Error estimates of the time-splitting methods for the nonlinear Schrödinger equation with semi-smooth nonlinearity
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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^1/2+σ + h^1+2σ) and O(τ + h^2) in L^2-norm for 0<σ≤1/2 and σ≥1/2, respectively, and an error bound at O(τ^1/2 + h) in H^1-norm for σ≥1/2, 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 semi-smooth nonlinearity in obtaining improved local truncation errors. Finally, numerical results are reported to demonstrate our error bounds.
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