Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-24T16:22:28.840Z Has data issue: false hasContentIssue false
  • English
  • Français

An Adaptive Mesh Refinement Strategy for Immersed Boundary/Interface Methods

Published online by Cambridge University Press:  20 August 2015

Zhilin Li  and
Peng Song
Zhilin Li*
Affiliation:
Center for Research in Scientific Computation & Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA; and Nanjing Normal University, China
Peng Song*
Affiliation:
Operations Research Program, North Carolina State University, Raleigh, NC 27695, USA
*
Corresponding author.Email:[email protected]
Email:[email protected]
Get access

Abstract

An adaptive mesh refinement strategy is proposed in this paper for the Immersed Boundary and Immersed Interface methods for two-dimensional elliptic interface problems involving singular sources. The interface is represented by the zero level set of a Lipschitz function ϕ(x,y). Our adaptive mesh refinement is done within a small tube of |ϕ(x,y)|≤δ with finer Cartesian meshes. The discrete linear system of equations is solved by a multigrid solver. The AMR methods could obtain solutions with accuracy that is similar to those on a uniform fine grid by distributing the mesh more economically, therefore, reduce the size of the linear system of the equations. Numerical examples presented show the efficiency of the grid refinement strategy.

Keywords

52B1065D1868U0568U07Adaptive mesh refinementimmersed boundary methodimmersed interface methodelliptic interface problemCartesian grid methodlevel set representationsingular sources
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 2 , August 2012 , pp. 515 - 527
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Beale, J. T. and Layton, A.T.. On the accuracyof finite difference methods for elliptic problems with interfaces. Math. Comput. Sci., 1 (2006), 91–119.Google Scholar
[2]Berger, M. and Colella, P.. Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys., 82 (1989), 64–84.Google Scholar
[3]Berger, M. and Rigoutsos, I.. An algorithm for point clustering and grid generation. IEEE on Systems, Man, and Cybernetics, 21 (1991), 1278–1286.CrossRefGoogle Scholar
[4]Deng, S., Ito, K., and Li, Z.. Three dimensional elliptic solvers for interface problems and applications. J. Comput. Phys., 184 (2003), 215–243.CrossRefGoogle Scholar
[5]Griffith, B. E.. A comparison of two adaptive versions of the immersed boundary method. Technical report, New York University, 2009, (preprint submitted to Elsevier Science).Google Scholar
[6]Griffith, B. E., Hornung, R. D., Mcqueen, D. M., and Peskin, C. S.. An adaptive, formally second order accurate version of the immersed boundary method. J. Comput. Phys., 223 (2007), 10–49.CrossRefGoogle Scholar
[7]Griffith, B. E. and Peskin, C. S.. On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems. J. Comput. Phys., 208 (2005), 75–105.CrossRefGoogle Scholar
[8]Hou, T., Li, Z., Osher, S., and Zhao, H.. A hybrid method for moving interface problems with application to the Hele-Shaw flow. J. Comput. Phys., 134 (1997), 236–252.Google Scholar
[9]Hunter, J., Li, Z., and Zhao, H.. Autophobic spreading of drops. J. Comput. Phys., 183 (2002), 335–366.CrossRefGoogle Scholar
[10]Ito, K., Kyei, Y., and Li, Z.. Higher-order, Cartesian grid based finite difference schemes for elliptic equations on irregular domains. SIAM J. Sci. Comput., 27 (2005), 346–367.CrossRefGoogle Scholar
[11]LeVeque, R. J. and Li, Z.. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal., 31 (1994), 1019–1044.Google Scholar
[12]Li, Z. and Ito, K.. Maximum principle preserving schemes for interface problems with discontinuous coefficients. SIAM J. Sci. Comput., 23 (2001), 1225–1242.Google Scholar
[13]Li, Z. and Ito, K.. The Immersed Interface Method –Numerical Solutions of PDEs Involving Interfaces and Irregular Domains. SIAM Frontier Series in Applied mathematics, FR 33, 2006.Google Scholar
[14]Li, Z., Ito, K., and Lai, M-C.. An augmented approach for Stokes equations with a discontinuous viscosity and singular forces. Computers and Fluids, 36 (2007), 622–635.Google Scholar
[15]Li, Z. and Lai, M-C.. The immersed interface method for the Navier-Stokes equations with singular forces. J. Comput. Phys., 171 (2001), 822–842.CrossRefGoogle Scholar
[16]Li, Z., Wan, X., Ito, K., and Lubkin, S.. An augmented pressure boundary condition for a Stokes flow with a non-slip boundary condition. Commun. Comput. Phys., 1 (2006), 874–885.Google Scholar
[17]Li, Z., Wang, W-C., Chern, I-L., and Lai, M-C.. New formulations for interface problems in polar coordinates. SIAM J. Sci. Comput., 25 (2003), 224–245.Google Scholar
[18]Losasso, F., Fedkiw, R., and Osher, S.. Spatially adaptive techniques for level set methods and incompressible flow. Computers and Fluids, 35 (2006), 995–1010.Google Scholar
[19]Peskin, C. S.. The immersed boundary method. Acta Numerica, 11 (2002), 479–517.Google Scholar
[20]Peskin, C. S. and McQueen, D. M.. A three-dimensional computational method for blood flow in the heart: (i) immersed elastic fibers in a viscous incompressible fluid. J. Comput. Phys., 81 (1989), 372–405.Google Scholar
[21]Peskin, C. S. and McQueen, D. M.. A three-dimensional computational method for blood flow in the heart: (ii) contractile fibers. J. Comput. Phys., 82 (1989), 289–297.Google Scholar
[22]Peskin, C. S. and McQueen, D. M.. A general method for the computer simulation of biological systems interacting with fluids. Symposia of the Society for Experimental Biology, 49 (1995), 265.Google Scholar
[23]Peskin, C. S. and Printz, B. F.. Improved volume conservation in the computation of flows with immersed elastic boundaries. J. Comput. Phys., 105 (1993), 33–46.Google Scholar
[24]Roma, A., Peskin, C. S., and Berger, M.. An adaptive version of the immersed boundary method. J. Comput. Phys., 153 (1999), 509–534.Google Scholar
[25]Roma, A. M.. A multilevel self adaptive version of the immersed boundary method. PhD thesis, New York University, 1996.Google Scholar
[26]Ruge, J. W. and Stuben, K.. Algebraic multigrid in Multigrid Methods. Frontiers in Applied Mathematics, Volume 3, SIAM, Philadelphia, 1987.Google Scholar
[27]Strain, J.. Fast tree-based redistancing for level set computations. J. Comput. Phys., 152 (1999), 664–686.Google Scholar
[28]Sussman, M., Almgren, A., Bell, J. B., Colella, P., Howell, L. H, and Welcome, M. L.. An adaptive level set approach for incompressible two-phase flows. J. Comput. Phys., 148 (1999), 81–124.Google Scholar
[29]Wu, C-T., Li, Z., and Lai, M-C.. Adaptive mesh refinement for elliptic interface problems using the non-conforming immersed finite element method. Int. J. Numer. Anal. Model., 1 (2010), 466–483.Google Scholar