Skip to main content
Log in

High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We present a review of some of the state-of-the-art numerical methods for solving the Stefan problem and the Poisson and the diffusion equations on irregular domains using (i) the level-set method for representing the (possibly moving) irregular domain’s boundary, (ii) the ghost-fluid method for imposing the Dirichlet boundary condition at the irregular domain’s boundary and (iii) a quadtree/octree node-based adaptive mesh refinement for capturing small length scales while significantly reducing the memory and CPU footprint. In addition, we highlight common misconceptions and describe how to properly implement these methods. Numerical experiments illustrate quantitative and qualitative results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Algorithm 1
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Algorithm 2
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27

Similar content being viewed by others

Notes

  1. For stiff problems, one may prefer the first-order accurate implicit Euler method.

  2. One may prefer a Newton’s form for constructing the interpolant \(\tilde{u}(x)\).

  3. We use a time step of Δtx 3/2 and Δtx 2 to emulate a third- and a fourth-order scheme in time.

  4. The tangential component of a velocity field changes a curve’s parameterization (if any), not its location.

  5. In practice a third-order accurate scheme in time should be chosen.

  6. The definition of the velocity does not depend on ϕ.

References

  1. Adalsteinsson, D., Sethian, J.: A fast level set method for propagating interfaces. J. Comput. Phys. 118, 269–277 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adalsteinsson, D., Sethian, J.: The fast construction of extension velocities in level set methods. J. Comput. Phys. 148, 2–22 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aftosmis, M.J., Berger, M.J., Melton, J.E.: Adaptive Cartesian mesh generation. In: CRC Handbook of Mesh Generation (1998) (Contributed chapter)

    Google Scholar 

  4. Ahn, H.T., Shashkov, M.: Adaptive moment-of-fluid method. J. Comput. Phys. 228(8), 2792–2821 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alexiades, V., Solomon, A.D.: Mathematical Modeling of Melting and Freezing Processes. Hemisphere, Washington (1993)

    Google Scholar 

  6. Alexiades, V., Solomon, A.D., Wilson, D.G.: The formation of a solid nucleus in supercooled liquid, I. J. Non-Equilib. Thermodyn. 13, 281–300 (1988)

    Article  MATH  Google Scholar 

  7. Almgren, R.: Variational algorithms and pattern formation in dendritic solidification. J. Comput. Phys. 106(2), 337–354 (1993)

    MathSciNet  MATH  Google Scholar 

  8. Aslam, T.: A partial differential equation approach to multidimensional extrapolation. J. Comput. Phys. 193, 349–355 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Aulisa, E., Manservisi, S., Scardovelli, R., Zaleski, S.: A geometrical area-preserving volume-of-fluid advection method. J. Comput. Phys. 192(1), 355–364 (2003)

    Article  MATH  Google Scholar 

  10. Azarenok, B.N.: A method of constructing adaptive hexahedral moving grids. J. Comput. Phys. 226(1), 1102–1121 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bedrossian, J., von Brecht, J.H., Zhu, S., Sifakis, E., Teran, J.M.: A second order virtual node method for elliptic problems with interfaces and irregular domains. J. Comput. Phys. 229(18), 6405–6426 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Benson, D.: Computational methods in Lagrangian and Eulerian hydrocodes. Comput. Methods Appl. Mech. Eng. 99, 235–394 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Benson, D.: Volume of fluid interface reconstruction methods for multimaterial problems. Appl. Mech. Rev. 52, 151–165 (2002)

    Article  Google Scholar 

  14. Berger, M., Colella, P.: Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82, 64–84 (1989)

    Article  MATH  Google Scholar 

  15. Berger, M., Oliger, J.: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53, 484–512 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  16. Beyer, R., LeVeque, R.: Analysis of a one-dimensional model for the immersed boundary method. SIAM J. Numer. Anal. 29, 332 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)

