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High-order implicit weighted compact nonlinear scheme for nonlinear coupled viscous Burgers’ equations

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  • Zhang, Xu
  • Jiang, Yanqun
  • Hu, Yinggang
  • Chen, Xun

Abstract

In this paper a high-order implicit weighted compact nonlinear scheme for nonlinear coupled viscous Burgers’ equations is presented. The fifth-order weighted compact nonlinear scheme is used for the spatial discretization, while the third-order diagonal implicit Runge–Kutta method is used for the time discretization. The generated nonlinear system is solved by the Jacobian-free Newton–Krylov nonlinear solver, which is composed of the outer Newton iteration method and the inner Krylov subspace iteration method. Stability analysis shows that the presented implicit weighted compact nonlinear scheme is unconditionally stable. Numerical results indicate that the implicit scheme can achieve the designed third-order accuracy in time and has a great advantage in the computation efficiency compared to the third-order explicit total variation diminishing Runge–Kutta weighted essentially non-oscillatory scheme. In addition, the implicit scheme can capture discontinuities and shock waves with high resolution and can solve Burgers’ equations with all kinds of Reynolds numbers.

Suggested Citation

  • Zhang, Xu & Jiang, Yanqun & Hu, Yinggang & Chen, Xun, 2022. "High-order implicit weighted compact nonlinear scheme for nonlinear coupled viscous Burgers’ equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 196(C), pages 151-165.
    • Handle: RePEc:eee:matcom:v:196:y:2022:i:c:p:151-165
    DOI: 10.1016/j.matcom.2022.01.009
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    References listed on IDEAS

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    1. Soliman, A.A., 2009. "On the solution of two-dimensional coupled Burgers’ equations by variational iteration method," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1146-1155.
    2. Guo, Yan & Shi, Yu-feng & Li, Yi-min, 2016. "A fifth-order finite volume weighted compact scheme for solving one-dimensional Burgers’ equation," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 172-185.
    3. Botti, L., 2015. "A choice of forcing terms in inexact Newton iterations with application to pseudo-transient continuation for incompressible fluid flow computations," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 713-737.
    4. Jiang, Yanqun & Chen, Xun & Fan, Rong & Zhang, Xu, 2021. "High order semi-implicit weighted compact nonlinear scheme for viscous Burgers’ equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 607-621.
    5. Doğan Kaya, 2001. "An explicit solution of coupled viscous Burgers' equation by the decomposition method," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 27, pages 1-6, January.
    6. Chen, Changkai & Zhang, Xiaohua & Liu, Zhang, 2020. "A high-order compact finite difference scheme and precise integration method based on modified Hopf-Cole transformation for numerical simulation of n-dimensional Burgers’ system," Applied Mathematics and Computation, Elsevier, vol. 372(C).
    7. Zhao, Guoyan & Sun, Mingbo & Xie, Songbai & Wang, Hongbo, 2018. "Numerical dissipation control in an adaptive WCNS with a new smoothness indicator," Applied Mathematics and Computation, Elsevier, vol. 330(C), pages 239-253.
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    Cited by:

    1. Kaushik, Sonali & Kumar, Rajesh, 2023. "Optimized decomposition method for solving multi-dimensional Burgers’ equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 326-350.

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