Error estimates of the time-splitting methods for the nonlinear Schrödinger equation with semi-smooth nonlinearity

TL;DR

This work analyzes the Lie–Trotter time-splitting sine pseudospectral (TSSP) method for the nonlinear Schrödinger equation with a semi-smooth nonlinearity $f(\rho)=\beta\rho^\sigma$. By introducing a local regularization near $\rho=0$ and proving unconditional $L^2$-stability (and $l^\infty$-stability under a mild CFL condition), the authors derive rigorous error bounds that depend on the nonlinearity exponent $\sigma$: for $0<\sigma\le 1/2$, the $L^2$-error is $O(\tau^{1/2+\sigma}+h^{1+2\sigma})$ without CFL restrictions; for $\sigma\ge 1/2$, the $L^2$-error is $O(\tau+h^2)$ and the $H^1$-error is $O(\tau^{1/2}+h)$ under a mild CFL constraint; and for $1/2<\sigma<1$ with $H^3$-regularity, the $H^1$-error is $O(\tau^{\sigma}+h^{2\sigma})$. The analysis combines detailed operator estimates for the nonlinear term, precise local truncation error calculation with the regularized nonlinearity, and stability arguments to propagate errors without excessive regularity assumptions. Numerical results in 1D validate the theoretical rates and illustrate the scheme’s robustness across a range of $\sigma$ values. Overall, the paper provides sharp, regime-dependent error guarantees for a mass-conserving, time-symmetric, fully discrete scheme applied to NLSEs with non-integer nonlinearities.

Abstract

We establish error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schrödinger equation (NLSE) with semi-smooth nonlinearity $ f(ρ) = ρ^σ$, where $ρ=|ψ|^2$ is the density with $ψ$ the wave function and $σ>0$ is the exponent of the semi-smooth nonlinearity. Under the assumption of $ H^2 $-solution of the NLSE, we prove error bounds at $ O(τ^{\frac{1}{2}+σ} + h^{1+2σ}) $ and $ O(τ+ h^{2}) $ in $ L^2 $-norm for $0<σ\leq\frac{1}{2}$ and $σ\geq\frac{1}{2}$, respectively, and an error bound at $ O(τ^\frac{1}{2} + h) $ in $ H^1 $-norm for $σ\geq \frac{1}{2}$, where $h$ and $τ$ are the mesh size and time step size, respectively. In addition, when $\frac{1}{2}<σ<1$ and under the assumption of $ H^3 $-solution of the NLSE, we show an error bound at $ O(τ^σ + h^{2σ}) $ in $ H^1 $-norm. Two key ingredients are adopted in our proof: one is to adopt an unconditional $ L^2 $-stability of the numerical flow in order to avoid an a priori estimate of the numerical solution for the case of $ 0 < σ\leq \frac{1}{2}$, and to establish an $ l^\infty $-conditional $ H^1 $-stability to obtain the $ l^\infty $-bound of the numerical solution by using the mathematical induction and the error estimates for the case of $ σ\ge \frac{1}{2}$; and the other one is to introduce a regularization technique to avoid the singularity of the semi-smooth nonlinearity in obtaining improved local truncation errors. Finally, numerical results are reported to demonstrate our error bounds.

Figures (4)

• Figure 5.1: (a) density $|\psi(x, 1)|^2$ with different $\sigma >0$ and (b) relative errors of the energy divided by $\tau$ up to $t=8$ with $\sigma = 0.1$ for the Type I initial datum \ref{['typeI_ini']} with $\beta = -10$.
• Figure 5.2: Temporal errors (a) and spatial errors (b) in $L^2$-norm for $\sigma=0.1, 0.2, 0.3, 0.4$ with Type I initial datum \ref{['typeI_ini']}.
• Figure 5.3: Temporal errors (a) and spatial errors (b) in $L^2$-norm and $H^1$-norm for $\sigma=0.5$ with Type II initial data \ref{['typeII_ini']}.
• Figure 5.4: Temporal errors (a) and spatial errors (b) in $H^1$-norm for $\sigma = 0.1, 0.25, 0.5, 0.75$ with Type I initial datum \ref{['typeI_ini']}.

Theorems & Definitions (53)

• Remark 2.1
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Lemma 2.4
• proof
• Theorem 2.5
• Corollary 2.6
• Theorem 2.7