    Article  Google Scholar 

  18. Bo, W., Liu, X., Glimm, J., Li, X.: A robust front tracking method: verification and application to simulation of the primary breakup of a liquid jet. SIAM J. Sci. Comput. 33(4), 1505–1524 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bornia, G., Cervone, A., Manservisi, S., Scardovelli, R., Zaleski, S.: On the properties and limitations of the height function method in two-dimensional Cartesian geometry. J. Comput. Phys. 230(4), 851–862 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Braun, R., Murray, M.: Adaptive phase-field computations of dendritic crystal growth. J. Cryst. Growth 174, 41 (1997)

    Article  Google Scholar 

  21. Brun, E., Guittet, A., Gibou, F.: A local level-set method using a hash table data structure. J. Comput. Phys. (2011). doi:10.1016/j.jcp.2011.12.001

    Google Scholar 

  22. Burstedde, C., Wilcox, L.C., Ghattas, O.: p4est: scalable algorithms for parallel adaptive mesh refinement on forests of octrees. SIAM J. Sci. Comput. 33(3), 1103–1133 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Caflisch, R., Gyure, M., Merriman, B., Osher, S., Ratsch, C., Vvedensky, D., Zinck, J.: Island dynamics and the level set method for epitaxial growth. Appl. Math. Lett. 12, 13 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Caiden, R., Fedkiw, R., Anderson, C.: A numerical method for two phase flow consisting of separate compressible and incompressible regions. J. Comput. Phys. 166, 1–27 (2001)

    Article  MATH  Google Scholar 

  25. Ceniceros, H.D., Nós, R.L., Roma, A.M.: Three-dimensional, fully adaptive simulations of phase-field fluid models. J. Comput. Phys. 229(17), 6135–6155 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chen, H., Min, C., Gibou, F.: A second-order accurate FDM for the heat equation on irregular domains and adaptive grids. In: Proceedings of the Materials Research Society Symposium, San Francisco, CA, USA, vol. 910, pp. 907–910 (2006)

    Google Scholar 

  27. Chen, H., Min, C., Gibou, F.: A supra-convergent finite difference scheme for the Poisson and heat equations on irregular domains and non-graded adaptive Cartesian grids. J. Sci. Comput. 31(1–2), 19–60 (2007)

    Article  MathSciNet  Google Scholar 

  28. Chen, H., Min, C., Gibou, F.: A numerical scheme for the Stefan problem on adaptive Cartesian grids with supralinear convergence rate. J. Comput. Phys. 228(16), 5803–5818 (2009)

    Article  MATH  Google Scholar 

  29. Chen, S., Merriman, B., Osher, S., Smereka, P.: A simple level set method for solving Stefan problems. J. Comput. Phys. 135, 8–29 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Chéné, A., Min, C., Gibou, F.: Second-order accurate computation of curvatures in a level set framework using novel high-order reinitialization schemes. J. Sci. Comput. 35(2–3), 114–131 (2007)

    Google Scholar 

  31. Cheng, L.-T., Tsai, Y.-H.: Redistancing by flow of time dependent eikonal equation redistancing by flow of time dependent eikonal equation redistancing by flow of time dependent eikonal equation redistancing by flow of time dependent eikonal equation. J. Comput. Phys. 227, 4002–4017 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Crockett, R., Colella, P., Graves, D.: A Cartesian grid embedded boundary method for solving the Poisson and heat equations with discontinuous coefficients in three dimensions. J. Comput. Phys. 230(7), 2451–2469 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Davis, S.: Theory of Solidification. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  34. DeBar, R.: Fundamentals of the KRAKEN code. Technical report, Lawrence Livermore National Laboratory (UCID-17366) (1974)

  35. Can, D., Prosperetti, A.: A level set method for vapor bubble dynamics. J. Comput. Phys. 231, 1533–1552 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Dyadechko, V., Shashkov, M.: Moment-of-fluid interface reconstruction. Technical report, Los Alamos National Laboratory (LA-UR-05-7571) (2006)

