arXiv subjects

Zhuangzhi Xu

Publications and source records attributed to Zhuangzhi Xu.

2 recordsLinked to original sources

arXiv2020

Mass- and energy-preserving exponential Runge-Kutta methods for the nonlinear Schrödinger equation

In this paper, a family of arbitrarily high-order structure-preserving exponential Runge-Kutta methods are developed for the nonlinear Schrödinger equation by combining the scalar auxiliary variable approach with the exponential Runge-Kutta method. By introducing an auxiliary variable, we first transform the original model into an equivalent system which admits both mass and modified energy conservation laws. Then applying the Lawson method and the symplectic Runge-Kutta method in time, we derive a class of mass- and energy-preserving time-discrete schemes which are arbitrarily high-order in time. Numerical experiments are addressed to demonstrate the accuracy and effectiveness of the newly proposed schemes.

math.NA↗

arXiv2019

Optimal error estimate of a conservative Fourier pseudo-spectral method for the space fractional nonlinear Schrödinger equation

In this paper, we consider the error analysis of a conservative Fourier pseudo-spectral method that conserves mass and energy for the space fractional nonlinear Schrödinger equation. We give a new fractional Sobolev norm that can construct the discrete fractional Sobolev space, and we also can prove some important lemmas for the new fractional Sobolev norm. Based on these lemmas and energy method, a priori error estimate for the method can be established. Then, we are able to prove that the Fourier pseudo-spectral method is unconditionally convergent with order $O(τ^{2}+N^{α/2-r})$ in the discrete $L^{\infty}$ norm, where $τ$ is the time step and $N$ is the number of collocation points used in the spectral method. Numerical examples are presented to verify the theoretical analysis.

math.AP↗