2012, 17(4): 1139-1153. doi: 10.3934/dcdsb.2012.17.1139

A partially implicit hybrid method for computing interface motion in Stokes flow

1. 

Department of Mathematics, Duke University, Durham, NC 27708, United States

Received  September 2010 Revised  August 2011 Published  February 2012

We present a partially implicit hybrid method for simulating the motion of a stiff interface immersed in Stokes flow, in free space or in a rectangular domain with boundary conditions. We assume the interface is a closed curve which remains in the interior of the computational region. The implicit time integration is based on the small-scale decomposition approach and does not require the iterative solution of a system of nonlinear equations. First-order and second-order versions of the time-stepping method are derived systematically, and numerical results indicate that both methods are substantially more stable than explicit methods. At each time level, the Stokes equations are solved using a hybrid approach combining nearly singular integrals on a band of mesh points near the interface and a mesh-based solver. The solutions are second-order accurate in space and preserve the jump discontinuities across the interface. Finally, the hybrid method can be used as an alternative to adaptive mesh refinement to resolve boundary layers that are frequently present around a stiff immersed interface.
Citation: Anita T. Layton, J. Thomas Beale. A partially implicit hybrid method for computing interface motion in Stokes flow. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1139-1153. doi: 10.3934/dcdsb.2012.17.1139
References:
[1]

C. R. Anderson, A method of local corrections for computing the velocity field due to a distribution of vortex blobs,, J. Comp. Phys., 62 (1986), 111. doi: 10.1016/0021-9991(86)90102-6.

[2]

J. T. Beale, T. Y. Hou and J. S. Lowengrub, Growth rates for the linearized motion of fluid interfaces away from equilibrium,, Comm. Pure Appl. Math., 46 (1993), 1269. doi: 10.1002/cpa.3160460903.

[3]

J. T. Beale and M.-C. Lai, A method for computing nearly singular integrals,, SIAM J. Numer. Anal., 38 (2001), 1902. doi: 10.1137/S0036142999362845.

[4]

J. T. Beale and A. T. Layton, On the accuracy of finite difference methods for elliptic problems with interfaces,, Comm. Appl. Math. Comput. Sci., 1 (2006), 91. doi: 10.2140/camcos.2006.1.91.

[5]

M. J. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations,, J. Comp. Phys., 53 (1984), 484. doi: 10.1016/0021-9991(84)90073-1.

[6]

H. D. Ceniceros, J. E. Fisher and A. M. Roma, Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method,, J. Comput. Phys., 228 (2009), 7137. doi: 10.1016/j.jcp.2009.05.031.

[7]

N. G. Cogan, R. Cortez and L. J. Fauci, Modeling physiological resistance in bacterial biofilms,, Bull. Math. Biol., 67 (2005), 831. doi: 10.1016/j.bulm.2004.11.001.

[8]

R. Cortez, The method of regularized Stokeslets,, SIAM J. Sci. Comput., 23 (2001), 1204. doi: 10.1137/S106482750038146X.

[9]

L. J. Fauci and A. L. Folgelson, Truncated Newton methods and the modeling of complex immersed elastic structures,, Comm. Pure Appl. Math., 66 (1993), 787. doi: 10.1002/cpa.3160460602.

[10]

T. Y. Hou, J. S. Lowengrub and M. J. Shelley, Removing the stiffness from interfacial flows with surface tension,, J. Comput. Phys., 114 (1994), 312. doi: 10.1006/jcph.1994.1170.

[11]

T. Y. Hou and Z. Shi, An efficient semi-implicit immersed boundary method for the Navier-Stokes equations,, J. Comput. Phys., 227 (2008), 8968. doi: 10.1016/j.jcp.2008.07.005.

[12]

T. Y. Hou and Z. Shi, Removing the stiffness of elastic force from the immersed boundary method for the 2D Stokes equations,, J. Comput. Phys., 227 (2008), 9138. doi: 10.1016/j.jcp.2008.03.002.

[13]

M. C. A. Kropinski, An efficient numerical method for studying interfacial motion in two-dimensional creeping flows,, J. Comput. Phys., 171 (2001), 479. doi: 10.1006/jcph.2001.6787.

[14]

L. Lee and R. J. LeVeque, An immersed interface method for the incompressible Navier-Stokes equations,, SIAM J. Sci. Comp., 25 (2003), 832. doi: 10.1137/S1064827502414060.

[15]

R. J. LeVeque and Z. Li, Immersed interface methods for Stokes flow with elastic boundaries or surface tension,, SIAM J. Sci. Comput., 18 (1997), 709. doi: 10.1137/S1064827595282532.

[16]

Z. Li and M.-C. Lai, The immersed interface method for the Navier-Stokes equations with singular forces,, J. Comput. Phys., 171 (2001), 822. doi: 10.1006/jcph.2001.6813.

[17]

Z. Li and S. R. Lubkin, Numerical analysis of interfacial two-dimensional Stokes flow with discontinuous viscosity and variable surface tension,, Int. J. Numer. Meth. Fluids, 37 (2001), 525. doi: 10.1002/fld.185.