  37. Elder, K., Grant, M., Provatas, N., Kosterlitz, J.: Sharp interface limits of phase-field models. SIAM J. Appl. Math. 64, 21604 (2001)

    Google Scholar 

  38. Enright, D., Fedkiw, R., Ferziger, J., Mitchell, I.: A hybrid particle level set method for improved interface capturing. J. Comput. Phys. 183, 83–116 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  39. Enright, D., Marschner, S., Fedkiw, R.: Animation and rendering of complex water surfaces. ACM Trans. Graph. 21(3), 736–744 (2002)

    Article  Google Scholar 

  40. Enright, D., Nguyen, D., Gibou, F., Fedkiw, R.: Using the particle level set method and a second order accurate pressure boundary condition for free surface flows. In: Proc. 4th ASME-JSME Joint Fluids Eng. Conf. ASME, New York (2003)

    Google Scholar 

  41. Estep, D., Tavener, S., Wildey, T.: A posteriori error estimation and adaptive mesh refinement for a multiscale operator decomposition approach to fluid–solid heat transfer. J. Comput. Phys. 229(11), 4143–4158 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  42. Fedkiw, R., Aslam, T., Merriman, B., Osher, S.: A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152, 457–492 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  43. Fedkiw, R., Aslam, T., Xu, S.: The ghost fluid method for deflagration and detonation discontinuities. J. Comput. Phys. 154, 393–427 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  44. Gibou, F., Chen, L., Nguyen, D., Banerjee, S.: A level set based sharp interface method for the multiphase incompressible Navier–Stokes equations with phase change. J. Comput. Phys. 222(2), 536–555 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  45. Gibou, F., Fedkiw, R.: 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)

    Article  MathSciNet  MATH  Google Scholar 

  46. Gibou, F., Fedkiw, R., Caflisch, R., Osher, S.: A level set approach for the numerical simulation of dendritic growth. J. Sci. Comput. 19, 183–199 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  47. Gibou, F., Fedkiw, R., Cheng, L.-T., Kang, M.: A second-order-accurate symmetric discretization of the Poisson equation on irregular domains. J. Comput. Phys. 176, 205–227 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  48. Glassman, I., Yetter, R.: Combustion. Academic Press, New York (2008)

    Google Scholar 

  49. Golub, G., Loan, C.: Matrix Computations. John Hopkins University Press, Baltimore (1989)

    MATH  Google Scholar 

  50. Greengard, L., Lee, J.-Y.: Electrostatics and heat conduction in high contrast composite materials. J. Comput. Phys. 211(1), 64–76 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  51. Griffith, B.E., Hornung, R.D., McQueen, D.M., Peskin, C.S.: An adaptive, formally second order accurate version of the immersed boundary method. J. Comput. Phys. 223(1), 10–49 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  52. Heinrich, J., Zhao, P.: Front tracking finite element method for dendritic solidification. J. Comput. Phys. 173, 765–796 (2001)

    Article  MATH  Google Scholar 

  53. Hellrung, J., Wang, L., Sifakis, E., Teran, J.M.: A second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions. J. Comput. Phys. 231(4), 2015–2048 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  54. Helmsen, J., Puckett, E.G., Colella, P., Dorr, M.: Two new methods for simulating photolithography development in 3D. Proc. SPIE 2726, 253–261 (1996)

    Article  Google Scholar 

  55. Hirt, C., Nichols, B.: Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201–225 (1981)

    Article  MATH  Google Scholar 

  56. Hou, S., Liu, X.-D.: A numerical method for solving variable coefficient elliptic equation with interfaces. J. Comput. Phys. 202(2), 411–445 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  57. Incropera, F.P., DeWitt, D.P.: Fundamentals of Heat and Mass Transfer. Wiley, New York (2007)

