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A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential
A space-time discontinuous Galerkin spectral element method for the Stefan problem
Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA |
A novel space-time discontinuous Galerkin (DG) spectral element method is presented to solve the one dimensional Stefan problem in an Eulerian coordinate system. This method employs the level set procedure to describe the time-evolving interface. To deal with the prior unknown interface, a backward transformation and a forward transformation are introduced in the space-time mesh. By combining an Eulerian description, i.e., a fixed frame of reference, with a Lagrangian description, i.e., a moving frame of reference, the issue of dealing with implicitly defined arbitrary shaped space-time elements is avoided. The backward transformation maps the unknown time-varying interface in the fixed frame of reference to a known stationary interface in the moving frame of reference. In the moving frame of reference, the transformed governing equations, written in the space-time framework, are discretized by a DG spectral element method in each space-time slab. The forward transformation is used to update the level set function and then to project the solution in each phase back from the moving frame of reference to the fixed Eulerian grid. Two options for calculating the interface velocity are presented, and both options exhibit spectral accuracy. Benchmark tests indicate that the method converges with spectral accuracy in both space and time for the temperature distribution and the interface velocity. A Picard iteration algorithm is introduced in order to solve the nonlinear algebraic system of equations and it is found that just a few iterations lead to convergence.
References:
[1] |
N. Al-Rawahi and G. Tryggvason, Numerical simulation of dendritic solidification with convection: Two-dimensional geometry, J. Comput. Phys., 180 (2002), 471-496. Google Scholar |
[2] |
N. Al-Rawahi and G. Tryggvason, Numerical simulation of dendritic solidification with convection: Three-dimensional flow, J. Comput. Phys., 194 (2004), 677-696. Google Scholar |
[3] |
V. Alexiades and A. D. Solomon, Mathematical Modelling of Melting and Freezing Processes Hemisphere Publishing Corporation, Washington, 1981. Google Scholar |
[4] |
E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney and D. Sorensen, LAPACK Users' Guide 3rd edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999. Google Scholar |
[5] |
T. D. Aslam,
A partial differential equation approach to multidimensional extrapolation, J. Comput. Phys., 193 (2004), 349-355.
doi: 10.1016/j.jcp.2003.08.001. |
[6] |
D. C. Assêncio and J. M. Teran,
A second order virtual node algorithm for Stokes flow problems with interfacial forces, discontinuous material properties and irregular domains, J. Comput. Phys., 250 (2013), 77-105.
doi: 10.1016/j.jcp.2013.04.041. |
[7] |
M. Azaï ez, F. Jelassi, M. Mint Brahim and J. Shen,
Two-phase Stefan problem with smoothed enthalpy, Commun. Math. Sci., 14 (2016), 1625-1641.
doi: 10.4310/CMS.2016.v14.n6.a8. |
[8] |
J. Bedrossian, J. H. von Brecht, S. Zhu, E. Sifakis and J. M. Teran,
A second order virtual node method for elliptic problems with interfaces and irregular domains, J. Comput. Phys., 229 (2010), 6405-6426.
doi: 10.1016/j.jcp.2010.05.002. |
[9] |
M. Benzi and M. A. Olshanskii,
An augmented Lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput., 28 (2006), 2095-2113.
doi: 10.1137/050646421. |
[10] |
C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains Berlin, Springer, 2006. |
[11] |
S. Chen, B. Merriman, S. Osher and P. Smereka,
A simple level set method for solving Stefan problems, J. Comput. Phys., 135 (1997), 8-29.
doi: 10.1006/jcph.1997.5721. |
[12] |
J. Chessa, P. Smolinski and T. Belytschko,
The extended finite element method (XFEM) for solidification problems, Int. J. Number. Meth. Eng., 53 (2002), 1959-1977.
