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Numerical treatment of Burgers' equation based on weakly L-stable generalized time integration formula with the NSFD scheme

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  • Rawani, Mukesh Kumar
  • Verma, Amit Kumar
  • Verma, Lajja

Abstract

In this study, we present a weakly L-stable convergent time integration formula of order N1>3 (N1∈Z is odd) to solve Burgers' equation. The time integration formula for the initial value problem w′(t)=f(t,w),w(t0)=η0 is formulated using backward explicit Taylor series approximation of order (N1−1) and Hermite approximation polynomial of order (N1−2). We convert the Burgers' equation into the initial value problem using the nonstandard finite difference scheme for spatial derivatives and implement the derived integration formula. The nonstandard finite difference scheme makes it possible to choose several denominator functions. The method's convergence, stability and truncation error are also discussed. To show the correctness and effectiveness of the proposed technique, we present numerical solutions, ‖e‖2 and ‖e‖∞ error norms in several tables and figures. Additionally, the numerical outcomes are compared with the results of some existing techniques and exact solutions.

Suggested Citation

  • Rawani, Mukesh Kumar & Verma, Amit Kumar & Verma, Lajja, 2024. "Numerical treatment of Burgers' equation based on weakly L-stable generalized time integration formula with the NSFD scheme," Applied Mathematics and Computation, Elsevier, vol. 467(C).
    • Handle: RePEc:eee:apmaco:v:467:y:2024:i:c:s0096300323006549
    DOI: 10.1016/j.amc.2023.128485
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    References listed on IDEAS

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    1. Jie Zhao & Hong Li & Zhichao Fang & Xue Bai, 2020. "Numerical Solution of Burgers’ Equation Based on Mixed Finite Volume Element Methods," Discrete Dynamics in Nature and Society, Hindawi, vol. 2020, pages 1-13, March.
    2. Saka, Bülent & Dağ, İdris, 2007. "Quartic B-spline collocation method to the numerical solutions of the Burgers’ equation," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 1125-1137.
    3. Kayenat, Sheerin & Verma, Amit Kumar, 2022. "On the choice of denominator functions and convergence of NSFD schemes for a class of nonlinear SBVPs," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 263-284.
    4. Pervaiz, Nosheen & Aziz, Imran, 2020. "Haar wavelet approximation for the solution of cubic nonlinear Schrodinger equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
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