    Google Scholar 

  58. Jeong, J.-H., Goldenfeld, N., Dantzig, J.: Phase field model for three-dimensional dendritic growth with fluid flow. Phys. Rev. E 64, 41602 (2001)

    Article  Google Scholar 

  59. Ji, H., Lien, F.-S., Yee, E.: A new adaptive mesh refinement data structure with an application to detonation. J. Comput. Phys. 229(23), 8981–8993 (2010)

    Article  MATH  Google Scholar 

  60. Jiang, G.-S., Peng, D.: Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2126–2143 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  61. Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  62. Johansen, H.: Cartesian grid embedded boundary finite difference methods for elliptic and parabolic differential equations on irregular domains. PhD thesis, Berkeley (1997)

  63. Johansen, H., Colella, P.: A Cartesian grid embedded boundary method for Poisson’s equation on irregular domains. J. Comput. Phys. 147, 60–85 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  64. Juric, D., Tryggvason, G.: A front tracking method for dendritic solidification. J. Comput. Phys. 123, 127–148 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  65. Juric, D., Tryggvason, G.: Computations of boiling flows. Int. J. Multiph. Flow 24, 387–410 (1998)

    Article  MATH  Google Scholar 

  66. Kamkar, S.J., Wissink, A.M., Sankaran, V., Jameson, A.: Feature-driven Cartesian adaptive mesh refinement for vortex-dominated flows. J. Comput. Phys. 230(16), 6271–6298 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  67. Karma, A.: Phase-field formulation for quantitative modeling of alloy solidification. Phys. Rev. Lett. 87, 115701 (2001)

    Article  Google Scholar 

  68. Karma, A., Rappel, W.-J.: Phase-field modeling method for computationally efficient modeling of solidification with arbitrary interface kinetics. Phys. Rev. E 53, 3017–3020 (1996)

    Article  Google Scholar 

  69. Karma, A., Rappel, W.-J.: Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys. Rev. E 57, 4323–4349 (1997)

    Article  Google Scholar 

  70. Kim, Y.-T., Goldenfeld, N., Dantzig, J.: Computation of dendritic microstructures using a level set method. Phys. Rev. E 62, 2471–2474 (2000)

    Article  Google Scholar 

  71. Kreiss, H.O., Manteuffel, H.-O., Schwartz, T.A., Wendroff, B., White, A.B. Jr.: Supra-convergent schemes on irregular grids. Math. Comput. 47, 537–554 (1986)

    Article  MATH  Google Scholar 

  72. Laan, W.J., Jalba, A.C., Roerdink, J.B.T.M.: A memory and computation efficient sparse level-set method. J. Sci. Comput. 46(2), 243–264 (2010)

    Article  Google Scholar 

  73. Leal, L.G.: Advanced Transport Phenomena—Fluid Mechanics and Convective Transport Processes. Cambridge Series in Chemical Engineering (2007)

    Book  MATH  Google Scholar 

  74. Leveque, R.: High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Numer. Anal. 33, 627–665 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  75. LeVeque, R., Li, Z.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31, 1019–1044 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  76. Li, S., Petzold, L.: Adjoint sensitivity analysis for time-dependent partial differential equations with adaptive mesh refinement. J. Comput. Phys. 198(1), 310–325 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  77. Li, Z., Ito, K.: In: The Immersed Interface Method—Numerical Solutions of PDEs Involving Interfaces and Irregular Domains. SIAM Frontiers in Applied Mathematics, vol. 33 (2006)

    Chapter  Google Scholar 

  78. Liu, X.D., Fedkiw, R., Kang, M.: A boundary condition capturing method for Poisson’s equation on irregular domains. J. Comput. Phys. 154, 151 (2000)

    Article  MathSciNet  Google Scholar 

  79. Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 126, 202–212 (1996)