doi: 10.1002/nme.386. |
[13] |
B. Bernardo Cockburn, G. Karniadakis and C. -W. Shu (eds. ), Discontinuous Galerkin Methods: Theory, Computation, and Applications Lecture notes in computational science and engineering, Springer, Berlin, New York, 2000. Google Scholar |
[14] |
B. Cockburn, G. Kanschat, I. Perugia and D. Schötzau,
Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal., 39 (2001), 264-285.
doi: 10.1137/S0036142900371544. |
[15] |
B. Cockburn and C.-W. Shu,
The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440-2463.
doi: 10.1137/S0036142997316712. |
[16] |
H. Coxeter, Regular Polytopes Dover Publications, Inc., New York, 1973. |
[17] |
A. Criscione, D. Kintea, Ž. Tuković, S. Jakirlić, I. Roisman and C. Tropea, Crystallization of supercooled water: A level-set-based modeling of the dendrite tip velocity, Int. J. Heat Mass Transfer, 66 (2013), 830-837. Google Scholar |
[18] |
M. Farid, The moving boundary problems from melting and freezing to drying and frying of food, Chemical Engineering and Processing: Process Intensification, 41 (2002), 1-10. Google Scholar |
[19] |
R. P. Fedkiw, T. Aslam, B. Merriman and S. Osher,
A Non-oscillatory Eulerian approach to interfaces in multimaterial flows (the Ghost Fluid Method), J. Comput. Phys., 152 (1999), 457-492.
doi: 10.1006/jcph.1999.6236. |
[20] |
W. L. George and J. A. Warren, A parallel 3D dendritic growth simulator using the phase-field method, J. Comput. Phys., 177 (2002), 264-283. Google Scholar |
[21] |
F. Gibou and R. Fedkiw,
A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem, J. Comput. Phys., 202 (2005), 577-601.
doi: 10.1016/j.jcp.2004.07.018. |
[22] |
F. Gibou, R. Fedkiw, R. Caflisch and S. Osher,
A level set approach for the numerical simulation of dendritic growth, J. Sci. Comput., 19 (2003), 183-199.
doi: 10.1023/A:1025399807998. |
[23] |
F. Gibou, R. P. Fedkiw, L.-T. Cheng and M. Kang,
A second-order-accurate symmetric discretization of the Poisson equation on irregular domains, J. Comput. Phys., 176 (2002), 205-227.
doi: 10.1006/jcph.2001.6977. |
[24] |
S. C. Gupta, The Classical Stefan Problem: Basic Concepts, Modelling and Analysis vol. 45 of North-Holland Series in Applied Mathematics and Mechanics, Elsevier Science B. V., Amsterdam, 2003. |
[25] |
D. Han,
A decoupled unconditionally stable numerical scheme for the Cahn-Hilliard-HeleShaw system, J. Sci. Comput., 66 (2016), 1102-1121.
doi: 10.1007/s10915-015-0055-y. |
[26] |
D. Han and X. Wang,
A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation, J. Comput. Phys., 290 (2015), 139-156.
doi: 10.1016/j.jcp.2015.02.046. |
[27] |
D. Han and X. Wang,
Decoupled energy-law preserving numerical schemes for the Cahn-Hilliard-Darcy system, Numer. Methods Partial Differential Equations, 32 (2016), 936-954.
doi: 10.1002/num.22036. |
[28] |
J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications vol. 54 of Texts in Applied Mathematics, Springer, New York, 2008.
doi: 10.1007/978-0-387-72067-8. |
[29] |
E. Javierre, C. Vuik, F. J. Vermolen and S. van der Zwaag,
A comparison of numerical models for one-dimensional Stefan problems, J. Comput. Appl. Math., 192 (2006), 445-459.
doi: 10.1016/j.cam.2005.04.062. |
[30] |
M. Jemison, E. Loch, M. Sussman, M. Shashkov, M. Arienti, M. Ohta and Y. Wang,
A coupled level set-moment of fluid method for incompressible two-phase flows, J. Sci. Comput., 54 (2013), 454-491.
