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.
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Notes
For stiff problems, one may prefer the first-order accurate implicit Euler method.
One may prefer a Newton’s form for constructing the interpolant \(\tilde{u}(x)\).
We use a time step of Δt=Δx 3/2 and Δt=Δx 2 to emulate a third- and a fourth-order scheme in time.
The tangential component of a velocity field changes a curve’s parameterization (if any), not its location.
In practice a third-order accurate scheme in time should be chosen.
The definition of the velocity does not depend on ϕ.
References
Adalsteinsson, D., Sethian, J.: A fast level set method for propagating interfaces. J. Comput. Phys. 118, 269–277 (1995)
Adalsteinsson, D., Sethian, J.: The fast construction of extension velocities in level set methods. J. Comput. Phys. 148, 2–22 (1999)
Aftosmis, M.J., Berger, M.J., Melton, J.E.: Adaptive Cartesian mesh generation. In: CRC Handbook of Mesh Generation (1998) (Contributed chapter)
Ahn, H.T., Shashkov, M.: Adaptive moment-of-fluid method. J. Comput. Phys. 228(8), 2792–2821 (2009)
Alexiades, V., Solomon, A.D.: Mathematical Modeling of Melting and Freezing Processes. Hemisphere, Washington (1993)
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)
Almgren, R.: Variational algorithms and pattern formation in dendritic solidification. J. Comput. Phys. 106(2), 337–354 (1993)
Aslam, T.: A partial differential equation approach to multidimensional extrapolation. J. Comput. Phys. 193, 349–355 (2004)
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)
Azarenok, B.N.: A method of constructing adaptive hexahedral moving grids. J. Comput. Phys. 226(1), 1102–1121 (2007)
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)
Benson, D.: Computational methods in Lagrangian and Eulerian hydrocodes. Comput. Methods Appl. Mech. Eng. 99, 235–394 (1992)
Benson, D.: Volume of fluid interface reconstruction methods for multimaterial problems. Appl. Mech. Rev. 52, 151–165 (2002)
Berger, M., Colella, P.: Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82, 64–84 (1989)
Berger, M., Oliger, J.: Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53, 484–512 (1984)
Beyer, R., LeVeque, R.: Analysis of a one-dimensional model for the immersed boundary method. SIAM J. Numer. Anal. 29, 332 (1992)
Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)
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)
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)
Braun, R., Murray, M.: Adaptive phase-field computations of dendritic crystal growth. J. Cryst. Growth 174, 41 (1997)
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
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)
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)
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)
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)
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)
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)
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)
Chen, S., Merriman, B., Osher, S., Smereka, P.: A simple level set method for solving Stefan problems. J. Comput. Phys. 135, 8–29 (1997)
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)
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)
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)
Davis, S.: Theory of Solidification. Cambridge University Press, Cambridge (2001)
DeBar, R.: Fundamentals of the KRAKEN code. Technical report, Lawrence Livermore National Laboratory (UCID-17366) (1974)
Can, D., Prosperetti, A.: A level set method for vapor bubble dynamics. J. Comput. Phys. 231, 1533–1552 (2012)
Dyadechko, V., Shashkov, M.: Moment-of-fluid interface reconstruction. Technical report, Los Alamos National Laboratory (LA-UR-05-7571) (2006)
Elder, K., Grant, M., Provatas, N., Kosterlitz, J.: Sharp interface limits of phase-field models. SIAM J. Appl. Math. 64, 21604 (2001)
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)
Enright, D., Marschner, S., Fedkiw, R.: Animation and rendering of complex water surfaces. ACM Trans. Graph. 21(3), 736–744 (2002)
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)
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)
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)
Fedkiw, R., Aslam, T., Xu, S.: The ghost fluid method for deflagration and detonation discontinuities. J. Comput. Phys. 154, 393–427 (1999)
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)
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)
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)
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)
Glassman, I., Yetter, R.: Combustion. Academic Press, New York (2008)
Golub, G., Loan, C.: Matrix Computations. John Hopkins University Press, Baltimore (1989)
Greengard, L., Lee, J.-Y.: Electrostatics and heat conduction in high contrast composite materials. J. Comput. Phys. 211(1), 64–76 (2006)
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)
Heinrich, J., Zhao, P.: Front tracking finite element method for dendritic solidification. J. Comput. Phys. 173, 765–796 (2001)
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)
Helmsen, J., Puckett, E.G., Colella, P., Dorr, M.: Two new methods for simulating photolithography development in 3D. Proc. SPIE 2726, 253–261 (1996)
