Linearly Implicit High-Order Exponential Integrators Conservative Runge–Kutta Schemes for the Fractional Schrödinger Equation

Fractal and Fractional

Zhuangzhi Xu¹  Yanmin Zhao²  Qianqian Zheng²  Yayun Fu² 

[1] School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, China;School of Science, Xuchang University, Xuchang 461000, China;

关键词: fractional Schrödinger equation;    exponential integrators;    high-order;    linearly implicit energy-preserving;   

DOI  :  10.3390/fractalfract6050243

来源: DOAJ

【 摘 要 】

In this paper, a family of high-order linearly implicit exponential integrators conservative schemes is constructed for solving the multi-dimensional nonlinear fractional Schrödinger equation. By virtue of the Lawson transformation and the generalized scalar auxiliary variable approach, the equation is first reformulated to an exponential equivalent system with a modified energy. Then, we construct a semi-discrete conservative scheme by using the Fourier pseudo-spectral method to discretize the exponential system in space direction. After that, linearly implicit energy-preserving schemes which have high accuracy are given by applying the Runge–Kutta method to approximate the semi-discrete system in temporal direction and using the extrapolation method to the nonlinear term. As expected, the constructed schemes can preserve the energy exactly and implement efficiently with a large time step. Numerical examples confirm the constructed schemes have high accuracy, energy-preserving, and effectiveness in long-time simulation.

【 授权许可】

Unknown