doi: 10.1007/s10915-012-9614-7. |
[31] |
M. Jemison, M. Sussman and M. Arienti,
Compressible, multiphase semi-implicit method with moment of fluid interface representation, J. Comput. Phys., 279 (2014), 182-217.
doi: 10.1016/j.jcp.2014.09.005. |
[32] |
D. Juric and G. Tryggvason,
A front-tracking method for dendritic solidification, J. Comput. Phys., 123 (1996), 127-148.
doi: 10.1006/jcph.1996.0011. |
[33] |
A. Karma and W.-J. Rappel, Quantitative phase-field modeling of dendritic growth in two and three dimensions, Phys. Rev. E, 57 (1998), 4323-4349. Google Scholar |
[34] |
G. E. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods for CFD 2nd edition, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2013. |
[35] |
C. M. Klaij, J. J. W. van der Vegt and H. van der Ven,
Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations, J. Comput. Phys., 217 (2006), 589-611.
doi: 10.1016/j.jcp.2006.01.018. |
[36] |
D. A. Kopriva, Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers Scientific Computation, Springer, Berlin, 2009. Google Scholar |
[37] |
M. Kucharik, R. V. Garimella, S. P. Schofield and M. J. Shashkov,
A comparative study of interface reconstruction methods for multi-material ALE simulations, J. Comput. Phys., 229 (2010), 2432-2452.
doi: 10.1016/j.jcp.2009.07.009. |
[38] |
B. Li and S. Da-Wen, Novel methods for rapid freezing and thawing of foods -a review, Journal of Food Engineering, 54 (2002), 175-182. Google Scholar |
[39] |
Z. Li,
Immersed interface methods for moving interface problems, Numer. Algorithms, 14 (1997), 269-293.
doi: 10.1023/A:1019173215885. |
[40] |
Z. Li and M.-C. Lai,
The immersed interface method for the Navier-Stokes equations with singular forces, J. Comput. Phys., 171 (2001), 822-842.
doi: 10.1006/jcph.2001.6813. |
[41] |
R. Loubère, P.-H. Maire, M. Shashkov, J. Breil and S. Galera,
ReALE: a reconnection-based arbitrary-Lagrangian-Eulerian method, J. Comput. Phys., 229 (2010), 4724-4761.
doi: 10.1016/j.jcp.2010.03.011. |
[42] |
P. G. Martinsson,
A direct solver for variable coefficient elliptic PDEs discretized via a composite spectral collocation method, J. Comput. Phys., 242 (2013), 460-479.
doi: 10.1016/j.jcp.2013.02.019. |
[43] |
M. N. J. Moore, Riemann-hilbert problems for the shapes formed by bodies dissolving, melting, and eroding in fluid flows, 2016, Accepted Communications in Pure and Applied Mathematics. Google Scholar |
[44] |
S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces vol. 153 of Applied Mathematical Sciences, Springer, New York, N. Y., 2003.
doi: 10.1007/b98879. |
[45] |
C. S. Peskin,
Numerical analysis of blood flow in the heart, J. Comput. Phys., 25 (1977), 220-252.
|
[46] |
S. Rhebergen and B. Cockburn,
A space-time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains, J. Comput. Phys., 231 (2012), 4185-4204.
doi: 10.1016/j.jcp.2012.02.011. |
[47] |
S. Rhebergen, B. Cockburn and J. J. W. van der Vegt,
A space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations, J. Comput. Phys., 233 (2013), 339-358.
doi: 10.1016/j.jcp.2012.08.052. |
[48] |
E. M. Ronquist and A. T. Patera,
A Legendre spectral element method for the Stefan problem, Int. J. Number. Meth. Eng., 24 (1987), 2273-2299.
doi: 10.1002/nme.1620241204. |
[49] |
B. Šarler, Stefan's work on solid-liquid phase changes, Engineering Analysis with Boundary Elements, 16 (1995), 83-92. Google Scholar |
[50] |
R. I. Saye,
High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles, SIAM J. Sci. Comput., 37 (2015), A993-A1019.
