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Numerical solution to the Poisson equation under the spherical coordinate system with Bi-CGSTAB method

  • 1.

    School of Computer Science and Technology, University of Science and Technology of China, Hefei 230027, China

  • 2.

    School of Mathematics, Hefei University of Technology, Hefei 230009, China

  • 3.

    School of Engineering Science,University of Science and Technology of China, Hefei 230027, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2017.08.009
  • Received Date: 11 November 2015
  • Rev Recd Date: 24 March 2016
  • Publish Date: 31 August 2017
    Abstract

  • Abstract

    Solving the Poisson equations in spherical coordinate system is a key problem in computational fluid dynamics. Hence, using a conjugate gradient method named Bi-CGSTAB, the typical Poisson equation whose right-hand-side source term is-1 with zero boundary value was solved to give a numerical solution in a domain similar to that in round jet calculation Ω:{ r∈[7,52],  θ∈[-θb,θb], φ∈ [0,2π], θb=arctan(1/14)}. The numerical solution and its relevant residual of discretized equation were discussed.

    Abstract

    Solving the Poisson equations in spherical coordinate system is a key problem in computational fluid dynamics. Hence, using a conjugate gradient method named Bi-CGSTAB, the typical Poisson equation whose right-hand-side source term is-1 with zero boundary value was solved to give a numerical solution in a domain similar to that in round jet calculation Ω:{ r∈[7,52],  θ∈[-θb,θb], φ∈ [0,2π], θb=arctan(1/14)}. The numerical solution and its relevant residual of discretized equation were discussed.
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    • References(6)
  • [1]
    朱祚金.柱和柱群绕流的数值模拟[D]. 上海：上海交通大学，1990.
    [2]
    BAKER T J. Potential flow calculation by approximation factorization method[J]. Journal of Computational Physics, 1981, 42(1): 1-19.
    [3]
    NIU J L, ZHU Z J. Numerical study of three-dimensional flows around two identical square cylinders in staggered arrangement[J]. Physics of Fluid, 2006, 18(4): 044106(1-14).
    [4]
    SUN X W, CHAN C K, MEI B W,et al. LES of convective heat transfer and incompressible fluid flow past a square cylinder[J]. Numerical Heat Transfer, Part A: Applications, 2016, 69(10): 1106-1124.
    [5]
    VUIK C, SEGAL A, MEIJERINKY J A. An efficient preconditioned CG method for the solution of a class of layered problems with extreme contrasts in the coefficients[J]. Journal of Computational Physics, 1999, 152(1): 385-403.
    [6]
    VAN DER VORST H A. BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems[J]. SIAM Journal on Scientific and Statistical Computing, 1992, 13(2): 631-644.
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    [1]
    朱祚金.柱和柱群绕流的数值模拟[D]. 上海：上海交通大学，1990.
    [2]
    BAKER T J. Potential flow calculation by approximation factorization method[J]. Journal of Computational Physics, 1981, 42(1): 1-19.
    [3]
    NIU J L, ZHU Z J. Numerical study of three-dimensional flows around two identical square cylinders in staggered arrangement[J]. Physics of Fluid, 2006, 18(4): 044106(1-14).
    [4]
    SUN X W, CHAN C K, MEI B W,et al. LES of convective heat transfer and incompressible fluid flow past a square cylinder[J]. Numerical Heat Transfer, Part A: Applications, 2016, 69(10): 1106-1124.
    [5]
    VUIK C, SEGAL A, MEIJERINKY J A. An efficient preconditioned CG method for the solution of a class of layered problems with extreme contrasts in the coefficients[J]. Journal of Computational Physics, 1999, 152(1): 385-403.
    [6]
    VAN DER VORST H A. BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems[J]. SIAM Journal on Scientific and Statistical Computing, 1992, 13(2): 631-644.
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