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L1/LDG method for the generalized time-fractional Burgers equation

Author

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  • Li, Changpin
  • Li, Dongxia
  • Wang, Zhen

Abstract

In this paper, we study the generalized time fractional Burgers equation, where the time fractional derivative is in the sense of Caputo with derivative order in (0,1). If its solution u(x,t) has strong regularity, for example u(⋅,t)∈C2[0,T] for a given time T, then we use the L1 scheme on uniform meshes to approximate the Caputo time-fractional derivative, and use the local discontinuous Galerkin (LDG) method to approach the space derivative. However, the solution u(x,t) likely behaves a certain regularity at the starting time, i.e., ∂u∂t and ∂2u∂2t can blow up as t→0+ albeit u(⋅,t)∈C[0,T] for a given time T. In this case, we use the L1 scheme on non-uniform meshes to approximate the Caputo time-fractional derivative, and use the LDG method to discretize the spatial derivative. The fully discrete schemes for both situations are established and analyzed. It is shown that the derived schemes are numerically stable and convergent. Finally, several numerical experiments are provided which support the theoretical analysis.

Suggested Citation

  • Li, Changpin & Li, Dongxia & Wang, Zhen, 2021. "L1/LDG method for the generalized time-fractional Burgers equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 357-378.
    • Handle: RePEc:eee:matcom:v:187:y:2021:i:c:p:357-378
    DOI: 10.1016/j.matcom.2021.03.005
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    References listed on IDEAS

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    1. Zhang, Jinghua & Liu, Fawang & Lin, Zeng & Anh, Vo, 2019. "Analytical and numerical solutions of a multi-term time-fractional Burgers’ fluid model," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 1-12.
    2. Li, Changpin & Wang, Zhen, 2020. "The discontinuous Galerkin finite element method for Caputo-type nonlinear conservation law," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 169(C), pages 51-73.
    3. Li, Changpin & Wang, Zhen, 2021. "Non-uniform L1/discontinuous Galerkin approximation for the time-fractional convection equation with weak regular solution," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 838-857.
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    Cited by:

    1. Zhang, Qifeng & Sun, Cuicui & Fang, Zhi-Wei & Sun, Hai-Wei, 2022. "Pointwise error estimate and stability analysis of fourth-order compact difference scheme for time-fractional Burgers’ equation," Applied Mathematics and Computation, Elsevier, vol. 418(C).
    2. Peng, Xiangyi & Xu, Da & Qiu, Wenlin, 2023. "Pointwise error estimates of compact difference scheme for mixed-type time-fractional Burgers’ equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 702-726.

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    More about this item

    Keywords

    Caputo derivative; L1 scheme; Local discontinuous Galerkin method; Stability; Convergence;
    All these keywords.

    JEL classification:

    • L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance

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