doi: 10.1137/140966290. |
[51] |
J. A. Sethian and J. Strain,
Crystal growth and dendritic solidification, J. Comput. Phys., 98 (1992), 231-253.
doi: 10.1016/0021-9991(92)90140-T. |
[52] |
W. E. H. Sollie, O. Bokhove and J. J. W. van der Vegt,
Space-time discontinuous Galerkin finite element method for two-fluid flow, J. Comput. Phys., 230 (2011), 789-817.
doi: 10.1016/j.jcp.2010.10.019. |
[53] |
J. J. Sudirham, J. J. W. van der Vegt and R. M. J. van Damme,
Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains, Appl. Numer. Math., 56 (2006), 1491-1518.
doi: 10.1016/j.apnum.2005.11.003. |
[54] |
M. Sussman, K. M. Smith, M. Y. Hussaini, M. Ohta and R. Zhi-Wei,
A sharp interface method for incompressible two-phase flows, J. Comput. Phys., 221 (2007), 469-505.
doi: 10.1016/j.jcp.2006.06.020. |
[55] |
M. Sussman and M. Y. Hussaini,
A discontinuous spectral element method for the level set equation, Special issue in honor of the sixtieth birthday of Stanley Osher, J. Sci. Comput., 19 (2003), 479-500.
doi: 10.1023/A:1025328714359. |
[56] |
M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, Journal of Computational Physics, 114 (1994), 146-159. Google Scholar |
[57] |
L. Tan and N. Zabaras,
A level set simulation of dendritic solidification of multi-component alloys, J. Comput. Phys., 221 (2007), 9-40.
doi: 10.1016/j.jcp.2006.06.003. |
[58] |
H. Udaykumar, S. Marella and S. Krishnan, Sharp-interface simulation of dendritic growth with convection: Benchmarks, International Journal of Heat and Mass Transfer, 46 (2003), 2615-2627. Google Scholar |
[59] |
M. Vahab and G. Miller,
A front-tracking shock-capturing method for two gases, Communications in Applied Mathematics and Computational Science, 11 (2016), 1-35.
doi: 10.2140/camcos.2016.11.1. |
[60] |
M. Vahab, C. Pei, M. Y. Hussaini, M. Sussman and Y. Lian, An adaptive coupled level set and moment-of-fluid method for simulating droplet impact and solidification on solid surfaces with application to aircraft icing, in 54th AIAA Aerospace Sciences Meeting 2016, p1340. Google Scholar |
[61] |
J. J. W. van der Vegt and J. J. Sudirham,
A space-time discontinuous Galerkin method for the time-dependent Oseen equations,, Appl. Numer. Math., 58 (2008), 1892-1917.
doi: 10.1016/j.apnum.2007.11.010. |
[62] |
S. W. Welch and J. Wilson, A volume of fluid based method for fluid flows with phase change, J. Comput. Phys., 160 (2000), 662-682. Google Scholar |
show all references
References:
[1] |
N. Al-Rawahi and G. Tryggvason, Numerical simulation of dendritic solidification with convection: Two-dimensional geometry, J. Comput. Phys., 180 (2002), 471-496. Google Scholar |
[2] |
N. Al-Rawahi and G. Tryggvason, Numerical simulation of dendritic solidification with convection: Three-dimensional flow, J. Comput. Phys., 194 (2004), 677-696. Google Scholar |
[3] |
V. Alexiades and A. D. Solomon, Mathematical Modelling of Melting and Freezing Processes Hemisphere Publishing Corporation, Washington, 1981. Google Scholar |
[4] |
E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney and D. Sorensen, LAPACK Users' Guide 3rd edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999. Google Scholar |
[5] |
T. D. Aslam,
A partial differential equation approach to multidimensional extrapolation, J. Comput. Phys., 193 (2004), 349-355.
