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A reduced-order finite volume element formulation based on POD method and numerical simulation for two-dimensional solute transport problems

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  • Luo, Zhendong
  • Li, Hong
  • Sun, Ping
  • An, Jing
  • Navon, Ionel Michael

Abstract

Proper orthogonal decomposition (POD) method has been successfully used in the reduced-order modeling of complex systems. In this paper, we extend the applications of POD method, i.e., combine the classical finite volume element (FVE) method with the POD method to obtain a reduced-order FVE formulation with lower dimensions and sufficiently high accuracy for two-dimensional solute transport problems, which have real life practical applications. We then provide error estimates between the reduced-order POD FVE solutions and classical FVE solutions and we provide implementation of an extrapolation algorithm for solving the reduced-order FVE formulation. Thus, we provide the theoretical basis for practical applications. A numerical example is then used to ascertain that the results of numerical computation are consistent with the theoretical derivations. Moreover, it is shown that the reduced-order FVE formulation based on POD method is both feasible and efficient for solving two-dimensional solute transport problems.

Suggested Citation

  • Luo, Zhendong & Li, Hong & Sun, Ping & An, Jing & Navon, Ionel Michael, 2013. "A reduced-order finite volume element formulation based on POD method and numerical simulation for two-dimensional solute transport problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 89(C), pages 50-68.
    • Handle: RePEc:eee:matcom:v:89:y:2013:i:c:p:50-68
    DOI: 10.1016/j.matcom.2012.11.012
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    References listed on IDEAS

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    1. K. Kunisch & S. Volkwein, 1999. "Control of the Burgers Equation by a Reduced-Order Approach Using Proper Orthogonal Decomposition," Journal of Optimization Theory and Applications, Springer, vol. 102(2), pages 345-371, August.
    2. Li, Baibing & Martin, Elaine B. & Morris, A. Julian, 2002. "On principal component analysis in L1," Computational Statistics & Data Analysis, Elsevier, vol. 40(3), pages 471-474, September.
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    Cited by:

    1. Luo, Zhendong & Teng, Fei & Chen, Jing, 2018. "A POD-based reduced-order Crank–Nicolson finite volume element extrapolating algorithm for 2D Sobolev equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 146(C), pages 118-133.
    2. Bai, Feng & Wang, Yi, 2022. "A reduced order modeling method based on GNAT-embedded hybrid snapshot simulation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 199(C), pages 100-132.

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