[18]

A. Mayo, The fast solution of Poisson's and the biharmonic equations on irregular regions,, SIAM J. Numer. Anal., 21 (1984), 285. doi: 10.1137/0721021.

[19]

A. Mayo, Fast high order accurate solution of Laplace's equation on irregular regions,, SIAM J. Sci. Statist. Comput., 6 (1985), 144. doi: 10.1137/0906012.

[20]

A. Mayo and C. S. Peskin, An implicit numerical method for fluid dynamics problems with immersed elastic boundaries,, in, 141 (1993), 261.

[21]

Y. Mori and C. S. Peskin, Implicit second-order immersed boundary methods with boundary mass,, Comput. Methods Appl. Mech. Engin., 197 (2008), 2049. doi: 10.1016/j.cma.2007.05.028.

[22]

E. Newren, A. Fogelson, R. Guy and M. Kirby, A comparison of implicit solvers for the immersed boundary equations,, Comput. Methods Appl. Mech. Engin., 197 (2008), 2290. doi: 10.1016/j.cma.2007.11.030.

[23]

E. P. Newren, A. L. Fogelson, R. D. Guy and R. M. Kirby, Unconditionally stable discretizations of the immersed boundary equations,, J. Comput. Phys., 222 (2007), 702. doi: 10.1016/j.jcp.2006.08.004.

[24]

C. S. Peskin, Numerical analysis of blood flow in the heart,, J. Comput. Phys., 25 (1977), 220. doi: 10.1016/0021-9991(77)90100-0.

[25]

C. S. Peskin, The immersed boundary method,, Acta Numerica, 11 (2002), 479. doi: 10.1017/S0962492902000077.

[26]

C. S. Peskin and B. F. Printz, Improved volume conservation in the computation of flows with immersed elastic boundaries,, J. Comput. Phys., 105 (1993), 33. doi: 10.1006/jcph.1993.1051.

[27]

J. S. Sohn, Y.-H. Tseng, S. Li, A. Voigt and J. S. Lowengrub, Dynamics of multicomponent vesicles in a viscous fluid,, J. Comput. Phys., 229 (2010), 119. doi: 10.1016/j.jcp.2009.09.017.

[28]

J. M. Stockie and B. R. Wetton, Analysis of stiffness in the immersed boundary method and implications for time-stepping schemes,, J. Comput. Phys., 154 (1999), 41. doi: 10.1006/jcph.1999.6297.

[29]

A.-K. Tornberg and M. J. Shelley, Simulating the dynamics and interactions of flexible fibers in Stokes flows,, J. Comput. Phys., 196 (2004), 8. doi: 10.1016/j.jcp.2003.10.017.

[30]

C. Tu and C. S. Peskin, Stability and instability in the computation of flows with moving immersed boundaries: A comparison of three methods,, SIAM J. Sci. Statist. Comput., 13 (1992), 1361. doi: 10.1137/0913077.

[31]

S. K. Veerapaneni, D. Gueyffier, D. Zorin and G. Biros, A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D,, J. Comput. Phys., 228 (2009), 2334. doi: 10.1016/j.jcp.2008.11.036.

show all references

References:
[1]

C. R. Anderson, A method of local corrections for computing the velocity field due to a distribution of vortex blobs,, J. Comp. Phys., 62 (1986), 111. doi: 10.1016/0021-9991(86)90102-6.

[2]

J. T. Beale, T. Y. Hou and J. S. Lowengrub, Growth rates for the linearized motion of fluid interfaces away from equilibrium,, Comm. Pure Appl. Math., 46 (1993), 1269. doi: 10.1002/cpa.3160460903.

[3]

J. T. Beale and M.-C. Lai, A method for computing nearly singular integrals,, SIAM J. Numer. Anal., 38 (2001), 1902. doi: 10.1137/S0036142999362845.

[4]

J. T. Beale and A. T. Layton, On the accuracy of finite difference methods for elliptic problems with interfaces,, Comm. Appl. Math. Comput. Sci., 1 (2006), 91. doi: 10.2140/camcos.2006.1.91.

[5]

M. J. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations,, J. Comp. Phys., 53 (1984), 484. doi: 10.1016/0021-9991(84)90073-1.

[6]

H. D. Ceniceros, J. E. Fisher and A. M. Roma, Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method,, J. Comput. Phys., 228 (2009), 7137. doi: 10.1016/j.jcp.2009.05.031.

[7]

N. G. Cogan, R. Cortez and L. J. Fauci, Modeling physiological resistance in bacterial biofilms,, Bull. Math. Biol., 67 (2005), 831. doi: 10.1016/j.bulm.2004.11.001.

[8]

R. Cortez, The method of regularized Stokeslets,, SIAM J. Sci. Comput., 23 (2001), 1204. doi: 10.1137/S106482750038146X.

[9]

L. J. Fauci and A. L. Folgelson, Truncated Newton methods and the modeling of complex immersed elastic structures,, Comm. Pure Appl. Math., 66 (1993), 787. doi: 10.1002/cpa.3160460602.