doi: 10.1016/j.jcp.2003.08.001. |
[6] |
D. C. Assêncio and J. M. Teran,
A second order virtual node algorithm for Stokes flow problems with interfacial forces, discontinuous material properties and irregular domains, J. Comput. Phys., 250 (2013), 77-105.
doi: 10.1016/j.jcp.2013.04.041. |
[7] |
M. Azaï ez, F. Jelassi, M. Mint Brahim and J. Shen,
Two-phase Stefan problem with smoothed enthalpy, Commun. Math. Sci., 14 (2016), 1625-1641.
doi: 10.4310/CMS.2016.v14.n6.a8. |
[8] |
J. Bedrossian, J. H. von Brecht, S. Zhu, E. Sifakis and J. M. Teran,
A second order virtual node method for elliptic problems with interfaces and irregular domains, J. Comput. Phys., 229 (2010), 6405-6426.
doi: 10.1016/j.jcp.2010.05.002. |
[9] |
M. Benzi and M. A. Olshanskii,
An augmented Lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput., 28 (2006), 2095-2113.
doi: 10.1137/050646421. |
[10] |
C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains Berlin, Springer, 2006. |
[11] |
S. Chen, B. Merriman, S. Osher and P. Smereka,
A simple level set method for solving Stefan problems, J. Comput. Phys., 135 (1997), 8-29.
doi: 10.1006/jcph.1997.5721. |
[12] |
J. Chessa, P. Smolinski and T. Belytschko,
The extended finite element method (XFEM) for solidification problems, Int. J. Number. Meth. Eng., 53 (2002), 1959-1977.
doi: 10.1002/nme.386. |
[13] |
B. Bernardo Cockburn, G. Karniadakis and C. -W. Shu (eds. ), Discontinuous Galerkin Methods: Theory, Computation, and Applications Lecture notes in computational science and engineering, Springer, Berlin, New York, 2000. Google Scholar |
[14] |
B. Cockburn, G. Kanschat, I. Perugia and D. Schötzau,
Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal., 39 (2001), 264-285.
doi: 10.1137/S0036142900371544. |
[15] |
B. Cockburn and C.-W. Shu,
The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440-2463.
doi: 10.1137/S0036142997316712. |
[16] |
H. Coxeter, Regular Polytopes Dover Publications, Inc., New York, 1973. |
[17] |
A. Criscione, D. Kintea, Ž. Tuković, S. Jakirlić, I. Roisman and C. Tropea, Crystallization of supercooled water: A level-set-based modeling of the dendrite tip velocity, Int. J. Heat Mass Transfer, 66 (2013), 830-837. Google Scholar |
[18] |
M. Farid, The moving boundary problems from melting and freezing to drying and frying of food, Chemical Engineering and Processing: Process Intensification, 41 (2002), 1-10. Google Scholar |
[19] |
R. P. Fedkiw, T. Aslam, B. Merriman and S. Osher,
A Non-oscillatory Eulerian approach to interfaces in multimaterial flows (the Ghost Fluid Method), J. Comput. Phys., 152 (1999), 457-492.
doi: 10.1006/jcph.1999.6236. |
[20] |
W. L. George and J. A. Warren, A parallel 3D dendritic growth simulator using the phase-field method, J. Comput. Phys., 177 (2002), 264-283. Google Scholar |
[21] |
F. Gibou and R. Fedkiw,
A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem, J. Comput. Phys., 202 (2005), 577-601.
doi: 10.1016/j.jcp.2004.07.018. |
[22] |
F. Gibou, R. Fedkiw, R. Caflisch and S. Osher,
A level set approach for the numerical simulation of dendritic growth, J. Sci. Comput., 19 (2003), 183-199.
doi: 10.1023/A:1025399807998. |
[23] |
F. Gibou, R. P. Fedkiw, L.-T. Cheng and M. Kang,
A second-order-accurate symmetric discretization of the Poisson equation on irregular domains, J. Comput. Phys., 176 (2002), 205-227.
