A Fourth-Order Iterative Solver for the Singular Poisson Equation

  • Stéphane Abide
  • Xavier Chesneau
  • Belkacem Zeghmati
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)


A compact fourth-order finite difference scheme solver devoted to the singular-Poisson equation is proposed and verified. The solver is based on a mixed formulation: the Poisson equation is splitted into a system of partial differential equations of the first order. This system is then discretized using a fourth-order compact scheme. This leads to a sparse linear system but introduces new variables related to the gradient of an unknow function. The Schur factorization allows us to work on a linear sub-problem for which a conjugated-gradient preconditioned by an algebraic multigrid method is proposed.Numerical results show that the new proposed Poisson solver is efficient while retaining the fourth-order compact accuracy.


Large Eddy Simulation Poisson Equation Mixed Formulation Stagger Grid Compact Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ferziger, J.H., Perić, M.: Computational Methods for Fluid Dynamics. Springer-verlag edn. Springer (2002)Google Scholar
  2. 2.
    Lele, S.K.: Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics 103(1), 16–42 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boersma, B.J.: A 6th order staggered compact finite difference method for the incompressible Navier-Stokes and scalar transport equations. Journal of Computational Physics 230(12), 4940–4954 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brüger, A., Gustafsson, B., Lötstedt, P., Nilsson, J.: High-order accurate solution of the incompressible Navier-Stokes equations. Journal of Computational Physics 203(1), 49–71 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Demuren, A.O., Wilson, R.V., Carpenter, M.: Higher order compact schemes for numerical simulation of incompressible flows, part 1: Theroretical development. Numerical Heat Transfer Part B Fundamentals 39, 207–230 (2001)CrossRefGoogle Scholar
  6. 6.
    Schiestel, R., Viazzo, S.: A hermitian-fourier numerical method for solving the incompressible Navier-Stokes equations. Computers & Fluids 24(6), 739–752 (1995)CrossRefzbMATHGoogle Scholar
  7. 7.
    Knikker, R.: Study of a staggered fourth-order compact scheme for unsteady incompressible viscous flows. International Journal for Numerical Methods in Fluids 59(10), 1063–1092 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Collatz, L.: The Numerical Treatment of Differential Equations. Springer, Berlin (1960)CrossRefzbMATHGoogle Scholar
  9. 9.
    Spotz, W.F., Carey, G.F.: A high-order compact formulation for the 3D Poisson equation. Numerical Methods for Partial Differential Equations 12(2), 235–243 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ge, L., Zhang, J.: High Accuracy Iterative Solution of Convection Diffusion Equation with Boundary Layers on Nonuniform Grids. Journal of Computational Physics 171(2), 560–578 (2001)CrossRefzbMATHGoogle Scholar
  11. 11.
    Spotz, W.F.: Formulation and experiments with high-order compact schemes for nonuniform grids. International Journal of Numerical Methods for Heat Fluid Flow 8(3), 288–303 (1998)CrossRefzbMATHGoogle Scholar
  12. 12.
    Gupta, M.M., Zhang, J.: High accuracy multigrid solution of the 3D convection diffusion equation. Applied Mathematics and Computation 113 (2000)Google Scholar
  13. 13.
    Zhuang, Y., Sun, X.H.: A High-Order Fast Direct Solver for Singular Poisson Equations. Journal of Computational Physics 171(1), 79–94 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Vedy, E., Viazzo, S., Schiestel, R.: A high-order finite difference method for incompressible fluid turbulence simulations. International Journal for Numerical Methods in Fluids 42(11), 1155–1188 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Abide, S., Viazzo, S.: A 2D compact fourth-order projection decomposition method. Journal of Computational Physics 206(1), 252–276 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Carey, G.F., Spotz, W.F.: Higher-order compact mixed methods. Communications in Numerical Methods in Engineering 13(7), 553–564 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nagarajan, S., Lele, S.K., Ferziger, J.H.: A robust high-order compact method for large eddy simulation. Journal of Computational Physics 191(2), 392–419 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Notay, Y.: An aggregation-based algebraic multigrid method. Electronic Transactions on Numerical Analysis 37, 123–146 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Stéphane Abide
    • 1
  • Xavier Chesneau
    • 1
  • Belkacem Zeghmati
    • 1
  1. 1.LAboratoire de Mathématiques et PhySique, EA 4217Univ. Perpignan via DomitiaPerpignanFrance

Personalised recommendations