    Article  MathSciNet  Google Scholar 

  80. Losasso, F., Fedkiw, R., Osher, S.: Spatially adaptive techniques for level set methods and incompressible flow. Comput. Fluids 35, 995–1010 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  81. Losasso, F., Gibou, F., Fedkiw, R.: Simulating water and smoke with an octree data structure. In: ACM Trans. Graph., pp. 457–462 (2004)

    Google Scholar 

  82. Manteuffel, T., White, A.: The numerical solution of second-order boundary value problems on nonuniform meshes. Math. Comput. 47(176), 511–535 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  83. Marques, A.N., Nave, J.-C., Rosales, R.R.: A correction function method for Poisson problems with interface jump conditions. J. Comput. Phys. 230(20), 7567–7597 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  84. Mayo, A.: The fast solution of Poisson’s and the biharmonic equations on irregular regions. SIAM J. Numer. Anal. 21, 285–299 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  85. McCorquodale, P., Colella, P., Grote, D., Vay, J.-L.: A node-centered local refinement algorithm for Poisson’s equation in complex geometries. J. Comput. Phys. 201, 34–60 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  86. Meiron, D.: Selection of steady-states in the two-dimensional symmetric model of dendritic growth. Phys. Rev. A 33, 2704 (1986)

    Article  Google Scholar 

  87. Min, C.: Local level set method in high dimension and codimension. J. Comput. Phys. 200, 368–382 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  88. Min, C.: On reinitializing level set functions. J. Comput. Phys. 229(8), 2764–2772 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  89. Min, C., Gibou, F.: A second order accurate projection method for the incompressible Navier–Stokes equations on non-graded adaptive grids. J. Comput. Phys. 219(2), 912–929 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  90. Min, C., Gibou, F.: A second order accurate level set method on non-graded adaptive Cartesian grids. J. Comput. Phys. 225(1), 300–321 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  91. Min, C., Gibou, F., Ceniceros, H.: A supra-convergent finite difference scheme for the variable coefficient Poisson equation on non-graded grids. J. Comput. Phys. 218(1), 123–140 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  92. Miniati, F., Colella, P.: Block structured adaptive mesh and time refinement for hybrid, hyperbolic, n-body systems. J. Comput. Phys. 227(1), 400–430 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  93. Moore, D.: The cost of balancing generalized quadtrees. In: Proceedings of the Third ACM Symposium on Solid Modeling and Applications, pp. 305–312 (1995)

    Chapter  Google Scholar 

  94. Nestler, B., Danilov, D., Galenko, P.: Crystal growth of pure substances: phase-field simulations in comparison with analytical and experimental results. J. Comput. Phys. 207, 221–239 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  95. Ng, Y.T., Min, C., Gibou, F.: An efficient fluid–solid coupling algorithm for single-phase flows. J. Comput. Phys. 228(23), 8807–8829 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  96. Nguyen, D., Fedkiw, R., Jensen, H.: Physically based modeling and animation of fire. ACM Trans. Graph. 29, 721–728 (2002)

    Google Scholar 

  97. Nguyen, D., Fedkiw, R., Kang, M.: A boundary condition capturing method for incompressible flame discontinuities. J. Comput. Phys. 172, 71–98 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  98. Nguyen, D., Gibou, F., Fedkiw, R.: A fully conservative ghost fluid method and stiff detonation waves. In: 12th Int. Detonation Symposium, San Diego, CA (2002)

    Google Scholar 

  99. Nielsen, M.B., Museth, K.: Dynamic tubular grid: an efficient data structure and algorithms for high resolution level sets. J. Sci. Comput. 26(3), 261–299 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  100. Noh, W., Woodward, P.: SLIC (simple line interface calculation). In: 5th International Conference on Numerical Methods in Fluid Dynamics, pp. 330–340 (1976)

    Google Scholar 

  101. Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, Berlin (2002)

