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Adaptive spectral solver for Riesz fractional reaction–diffusion equations via penalized minimum residual iteration

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  • Guan, Chaoyue
  • Zhang, Jian

Abstract

A high-order solver is presented for two-dimensional Riesz fractional nonlinear reaction–diffusion equations. It employs a midpoint starter and a three-point backward differentiation formula (BDF2) to achieve second-order temporal accuracy, together with a weighted Jacobi spectral approximation that delivers nearly exponential spatial convergence for analytic solutions. After Newton linearization, each correction is obtained via a penalized Levenberg–Marquardt minimum residual method (PLM-MRM). This iteration adaptively enforces boundary conditions without requiring boundary-fitted basis functions. We establish stability and rigorous a priori error bounds. Numerical experiments over a wide range of fractional orders confirm these rates and drive the residual to machine precision within a few PLM-MRM sweeps. Compared with a conventional LM update, global errors are reduced by up to 35%, and by one to two orders of magnitude relative to Galerkin-BDF or Crank–Nicolson (CN) baselines. For a given accuracy, the scheme allows time steps up to about four times larger than a recent fourth-order CN method.

Suggested Citation

  • Guan, Chaoyue & Zhang, Jian, 2026. "Adaptive spectral solver for Riesz fractional reaction–diffusion equations via penalized minimum residual iteration," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 241(PB), pages 431-451.
    • Handle: RePEc:eee:matcom:v:241:y:2026:i:pb:p:431-451
    DOI: 10.1016/j.matcom.2025.10.027
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    References listed on IDEAS

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    1. Cheng, Xiujun & Duan, Jinqiao & Li, Dongfang, 2019. "A novel compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction–diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 452-464.
    2. Xing, Zhiyong & Wen, Liping, 2019. "Numerical analysis and fast implementation of a fourth-order difference scheme for two-dimensional space-fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 155-166.
    3. Abdelkawy, M.A. & Alyami, S.A., 2021. "Legendre-Chebyshev spectral collocation method for two-dimensional nonlinear reaction-diffusion equation with Riesz space-fractional," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    4. Qu, Wei & Li, Zhi, 2021. "Fast direct solver for CN-ADI-FV scheme to two-dimensional Riesz space-fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 401(C).
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