June  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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[25]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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