Hirt, C., Nichols, B.: Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201–225 (1981)
Hou, S., Liu, X.-D.: A numerical method for solving variable coefficient elliptic equation with interfaces. J. Comput. Phys. 202(2), 411–445 (2005)
Incropera, F.P., DeWitt, D.P.: Fundamentals of Heat and Mass Transfer. Wiley, New York (2007)
Jeong, J.-H., Goldenfeld, N., Dantzig, J.: Phase field model for three-dimensional dendritic growth with fluid flow. Phys. Rev. E 64, 41602 (2001)
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)
Jiang, G.-S., Peng, D.: Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2126–2143 (2000)
Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
Johansen, H.: Cartesian grid embedded boundary finite difference methods for elliptic and parabolic differential equations on irregular domains. PhD thesis, Berkeley (1997)
Johansen, H., Colella, P.: A Cartesian grid embedded boundary method for Poisson’s equation on irregular domains. J. Comput. Phys. 147, 60–85 (1998)
Juric, D., Tryggvason, G.: A front tracking method for dendritic solidification. J. Comput. Phys. 123, 127–148 (1996)
Juric, D., Tryggvason, G.: Computations of boiling flows. Int. J. Multiph. Flow 24, 387–410 (1998)
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)
Karma, A.: Phase-field formulation for quantitative modeling of alloy solidification. Phys. Rev. Lett. 87, 115701 (2001)
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)
Karma, A., Rappel, W.-J.: Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys. Rev. E 57, 4323–4349 (1997)
Kim, Y.-T., Goldenfeld, N., Dantzig, J.: Computation of dendritic microstructures using a level set method. Phys. Rev. E 62, 2471–2474 (2000)
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)
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)
Leal, L.G.: Advanced Transport Phenomena—Fluid Mechanics and Convective Transport Processes. Cambridge Series in Chemical Engineering (2007)
Leveque, R.: High-resolution conservative algorithms for advection in incompressible flow. SIAM J. Numer. Anal. 33, 627–665 (1996)
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)
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)
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)
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)
Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 126, 202–212 (1996)
Losasso, F., Fedkiw, R., Osher, S.: Spatially adaptive techniques for level set methods and incompressible flow. Comput. Fluids 35, 995–1010 (2006)
Losasso, F., Gibou, F., Fedkiw, R.: Simulating water and smoke with an octree data structure. In: ACM Trans. Graph., pp. 457–462 (2004)
Manteuffel, T., White, A.: The numerical solution of second-order boundary value problems on nonuniform meshes. Math. Comput. 47(176), 511–535 (1986)
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)
Mayo, A.: The fast solution of Poisson’s and the biharmonic equations on irregular regions. SIAM J. Numer. Anal. 21, 285–299 (1984)
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)
Meiron, D.: Selection of steady-states in the two-dimensional symmetric model of dendritic growth. Phys. Rev. A 33, 2704 (1986)
Min, C.: Local level set method in high dimension and codimension. J. Comput. Phys. 200, 368–382 (2004)
Min, C.: On reinitializing level set functions. J. Comput. Phys. 229(8), 2764–2772 (2010)
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)
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)
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)
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)
Moore, D.: The cost of balancing generalized quadtrees. In: Proceedings of the Third ACM Symposium on Solid Modeling and Applications, pp. 305–312 (1995)
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)
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)
Nguyen, D., Fedkiw, R., Jensen, H.: Physically based modeling and animation of fire. ACM Trans. Graph. 29, 721–728 (2002)
Nguyen, D., Fedkiw, R., Kang, M.: A boundary condition capturing method for incompressible flame discontinuities. J. Comput. Phys. 172, 71–98 (2001)
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)
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)
Noh, W., Woodward, P.: SLIC (simple line interface calculation). In: 5th International Conference on Numerical Methods in Fluid Dynamics, pp. 330–340 (1976)
Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, Berlin (2002)
Osher, S., Paragios, N.: Geometric Level Set Methods in Imaging, Vision, and Graphics. Springer, Berlin (2003)
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)
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)
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)
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)
Peskin, C.: Flow patterns around heart valves: a numerical method. J. Comput. Phys. 10, 252–271 (1972)
Peskin, C.: Numerical analysis of blood flow in the heart. J. Comput. Phys. 25, 220–252 (1977)
Peskin, C.: The immersed boundary method. Acta Numer. 11, 479–517 (2002)
Popinet, S.: Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190, 572–600 (2003)
Popinet, S.: An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228(16), 5838–5866 (2009)
