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 f(\rho ) = \rho ^\sigma , where ρ = | ψ | 2 \rho =|\psi |^2 is the density with ψ \psi the wave function and σ > 0 \sigma >0 is the exponent of the semi-smooth nonlinearity. Under the assumption of H 2 H^2 -solution of the NLSE, we prove error bounds at O ( τ 1 2 + σ + h 1 + 2 σ ) O(\tau ^{\frac {1}{2}+\sigma } + h^{1+2\sigma }) and O ( τ + h 2 ) O(\tau + h^{2}) in L 2 L^2 -norm for 0 > σ ≤ 1 2 0>\sigma \leq \frac {1}{2} and σ ≥ 1 2 \sigma \geq \frac {1}{2} , respectively, and an error bound at O ( τ 1 2 + h ) O(\tau ^\frac {1}{2} + h) in H 1 H^1 -norm for σ ≥ 1 2 \sigma \geq \frac {1}{2} , where h h and τ \tau are the mesh size and time step size, respectively. In addition, when 1 2 > σ > 1 \frac {1}{2}>\sigma >1 and under the assumption of H 3 H^3 -solution of the NLSE, we show an error bound at O ( τ σ + h 2 σ ) O(\tau ^{\sigma } + h^{2\sigma }) in H 1 H^1 -norm. Two key ingredients are adopted in our proof: one is to adopt an unconditional L 2 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 0 > \sigma \leq \frac {1}{2} , and to establish an l ∞ l^\infty -conditional H 1 H^1 -stability to obtain the l ∞ l^\infty -bound of the numerical solution by using the mathematical induction and the error estimates for the case of σ ≥ 1 2 \sigma \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.