High-Order Time-Stepping Methods for Two-Dimensional Space-Fractional Reaction-Diffusion Models

Abstract

In this paper, we developed novel fourth-order Runge-Kutta type exponential time differencing (ETD) A-stable and L-stable methods for space-fractional nonlinear reaction-diffusion equations with initial non-smooth or smooth data. Based on compact finite differences, a fourth-order technique is used for spatial discretization, while ETD is employed to discretize the time. Our novel numerical schemes have the benefit of explicitly handling the nonlinear term. The well-known issue of numerical instability related to computing the matrix exponential is addressed using the real single-pole rational approximation, namely the restricted Padé approximation approach. The corresponding ETD-A-stable and L-stable methods are obtained. Convergence, error estimates, and stability analysis of the suggested approaches are studied theoretically. Under a global Lipschitz continuity assumption, the unconditional

L^2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document}

numerical stability is established. Moreover, the convergence order of

\mathcal{O}(k^4)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {O}}(k^4)$$\end{document}

for the derived methods is also studied in the norm

L^2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^2$$\end{document}

. Numerical experiments demonstrate the advantages of the methods in computational accuracy, efficiency, and reliability.

Keywords

Riesz derivative / Reaction-diffusion equation / Allen-Cahn equation / Exponential time-differencing (ETD) method / The damped nonlinear fractional Schrödinger equations / Numerical method / Stability / 26A33 / 35K57 / 65M12 / 65M22

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