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Fast numerical schemes for nonlinear space-fractional multidelay reaction-diffusion equations by implicit integration factor methods

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  • Jian, Huan-Yan
  • Huang, Ting-Zhu
  • Ostermann, Alexander
  • Gu, Xian-Ming
  • Zhao, Yong-Liang

Abstract

In this article, a fast and efficient numerical method is constructed for solving nonlinear space-fractional multidelay reaction-diffusion equations. Firstly, we spatially discretize the equation using the weighted and shifted Grünwald–Letnikov difference (WSGD) formula. As a result, a nonlinear system of ordinary differential equations (ODEs) is obtained. Then, based on the fact that the implicit integration factor (IIF) method is an effective time-stepping scheme with good stability properties, we develop a multidelay IIF (MIIF) method to deal with the resulting ODE system. Compared with traditional numerical schemes, such as the backward differential formula (BDF) and the Crank–Nicolson scheme (CN), the proposed MIIF scheme has two main advantages: (1) MIIF can achieve high-order accuracy in time; (2) the temporal errors are smaller and the convergence orders observed with MIIF are more regular. In addition, in order to overcome the computational challenges, we propose some Krylov subspace methods to calculate the actions of the Toeplitz matrix exponentials in MIIF. Finally, numerical examples are presented to confirm the accuracy of the MIIF scheme and to demonstrate the considerable computational advantages of the proposed fast solving algorithms.

Suggested Citation

  • Jian, Huan-Yan & Huang, Ting-Zhu & Ostermann, Alexander & Gu, Xian-Ming & Zhao, Yong-Liang, 2021. "Fast numerical schemes for nonlinear space-fractional multidelay reaction-diffusion equations by implicit integration factor methods," Applied Mathematics and Computation, Elsevier, vol. 408(C).
    • Handle: RePEc:eee:apmaco:v:408:y:2021:i:c:s0096300321004495
    DOI: 10.1016/j.amc.2021.126360
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