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, respectively, and an error bound at O(τ, where h and τ are the mesh size and time step size, respectively. In addition, when 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 ≤ 1 2 , and to establish an l ∞ -conditional H 1 -stability to obtain the l ∞ -bound of the numerical solution by using the mathematical induction and the error estimates for the case of σ ≥ 1 2 ; and the other one is to introduce a regularization technique to avoid the singularity of the semismooth nonlinearity in obtaining improved local truncation errors. Finally, numerical results are reported to demonstrate our error bounds.
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