Convergence analysis of an L1-continuous Galerkin method for nonlinear time-space fractional Schrödinger equations

Abstract

This paper develops and analyses a finite difference/spectral-Galerkin scheme for the nonlinear fractional Schrödinger equations with Riesz space- and Caputo time-fractional derivatives. The $L1$ finite difference approximation is used for the discretization of the Caputo fractional derivative and the Legendre-Galerkin spectral method is used for the spatial approximation. Additionally, by using a proper form of discrete Grönwall inequality, the scheme is proved to be unconditionally stable and convergent with $2-\text{β}$ accuracy in time and spectral accuracy in space in case of smooth solutions. Finally, some numerical tests are preformed to distinguish the validity of our theoretical results.

KEYWORDS:

• Spectral approximation of Galerkin form
• $L1$-type scheme
• discrete Grönwall inequality
• time-space fractional Schrödinger equation
• theoretical analysis

AMS CLASSIFICATIONS:

• 65M06
• 65M12
• 65Q10
• 34K37

Acknowledgements

A. S. H. acknowledges RFBR Grant 19-01-00019 financial support.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The financial support was provided by RFBR [grant number 19-01-00019].