Finite Difference Schemes for Incompressible Flows on Fully Adaptive Grids

  • Frédéric Gibou
  • Chohong Min
  • Hector Ceniceros
Part of the International Series of Numerical Mathematics book series (ISNM, volume 154)


We describe a finite difference scheme for simulating incompressible flows on nonuniform meshes using quadtree/octree data structure. A semi- Lagrangian method is used to update the intermediate fluid velocity in a standard projection framework. Two Poisson solvers on fully adaptive grids are also described. The first one is cell-centered and yields first-order accurate solutions, while producing symmetric linear systems (see Losasso, Gibou and Fedkiw [15]). The second is node-based and yields second-order accurate solutions, while producing nonsymmetric linear systems (see Min, Gibou and Ceniceros [17]). A distinguishing feature of the node-based algorithm is that gradients are found to second-order accuracy as well. The schemes are fully adaptive, i.e., the difference of level between two adjacent cells can be arbitrary. Numerical results are presented in two and three spatial dimensions.


Poisson Equation Adaptive Mesh Adaptive Grid Nonuniform Mesh Nonsymmetric Linear System 
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]
    A. Almgren, J. Bell, P. Colella, L. Howell, and M. Welcome. A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations. J. Comput. Phys., 142:1–46, 1998.CrossRefMathSciNetGoogle Scholar
  2. [2]
    M. Berger and J. Oliger. Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys., 53:484–512, 1984.CrossRefMathSciNetGoogle Scholar
  3. [3]
    D. Brown, R. Cortez, and M. Minion. Accurate projection methods for the incompressible Navier-Stokes equations. J. Comput. Phys., 168:464–499, 2001.CrossRefMathSciNetGoogle Scholar
  4. [4]
    A. Chorin. A Numerical Method for Solving Incompressible Viscous Flow Problems. J. Comput. Phys., 2:12–26, 1967.CrossRefGoogle Scholar
  5. [5]
    F. Gibou and R. Fedkiw. A fourth-order accurate discretization for the laplace and heat equations on arbitrary domains, with applications to the Stefan problem. J, Comput. Phys., 202:577–601, 2005.CrossRefMathSciNetGoogle Scholar
  6. [6]
    F. Gibou, R. Fedkiw, L.-T. Cheng, and M. Kang. A second-order accurate symmetric discretization of the Poisson equation on irregular domains. J. Comput. Phys., 176:205–227, 2002.CrossRefMathSciNetGoogle Scholar
  7. [7]
    F. Ham, F. Lien, and A. Strong. A fully conservative second-order finite difference scheme for incompressible flow on nonuniform grids. J. Comput. Phys., 117:117–133, 2002.CrossRefGoogle Scholar
  8. [8]
    F. Harlow and J. Welch. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids, 8:2182–2189, 1965.CrossRefGoogle Scholar
  9. [9]
    A. Harten, B. Enquist, S. Osher, and S Chakravarthy. Uniformly high-order accurate essentially non-oscillatory schemes III. J. Comput. Phys., 71:231–303, 1987.CrossRefMathSciNetGoogle Scholar
  10. [10]
    H. Johansen and P. Colella. A Cartesian grid embedded boundary method for Poisson’s equation on irregular domains. J. Comput. Phys., 147:60–85, 1998.CrossRefMathSciNetGoogle Scholar
  11. [11]
    H.O. Kreiss, H.-O. Manteuffel, T.A. Schwartz, B. Wendroff, and A.B. White Jr. Supra-convergent schemes on irregular grids. Math. Comp., 47:537–554, 1986.CrossRefMathSciNetGoogle Scholar
  12. [12]
    K. Lipnikov, J. Morel, and M. Shashkov. Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes. J. Comput. Phys., 199:589–597, 2004.CrossRefGoogle Scholar
  13. [13]
    X.-D. Liu, S. Osher, and T. Chan. Weighted essentially non-oscillatory schemes. J. Comput. Phys., 126:202–212, 1996.CrossRefMathSciNetGoogle Scholar
  14. [14]
    F. Losasso, R. Fedkiw, and S. Osher. Spatially adaptive techniques for level set methods and incompressible flow. Computers and Fluids (in press).Google Scholar
  15. [15]
    F. Losasso, F. Gibou, and R. Fedkiw. Simulating water and smoke with an octree data structure. ACM Trans. Graph. (SIGGRAPH Proc.), pages 457–462, 2004.Google Scholar
  16. [16]
    T. Manteuffel and A. White. The numerical solution of second-order boundary value problems on nonuniform meshes. Math. Comput., 47(176):511–535, 1986.CrossRefMathSciNetGoogle Scholar
  17. [17]
    C. Min, F. Gibou, and H. Ceniceros. A simple second-order accurate finite difference scheme for the variable coefficient poisson equation on fully adaptive grids. CAM report 05-29, Submitted to J. Comput. Phys. Google Scholar
  18. [18]
    S. Popinet. Gerris: A tree-based adaptive solver for the incompressible euler equations in complex geometries. J. Comput. Phys., 190:572–600, 2003.CrossRefMathSciNetGoogle Scholar
  19. [19]
    H. Samet. The Design and Analysis of Spatial Data Structures. Addison-Wesley, New York, 1989.Google Scholar
  20. [20]
    C.-W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock capturing schemes II (two). J. Comput. Phys., 83:32–78, 1989.CrossRefMathSciNetGoogle Scholar
  21. [21]
    A. Staniforth and J. Cote. Semi-Lagrangian Integration Schemes for Atmospheric Models: A Review. Monthly Weather Review, 119:2206–2223, 1991.CrossRefGoogle Scholar
  22. [22]
    J. Strain. Fast tree-based redistancing for level set computations. J. Comput. Phys., 152:664–686, 1999.CrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Frédéric Gibou
    • 1
  • Chohong Min
    • 2
  • Hector Ceniceros
    • 2
  1. 1.Department of Mechanical Engineering & Department of Computer ScienceUCSBUSA
  2. 2.Department of MathematicsUCSBUSA

Personalised recommendations