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Compact θ-method for the generalized delay diffusion equation

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  • Zhang, Qifeng
  • Chen, Mengzhe
  • Xu, Yinghong
  • Xu, Dinghua

Abstract

The generalized diffusion equation with a delay has inherent complex nature because its analytical solutions are hard to obtain. Therefore, one has to seek numerical methods, especially the high-order accurate ones, for their approximate solutions. In this paper, we have established the results of the numerical asymptotic stability of the compact θ-method for the generalized delay diffusion equation. It shows that the compact θ-method is asymptotically stable if and only if (k+r)Δth2<10−cos(h)12(1+cos(h))(1−2θ) for θ∈[0,12) and is unconditionally asymptotically stable for θ∈[12,1], respectively. The convergent results in the maximum norm are studied according to the consistency analysis and Lax theorem. In the end, a series of numerical tests on stability and convergence are carried out to support our theoretical results.

Suggested Citation

  • Zhang, Qifeng & Chen, Mengzhe & Xu, Yinghong & Xu, Dinghua, 2018. "Compact θ-method for the generalized delay diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 316(C), pages 357-369.
    • Handle: RePEc:eee:apmaco:v:316:y:2018:i:c:p:357-369
    DOI: 10.1016/j.amc.2017.08.033
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    References listed on IDEAS

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    1. Liang, Hui, 2015. "Convergence and asymptotic stability of Galerkin methods for linear parabolic equations with delays," Applied Mathematics and Computation, Elsevier, vol. 264(C), pages 160-178.
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    Cited by:

    1. Zhang, Qifeng & Ren, Yunzhu & Lin, Xiaoman & Xu, Yinghong, 2019. "Uniform convergence of compact and BDF methods for the space fractional semilinear delay reaction–diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 91-110.
    2. Qin, Hongyu & Zhang, Qifeng & Wan, Shaohua, 2019. "The continuous Galerkin finite element methods for linear neutral delay differential equations," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 76-85.
    3. Ran, Maohua & Luo, Taibai & Zhang, Li, 2019. "Unconditionally stable compact theta schemes for solving the linear and semi-linear fourth-order diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 342(C), pages 118-129.
    4. 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).
    5. Qin, Hongyu & Wu, Fengyan & Ding, Deng, 2022. "A linearized compact ADI numerical method for the two-dimensional nonlinear delayed Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 412(C).

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