$ while maintaining optimal convergence orders, and are supported by extensive numerical tests that confirm sharpness. The work provides both theoretical guarantees and practical guidance for simulating NLSE with rough data in low-regularity settings, with potential extensions to related dispersive systems.

Abstract

We establish optimal error bounds on time-splitting methods for the nonlinear Schrödinger equation with low regularity potential and typical power-type nonlinearity $ f(ρ) = ρ^σ$, where $ ρ:=|ψ|^2 $ is the density with $ ψ$ the wave function and $ σ> 0 $ the exponent of the nonlinearity. For the first-order Lie-Trotter time-splitting method, optimal $ L^2 $-norm error bound is proved for $L^\infty$-potential and $ σ> 0 $, and optimal $H^1$-norm error bound is obtained for $ W^{1, 4} $-potential and $ σ\geq 1/2 $. For the second-order Strang time-splitting method, optimal $ L^2 $-norm error bound is established for $H^2$-potential and $ σ\geq 1 $, and optimal $H^1$-norm error bound is proved for $H^3$-potential and $ σ\geq 3/2 $ (or $σ= 1$). Compared to those error estimates of time-splitting methods in the literature, our optimal error bounds either improve the convergence rates under the same regularity assumptions or significantly relax the regularity requirements on potential and nonlinearity for optimal convergence orders. A key ingredient in our proof is to adopt a new technique called \textit{regularity compensation oscillation} (RCO), where low frequency modes are analyzed by phase cancellation, and high frequency modes are estimated by regularity of the solution. Extensive numerical results are reported to confirm our error estimates and to demonstrate that they are sharp.

Figures (8)

• Figure 5.1: Temporal errors of the LTFS method with different mesh sizes for \ref{['eq:NLSE_low_reg_poten']}: (a) $V=V_1 \in L^\infty(\Omega)$ and (b) $V = V_2 \in W^{1, 4}(\Omega) \cap H^1_\text{per}(\Omega)$.
• Figure 5.2: Temporal errors of the STFS method with different mesh sizes for \ref{['eq:NLSE_low_reg_poten']}: (a) $V=V_3 \in H^2_\text{per}(\Omega)$ and (b) $V = V_4 \in H^3_\text{per}(\Omega)$.
• Figure 5.3: Temporal errors of the LTFS method with different mesh sizes for \ref{['eq:NLSE_low_reg_nonl']}: (a) $\sigma = 0.1$ and (b) $\sigma = 0.5$.
• Figure 5.4: Temporal errors of the STFS method with different mesh sizes for \ref{['eq:NLSE_low_reg_nonl']}: (a) $\sigma = 1.1$ and (b) $\sigma = 1.5$.
• Figure 5.5: Temporal errors of the LTFS method for \ref{['eq:NLSE_low_reg_nonl']} with fixed mesh size $h=h_e$ and different values of $\sigma$: (a) $L^2$-error and (b) $H^1$-error.

Theorems & Definitions (53)

• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Proposition 3.4
• proof