[10]

T. Y. Hou, J. S. Lowengrub and M. J. Shelley, Removing the stiffness from interfacial flows with surface tension,, J. Comput. Phys., 114 (1994), 312. doi: 10.1006/jcph.1994.1170.

[11]

T. Y. Hou and Z. Shi, An efficient semi-implicit immersed boundary method for the Navier-Stokes equations,, J. Comput. Phys., 227 (2008), 8968. doi: 10.1016/j.jcp.2008.07.005.

[12]

T. Y. Hou and Z. Shi, Removing the stiffness of elastic force from the immersed boundary method for the 2D Stokes equations,, J. Comput. Phys., 227 (2008), 9138. doi: 10.1016/j.jcp.2008.03.002.

[13]

M. C. A. Kropinski, An efficient numerical method for studying interfacial motion in two-dimensional creeping flows,, J. Comput. Phys., 171 (2001), 479. doi: 10.1006/jcph.2001.6787.

[14]

L. Lee and R. J. LeVeque, An immersed interface method for the incompressible Navier-Stokes equations,, SIAM J. Sci. Comp., 25 (2003), 832. doi: 10.1137/S1064827502414060.

[15]

R. J. LeVeque and Z. Li, Immersed interface methods for Stokes flow with elastic boundaries or surface tension,, SIAM J. Sci. Comput., 18 (1997), 709. doi: 10.1137/S1064827595282532.

[16]

Z. Li and M.-C. Lai, The immersed interface method for the Navier-Stokes equations with singular forces,, J. Comput. Phys., 171 (2001), 822. doi: 10.1006/jcph.2001.6813.

[17]

Z. Li and S. R. Lubkin, Numerical analysis of interfacial two-dimensional Stokes flow with discontinuous viscosity and variable surface tension,, Int. J. Numer. Meth. Fluids, 37 (2001), 525. doi: 10.1002/fld.185.

[18]

A. Mayo, The fast solution of Poisson's and the biharmonic equations on irregular regions,, SIAM J. Numer. Anal., 21 (1984), 285. doi: 10.1137/0721021.

[19]

A. Mayo, Fast high order accurate solution of Laplace's equation on irregular regions,, SIAM J. Sci. Statist. Comput., 6 (1985), 144. doi: 10.1137/0906012.

[20]

A. Mayo and C. S. Peskin, An implicit numerical method for fluid dynamics problems with immersed elastic boundaries,, in, 141 (1993), 261.

[21]

Y. Mori and C. S. Peskin, Implicit second-order immersed boundary methods with boundary mass,, Comput. Methods Appl. Mech. Engin., 197 (2008), 2049. doi: 10.1016/j.cma.2007.05.028.

[22]

E. Newren, A. Fogelson, R. Guy and M. Kirby, A comparison of implicit solvers for the immersed boundary equations,, Comput. Methods Appl. Mech. Engin., 197 (2008), 2290. doi: 10.1016/j.cma.2007.11.030.

[23]

E. P. Newren, A. L. Fogelson, R. D. Guy and R. M. Kirby, Unconditionally stable discretizations of the immersed boundary equations,, J. Comput. Phys., 222 (2007), 702. doi: 10.1016/j.jcp.2006.08.004.

[24]

C. S. Peskin, Numerical analysis of blood flow in the heart,, J. Comput. Phys., 25 (1977), 220. doi: 10.1016/0021-9991(77)90100-0.

[25]

C. S. Peskin, The immersed boundary method,, Acta Numerica, 11 (2002), 479. doi: 10.1017/S0962492902000077.

[26]

C. S. Peskin and B. F. Printz, Improved volume conservation in the computation of flows with immersed elastic boundaries,, J. Comput. Phys., 105 (1993), 33. doi: 10.1006/jcph.1993.1051.

[27]

J. S. Sohn, Y.-H. Tseng, S. Li, A. Voigt and J. S. Lowengrub, Dynamics of multicomponent vesicles in a viscous fluid,, J. Comput. Phys., 229 (2010), 119. doi: 10.1016/j.jcp.2009.09.017.

[28]

J. M. Stockie and B. R. Wetton, Analysis of stiffness in the immersed boundary method and implications for time-stepping schemes,, J. Comput. Phys., 154 (1999), 41. doi: 10.1006/jcph.1999.6297.

[29]

A.-K. Tornberg and M. J. Shelley, Simulating the dynamics and interactions of flexible fibers in Stokes flows,, J. Comput. Phys., 196 (2004), 8. doi: 10.1016/j.jcp.2003.10.017.

[30]

C. Tu and C. S. Peskin, Stability and instability in the computation of flows with moving immersed boundaries: A comparison of three methods,, SIAM J. Sci. Statist. Comput., 13 (1992), 1361. doi: 10.1137/0913077.

[31]

S. K. Veerapaneni, D. Gueyffier, D. Zorin and G. Biros, A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D,, J. Comput. Phys., 228 (2009), 2334. doi: 10.1016/j.jcp.2008.11.036.

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