    Google Scholar 

  102. Osher, S., Paragios, N.: Geometric Level Set Methods in Imaging, Vision, and Graphics. Springer, Berlin (2003)

    MATH  Google Scholar 

  103. Osher, S., Sethian, J.A.: Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  104. Papac, J., Gibou, F., Ratsch, C.: Efficient symmetric discretization for the Poisson, heat and Stefan-type problems with Robin boundary conditions. J. Comput. Phys. 229(3), 875–889 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  105. Peng, D., Merriman, B., Osher, S., Zhao, H., Kang, M.: A PDE-based fast local level set method. J. Comput. Phys. 155, 410–438 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  106. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. In: Proceedings of IEEE Computer Society Workshop on Computer Vision, pp. 16–22 (1987)

    Google Scholar 

  107. Peskin, C.: Flow patterns around heart valves: a numerical method. J. Comput. Phys. 10, 252–271 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  108. Peskin, C.: Numerical analysis of blood flow in the heart. J. Comput. Phys. 25, 220–252 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  109. Peskin, C.: The immersed boundary method. Acta Numer. 11, 479–517 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  110. Popinet, S.: Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190, 572–600 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  111. Popinet, S.: An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228(16), 5838–5866 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  112. Provatas, N., Goldenfeld, N., Dantzig, J.: Efficient computation of dendritic microstructures using adaptive mesh refinement. Phys. Rev. Lett. 80, 3308 (1998)

    Article  Google Scholar 

  113. Provatas, N., Goldenfeld, N., Dantzig, J.: Adaptive mesh refinement computation of solidification microstructure using dynamic data structures. J. Comput. Phys. 148, 265 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  114. Purvis, J.W., Burkhalter, J.E.: Prediction of critical Mach number for store configurations. AIAA J. 17, 1170–1177 (1979)

    Article  Google Scholar 

  115. Qian, J., Tryggvason, G., Law, C.: A front tracking method for the motion of premixed flames. J. Comput. Phys. 144(1), 52–69 (1998)

    Article  Google Scholar 

  116. Ratsch, C., Gyure, M., Gibou, F., Petersen, M., Kang, M., Garcia, J., Vvedensky, D.: Level-set method for island dynamics in epitaxial growth. Phys. Rev. B 65, 195403 (2002)

    Article  Google Scholar 

  117. Renardy, M.: Prost: a parabolic reconstruction of surface tension for the volume-of-fluid method. J. Comput. Phys. 183(2), 400–421 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  118. Russo, G., Smereka, P.: A remark on computing distance functions. J. Comput. Phys. 163, 51–67 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  119. Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS, Boston (1996)

    MATH  Google Scholar 

  120. Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  121. Samet, H.: The Design and Analysis of Spatial Data Structures. Addison-Wesley, New York (1989)

    Google Scholar 

  122. Samet, H.: Applications of Spatial Data Structures: Computer Graphics, Image Processing and GIS. Addison-Wesley, New York (1990)

    Google Scholar 

  123. Sampath, R.S., Biros, G.: A parallel geometric multigrid method for finite elements on octree meshes. SIAM J. Sci. Comput. 32(3), 1361–1392 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  124. Sethian, J.: A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. 93, 1591–1595 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  125. Sethian, J.: Fast marching methods. SIAM Rev. 41, 199–235 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  126. Sethian, J., Strain, J.: Crystal growth and dendritic solidification. J. Comput. Phys. 98, 231–253 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  127. Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  128. Shen, C., Qiu, J.-M., Christlieb, A.: Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations. J. Comput. Phys. 230(10), 3780–3802 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  129. Shortley, G.H., Weller, R.: Numerical solution of Laplace’s equation. J. Appl. Phys. 9, 334–348 (1938)

    Article  Google Scholar 

  130. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys. 83, 32–78 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  131. Son, G., Dhir, V.: Numerical simulation of saturated film boiling on a horizontal surface. J. Heat Transf. 119, 525 (1997)