doi: 10.1006/jcph.2001.6977. |
[24] |
S. C. Gupta, The Classical Stefan Problem: Basic Concepts, Modelling and Analysis vol. 45 of North-Holland Series in Applied Mathematics and Mechanics, Elsevier Science B. V., Amsterdam, 2003. |
[25] |
D. Han,
A decoupled unconditionally stable numerical scheme for the Cahn-Hilliard-HeleShaw system, J. Sci. Comput., 66 (2016), 1102-1121.
doi: 10.1007/s10915-015-0055-y. |
[26] |
D. Han and X. Wang,
A second order in time, uniquely solvable, unconditionally stable numerical scheme for Cahn-Hilliard-Navier-Stokes equation, J. Comput. Phys., 290 (2015), 139-156.
doi: 10.1016/j.jcp.2015.02.046. |
[27] |
D. Han and X. Wang,
Decoupled energy-law preserving numerical schemes for the Cahn-Hilliard-Darcy system, Numer. Methods Partial Differential Equations, 32 (2016), 936-954.
doi: 10.1002/num.22036. |
[28] |
J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications vol. 54 of Texts in Applied Mathematics, Springer, New York, 2008.
doi: 10.1007/978-0-387-72067-8. |
[29] |
E. Javierre, C. Vuik, F. J. Vermolen and S. van der Zwaag,
A comparison of numerical models for one-dimensional Stefan problems, J. Comput. Appl. Math., 192 (2006), 445-459.
doi: 10.1016/j.cam.2005.04.062. |
[30] |
M. Jemison, E. Loch, M. Sussman, M. Shashkov, M. Arienti, M. Ohta and Y. Wang,
A coupled level set-moment of fluid method for incompressible two-phase flows, J. Sci. Comput., 54 (2013), 454-491.
doi: 10.1007/s10915-012-9614-7. |
[31] |
M. Jemison, M. Sussman and M. Arienti,
Compressible, multiphase semi-implicit method with moment of fluid interface representation, J. Comput. Phys., 279 (2014), 182-217.
doi: 10.1016/j.jcp.2014.09.005. |
[32] |
D. Juric and G. Tryggvason,
A front-tracking method for dendritic solidification, J. Comput. Phys., 123 (1996), 127-148.
doi: 10.1006/jcph.1996.0011. |
[33] |
A. Karma and W.-J. Rappel, Quantitative phase-field modeling of dendritic growth in two and three dimensions, Phys. Rev. E, 57 (1998), 4323-4349. Google Scholar |
[34] |
G. E. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods for CFD 2nd edition, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2013. |
[35] |
C. M. Klaij, J. J. W. van der Vegt and H. van der Ven,
Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations, J. Comput. Phys., 217 (2006), 589-611.
doi: 10.1016/j.jcp.2006.01.018. |
[36] |
D. A. Kopriva, Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers Scientific Computation, Springer, Berlin, 2009. Google Scholar |
[37] |
M. Kucharik, R. V. Garimella, S. P. Schofield and M. J. Shashkov,
A comparative study of interface reconstruction methods for multi-material ALE simulations, J. Comput. Phys., 229 (2010), 2432-2452.
doi: 10.1016/j.jcp.2009.07.009. |
[38] |
B. Li and S. Da-Wen, Novel methods for rapid freezing and thawing of foods -a review, Journal of Food Engineering, 54 (2002), 175-182. Google Scholar |
[39] |
Z. Li,
Immersed interface methods for moving interface problems, Numer. Algorithms, 14 (1997), 269-293.
doi: 10.1023/A:1019173215885. |
[40] |
Z. Li and M.-C. Lai,
The immersed interface method for the Navier-Stokes equations with singular forces, J. Comput. Phys., 171 (2001), 822-842.
