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 $f(\rho )={\rho }^{\sigma }$, where $\rho =|\psi {|}^2$ is the density with $\psi$ the wave function and $\sigma >0$ is the exponent of the semi-smooth nonlinearity. Under the assumption of $H^2$-solution of the NLSE, we prove error bounds at $O({\tau }^{\frac{1}{2}+\sigma }+h^{1+2\sigma })$ and $O(\tau +h^2)$ in $L^2$-norm for $0<\sigma \le \frac{1}{2}$ and $\sigma \ge \frac{1}{2}$, respectively, and an error bound at $O({\tau }^{\frac{1}{2}}+h)$ in $H^1$-norm for $\sigma \ge \frac{1}{2}$, where $h$ and $\tau$ are the mesh size and time step size, respectively. In addition, when $\frac{1}{2}<\sigma <1$ and under the assumption of $H^3$-solution of the NLSE, we show an error bound at $O({\tau }^{\sigma }+h^{2\sigma })$ 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 $0<\sigma \le \frac{1}{2}$, 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 $\sigma \ge \frac{1}{2}$; 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.

32 pages, 4 figures

References in corpus (3)

• Quantum liquid droplets in a mixture of Bose-Einstein condensates
• Collisional dynamics of symmetric two-dimensional quantum droplets
• Regularized numerical methods for the nonlinear Schrödinger equation with singular nonlinearity

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