    Article  Google Scholar 

  132. Son, G., Dhir, V.: Numerical simulation of film boiling near critical pressures with a level set method. J. Heat Transf. 120, 183 (1998)

    Article  Google Scholar 

  133. Son, G., Dhir, V.: A level set method for analysis of film boiling on an immersed liquid surface. Numer. Heat Transf. B 52, 153–177 (2007)

    Article  Google Scholar 

  134. Son, G., Dhir, V.: Numerical simulation of nucleate boiling on a horizontal surface at high heat fluxes. Int. J. Heat Mass Transf. 51, 2566–2582 (2008)

    Article  MATH  Google Scholar 

  135. Strain, J.: Fast tree-based redistancing for level set computations. J. Comput. Phys. 152, 664–686 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  136. Strain, J.: Tree methods for moving interfaces. J. Comput. Phys. 151, 616–648 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  137. Sun, P., Russell, R.D., Xu, J.: A new adaptive local mesh refinement algorithm and its application on fourth order thin film flow problem. J. Comput. Phys. 224(2), 1021–1048 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  138. Sussman, M., Almgren, A., Bell, J., Colella, P., Howell, L., Welcome, M.: An adaptive level set approach for incompressible two-phase flows. J. Comput. Phys. 148, 81–124 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  139. Sussman, M., Puckett, E.G.: A coupled level let and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows. J. Comput. Phys. 162, 301–337 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  140. Sussman, M., Puckett, E.G.: A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows. J. Comput. Phys. 162(2), 301–337 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  141. Sussman, M., Smereka, P., Osher, S.: A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146–159 (1994)

    Article  MATH  Google Scholar 

  142. Tan, L., Zabaras, N.: A level set simulation of dendritic solidification of multi-component alloys. J. Comput. Phys. 221(1), 9–40 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  143. Tan, L., Zabaras, N.: Modeling the growth and interaction of multiple dendrites in solidification using a level set method. J. Comput. Phys. 226(1), 131–155 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  144. Tang, H.-Z., Tang, T., Zhang, P.: An adaptive mesh redistribution method for nonlinear Hamilton-Jacobi equations in two- and three-dimensions. J. Comput. Phys. 188, 543–572 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  145. Theillard, M., Rycroft, C.H., Gibou, F.: A multigrid method on non-graded adaptive octree and quadtree Cartesian grids. J. Sci. Comput. (2012). doi:10.1007/s10915-012-9619-2

    Google Scholar 

  146. Tomar, G., Biswas, G., Sharma, A., Agrawal, A.: Numerical simulation of bubble growth in film boiling using a coupled level-set and volume-of-fluid method. Phys. Fluids 17, 112103 (2005)

    Article  Google Scholar 

  147. Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S., Jan, Y.-J.: A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169, 708–759 (2001)

    Article  MATH  Google Scholar 

  148. Tsai, Y.-H., Cheng, L.-T., Osher, S.: Fast sweeping algorithms for a class of Hamilton-Jacobi equations. SIAM J. Numer. Anal. 41(2), 673–694 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  149. Tsai, Y.-H.R.: Rapid and accurate computation of the distance function using grids. J. Comput. Phys. 178(1), 175–195 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  150. Tsitsiklis, J.: Efficient algorithms for globally optimal trajectories. IEEE Trans. Autom. Control 40, 1528–1538 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  151. Udaykumar, H., Kan, H.-C., Shyy, W., Tran-Son-Tay, R.: Multiphase dynamics in arbitrary geometries on fixed Cartesian grids. J. Comput. Phys. 137(2), 366–405 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  152. Udaykumar, H., Marella, S., Krishnan, S.: Sharp-interface simulation of dendritic growth with convection: benchmarks. Int. J. Heat Mass Transf. 46(14), 2615–2627 (2003)

    Article  MATH  Google Scholar 

  153. Udaykumar, H., Mittal, H., Shyy, W.: Computation of solid-liquid phase fronts in the sharp interface limit on fixed grids. J. Comput. Phys. 153, 535–574 (1999)

