A Velocity Decomposition Approach for Solving the Immersed Interface Problem with Dirichlet Boundary Conditions

  • Anita T. LaytonEmail author
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 155)


In a previous study, we presented a second-order accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness (Beale and Layton, J Comput Phys 228:3358–3367, 2009). The fluid flow was described by the Navier-Stokes equations with periodic boundary conditions, and the deformation of the moving interface exerts a singular force onto the fluid. In this study, we extend that method to Dirichlet boundary conditions. We decompose the velocity into three parts: a “Stokes” part, a “regular” part, and a “boundary correction” part. The “Stokes” part is determined by the Stokes equations and the singular interfacial force. The Stokes solution is obtained using the immersed interface method, which gives second-order accurate values by incorporating known jumps for the solution and its derivatives into a finite difference method. The regular part of the velocity is given by the Navier-Stokes equations with a body force resulting from the Stokes part, and with periodic boundary conditions. The regular velocity is obtained using a time-stepping method that combines the semi-Lagrangian method with the backward difference formula. Because the body force is continuous, jump conditions are not necessary. The boundary correction solution is described by the unforced Navier-Stokes equations, with Dirichet boundary conditions given by the difference between the Dirichlet boundary conditions of the overall Navier-Stokes solution, and the boundary values of the Stokes and regular velocities. Because the boundary correction solution is sufficiently smooth, jump conditions are also not necessary. Numerical results exhibit approximately second-order accuracy in time and space.


Dirichlet Boundary Condition Total Derivative Jump Condition Jump Discontinuity Immerse Boundary Method 
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.


  1. [1]
    Beale JT, Layton AT (2009) A velocity decomposition approach for moving interfaces in viscous fluids. J Comput Phys 228:3358–3367MathSciNetCrossRefGoogle Scholar
  2. [2]
    Lee L, LeVeque RJ (2003) An immersed interface method for the incompressible Navier-Stokes equations. SIAM J Sci Comput 25:832–856MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    LeVeque RJ, Li Z (1994) The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J Numer Anal 31:1019–1044MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    LeVeque RJ, Li Z (1997) Immersed interface methods for Stokes flow with elastic boundaries or surface tension. SIAM J Sci Comput 18(3):709–735MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Li Z, Lai M-C (2001) The immersed interface method for the Navier-Stokes equations with singular forces. J Comput Phys 171:822–842MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Peskin CS (2002) The immersed boundary method. Acta Numer 11:479–517MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Peskin CS, Printz BF (1993) Improved volume conservation in the computation of flows with immersed elastic boundaries. J Comput Phys 105:33–46MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Pozrikidis C (1992) Boundary integral and singularity methods for linearized viscous flow. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  9. [9]
    Tu C, Peskin CS (1992) Stability and instability in the computation of flows with moving immersed boundaries: a comparison of three methods. SIAM J Sci Stat Comput 13:1361–1376MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA

Personalised recommendations