Provatas, N., Goldenfeld, N., Dantzig, J.: Efficient computation of dendritic microstructures using adaptive mesh refinement. Phys. Rev. Lett. 80, 3308 (1998)
Provatas, N., Goldenfeld, N., Dantzig, J.: Adaptive mesh refinement computation of solidification microstructure using dynamic data structures. J. Comput. Phys. 148, 265 (1999)
Purvis, J.W., Burkhalter, J.E.: Prediction of critical Mach number for store configurations. AIAA J. 17, 1170–1177 (1979)
Qian, J., Tryggvason, G., Law, C.: A front tracking method for the motion of premixed flames. J. Comput. Phys. 144(1), 52–69 (1998)
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)
Renardy, M.: Prost: a parabolic reconstruction of surface tension for the volume-of-fluid method. J. Comput. Phys. 183(2), 400–421 (2002)
Russo, G., Smereka, P.: A remark on computing distance functions. J. Comput. Phys. 163, 51–67 (2000)
Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS, Boston (1996)
Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)
Samet, H.: The Design and Analysis of Spatial Data Structures. Addison-Wesley, New York (1989)
Samet, H.: Applications of Spatial Data Structures: Computer Graphics, Image Processing and GIS. Addison-Wesley, New York (1990)
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)
Sethian, J.: A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. 93, 1591–1595 (1996)
Sethian, J.: Fast marching methods. SIAM Rev. 41, 199–235 (1999)
Sethian, J., Strain, J.: Crystal growth and dendritic solidification. J. Comput. Phys. 98, 231–253 (1992)
Sethian, J.A.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999)
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)
Shortley, G.H., Weller, R.: Numerical solution of Laplace’s equation. J. Appl. Phys. 9, 334–348 (1938)
Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes II. J. Comput. Phys. 83, 32–78 (1989)
Son, G., Dhir, V.: Numerical simulation of saturated film boiling on a horizontal surface. J. Heat Transf. 119, 525 (1997)
Son, G., Dhir, V.: Numerical simulation of film boiling near critical pressures with a level set method. J. Heat Transf. 120, 183 (1998)
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)
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)
Strain, J.: Fast tree-based redistancing for level set computations. J. Comput. Phys. 152, 664–686 (1999)
Strain, J.: Tree methods for moving interfaces. J. Comput. Phys. 151, 616–648 (1999)
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)
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)
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)
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)
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)
Tan, L., Zabaras, N.: A level set simulation of dendritic solidification of multi-component alloys. J. Comput. Phys. 221(1), 9–40 (2007)
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)
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)
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
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)
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)
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)
Tsai, Y.-H.R.: Rapid and accurate computation of the distance function using grids. J. Comput. Phys. 178(1), 175–195 (2002)
Tsitsiklis, J.: Efficient algorithms for globally optimal trajectories. IEEE Trans. Autom. Control 40, 1528–1538 (1995)
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)
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)
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)
Unverdi, S.O., Tryggvason, G.: A front-tracking method for viscous, incompressible, multifluid flows. J. Comput. Phys. 100, 25–37 (1992)
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)
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
Welch, S., Wilson, J.: A volume of fluid based method for fluid flows with phase change. J. Comput. Phys. 160, 662–682 (2000)
Whitaker, S.: Introduction to Fluid Mechanics. Prentice Hall, New York (1968)
Hu, N.-A.A.X.-Y.: Scale separation for implicit large eddy simulation. J. Comput. Phys. 230(19), 7240–7249 (2011)
Xiu, D., Karniadakis, G.: A semi-Lagrangian high-order method for Navier-Stokes equations. J. Comput. Phys. 172, 658–684 (2001)
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)
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)
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)
Youngs, D.: An interface tracking method for a 3D Eulerian hydrodynamics code. Technical report, AWRE (44/92/35) (1984)
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)
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)
Zhao, H.: A fast sweeping method for eikonal equations. Math. Comput. 74, 603–627 (2004)
Zhao, P., Vénere, M., Heinrich, J., Poirier, D.: Modeling dendritic growth of a binary alloy. J. Comput. Phys. 188(2), 434–461 (2003)
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)
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.
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In honor of Stan Osher’s 70th birthday.
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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
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DOI: https://doi.org/10.1007/s10915-012-9660-1