    Article  MATH  Google Scholar 

  154. Unverdi, S.O., Tryggvason, G.: A front-tracking method for viscous, incompressible, multifluid flows. J. Comput. Phys. 100, 25–37 (1992)

    Article  MATH  Google Scholar 

  155. Wang, Z., Yang, J., Stern, F.: A new volume-of-fluid method with a constructed distance function on general structured grids. J. Comput. Phys. 231(9), 3703–3722 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  156. Weiser, A.: Local-mesh, local-order, adaptive finite element methods with a posteriori error estimators for elliptic parital differential equations. PhD thesis, Yale University, June 1981

  157. Welch, S., Wilson, J.: A volume of fluid based method for fluid flows with phase change. J. Comput. Phys. 160, 662–682 (2000)

    Article  MATH  Google Scholar 

  158. Whitaker, S.: Introduction to Fluid Mechanics. Prentice Hall, New York (1968)

    Google Scholar 

  159. Hu, N.-A.A.X.-Y.: Scale separation for implicit large eddy simulation. J. Comput. Phys. 230(19), 7240–7249 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  160. Xiu, D., Karniadakis, G.: A semi-Lagrangian high-order method for Navier-Stokes equations. J. Comput. Phys. 172, 658–684 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  161. Yang, G.Z., Zabaras, N.: The adjoint method for an inverse design problem in the directional solidification of binary alloys. J. Comput. Phys. 140, 432–452 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  162. Yang, X., James, A.J., Lowengrub, J., Zheng, X., Cristini, V.: An adaptive coupled level-set/volume-of-fluid interface capturing method for unstructured triangular grids. J. Comput. Phys. 217(2), 364–394 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  163. Yang, Y., Udaykumar, H.: Sharp interface Cartesian grid method iii: solidification of pure materials and binary solutions. J. Comput. Phys. 210(1), 55–74 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  164. Youngs, D.: An interface tracking method for a 3D Eulerian hydrodynamics code. Technical report, AWRE (44/92/35) (1984)

  165. Zabaras, N., Ganapathysubramanian, B., Tan, L.: Modelling dendritic solidification with melt convection using the extended finite element method. J. Comput. Phys. 218(1), 200–227 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  166. Zhang, Q., Liu, P.: Handling solid-fluid interfaces for viscous flows: explicit jump approximation vs. ghost cell approaches. J. Comput. Phys. 229(11), 4225–4246 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  167. Zhao, H.: A fast sweeping method for eikonal equations. Math. Comput. 74, 603–627 (2004)

    Article  Google Scholar 

  168. Zhao, P., Vénere, M., Heinrich, J., Poirier, D.: Modeling dendritic growth of a binary alloy. J. Comput. Phys. 188(2), 434–461 (2003)

    Article  MATH  Google Scholar 

  169. Zhu, L., Peskin, C.: Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method. J. Comput. Phys. 179, 452–468 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The research of F. Gibou was supported in part by ONR N00014-11-1-0027, NSF CHE 1027817, DOE DE-FG02-08ER15991, ICB W911NF-09-D-0001 and by the W.M. Keck Foundation. The research of C. Min was supported in part by the Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0028298) and by the Korea Research Foundation Grant funded by the Korean Government (KRF-2011-0013649). The research of R. Fedkiw was supported in part by ONR N00014-09-1-0101, ONR N-00014-11-1-0027, ARL AHPCRC W911NF-07-0027, NSF IIS-1048573, and the Intel Science and Technology Center for Visual Computing.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Frédéric Gibou.

Additional information

In honor of Stan Osher’s 70th birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gibou, F., Min, C. & Fedkiw, R. High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries. J Sci Comput 54, 369–413 (2013). https://doi.org/10.1007/s10915-012-9660-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-012-9660-1

Keywords

Navigation