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

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.