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Numerical Analysis of Convection–Diffusion Using a Modified Upwind Approach in the Finite Volume Method

Author

Listed:
  • Arafat Hussain

    (School of Mathematics and Statistics, Central South University, Changsha 410083, China)

  • Zhoushun Zheng

    (School of Mathematics and Statistics, Central South University, Changsha 410083, China)

  • Eyaya Fekadie Anley

    (School of Mathematics and Statistics, Central South University, Changsha 410083, China
    Department of Mathematics, College of Natural and Computational Science, Arba-Minch University, Arba-Minch 21, Ethiopia)

Abstract

The main focus of this study was to develop a numerical scheme with new expressions for interface flux approximations based on the upwind approach in the finite volume method. Our new proposed numerical scheme is unconditionally stable with second-order accuracy in both space and time. The method is based on the second-order formulation for the temporal approximation, and an upwind approach of the finite volume method is used for spatial interface approximation. Some numerical experiments have been conducted to illustrate the performance of the new numerical scheme for a convection–diffusion problem. For the phenomena of convection dominance and diffusion dominance, we developed a comparative study of this new upwind finite volume method with an existing upwind form and central difference scheme of the finite volume method. The modified numerical scheme shows highly accurate results as compared to both numerical schemes.

Suggested Citation

  • Arafat Hussain & Zhoushun Zheng & Eyaya Fekadie Anley, 2020. "Numerical Analysis of Convection–Diffusion Using a Modified Upwind Approach in the Finite Volume Method," Mathematics, MDPI, vol. 8(11), pages 1-21, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:11:p:1869-:d:436053
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    References listed on IDEAS

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    1. Mehdi Dehghan, 2005. "On the numerical solution of the one-dimensional convection-diffusion equation," Mathematical Problems in Engineering, Hindawi, vol. 2005, pages 1-14, January.
    2. Renwei Liu & Dongjie Wang & Xinyu Zhang & Wang Li & Bo Yu, 2013. "Comparison Study on the Performances of Finite Volume Method and Finite Difference Method," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-10, July.
    3. Wei, Tao & Xu, Mingtian, 2016. "An integral equation approach to the unsteady convection–diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 55-64.
    4. Wei Gu & Peng Wang, 2014. "A Crank-Nicolson Difference Scheme for Solving a Type of Variable Coefficient Delay Partial Differential Equations," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-6, February.
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    Cited by:

    1. Jen-Yi Chang & Ru-Yun Chen & Chia-Cheng Tsai, 2022. "Hermite Method of Approximate Particular Solutions for Solving Time-Dependent Convection-Diffusion-Reaction Problems," Mathematics, MDPI, vol. 10(2), pages 1-20, January.

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