doi: 10.1006/jcph.2001.6813. |
[41] |
R. Loubère, P.-H. Maire, M. Shashkov, J. Breil and S. Galera,
ReALE: a reconnection-based arbitrary-Lagrangian-Eulerian method, J. Comput. Phys., 229 (2010), 4724-4761.
doi: 10.1016/j.jcp.2010.03.011. |
[42] |
P. G. Martinsson,
A direct solver for variable coefficient elliptic PDEs discretized via a composite spectral collocation method, J. Comput. Phys., 242 (2013), 460-479.
doi: 10.1016/j.jcp.2013.02.019. |
[43] |
M. N. J. Moore, Riemann-hilbert problems for the shapes formed by bodies dissolving, melting, and eroding in fluid flows, 2016, Accepted Communications in Pure and Applied Mathematics. Google Scholar |
[44] |
S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces vol. 153 of Applied Mathematical Sciences, Springer, New York, N. Y., 2003.
doi: 10.1007/b98879. |
[45] |
C. S. Peskin,
Numerical analysis of blood flow in the heart, J. Comput. Phys., 25 (1977), 220-252.
|
[46] |
S. Rhebergen and B. Cockburn,
A space-time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains, J. Comput. Phys., 231 (2012), 4185-4204.
doi: 10.1016/j.jcp.2012.02.011. |
[47] |
S. Rhebergen, B. Cockburn and J. J. W. van der Vegt,
A space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations, J. Comput. Phys., 233 (2013), 339-358.
doi: 10.1016/j.jcp.2012.08.052. |
[48] |
E. M. Ronquist and A. T. Patera,
A Legendre spectral element method for the Stefan problem, Int. J. Number. Meth. Eng., 24 (1987), 2273-2299.
doi: 10.1002/nme.1620241204. |
[49] |
B. Šarler, Stefan's work on solid-liquid phase changes, Engineering Analysis with Boundary Elements, 16 (1995), 83-92. Google Scholar |
[50] |
R. I. Saye,
High-order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles, SIAM J. Sci. Comput., 37 (2015), A993-A1019.
doi: 10.1137/140966290. |
[51] |
J. A. Sethian and J. Strain,
Crystal growth and dendritic solidification, J. Comput. Phys., 98 (1992), 231-253.
doi: 10.1016/0021-9991(92)90140-T. |
[52] |
W. E. H. Sollie, O. Bokhove and J. J. W. van der Vegt,
Space-time discontinuous Galerkin finite element method for two-fluid flow, J. Comput. Phys., 230 (2011), 789-817.
doi: 10.1016/j.jcp.2010.10.019. |
[53] |
J. J. Sudirham, J. J. W. van der Vegt and R. M. J. van Damme,
Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains, Appl. Numer. Math., 56 (2006), 1491-1518.
doi: 10.1016/j.apnum.2005.11.003. |
[54] |
M. Sussman, K. M. Smith, M. Y. Hussaini, M. Ohta and R. Zhi-Wei,
A sharp interface method for incompressible two-phase flows, J. Comput. Phys., 221 (2007), 469-505.
doi: 10.1016/j.jcp.2006.06.020. |
[55] |
M. Sussman and M. Y. Hussaini,
A discontinuous spectral element method for the level set equation, Special issue in honor of the sixtieth birthday of Stanley Osher, J. Sci. Comput., 19 (2003), 479-500.
doi: 10.1023/A:1025328714359. |
[56] |
M. Sussman, P. Smereka and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, Journal of Computational Physics, 114 (1994), 146-159. Google Scholar |
[57] |
L. Tan and N. Zabaras,
A level set simulation of dendritic solidification of multi-component alloys, J. Comput. Phys., 221 (2007), 9-40.
doi: 10.1016/j.jcp.2006.06.003. |
[58] |
H. Udaykumar, S. Marella and S. Krishnan, Sharp-interface simulation of dendritic growth with convection: Benchmarks, International Journal of Heat and Mass Transfer, 46 (2003), 2615-2627. Google Scholar |
[59] |
M. Vahab and G. Miller,
A front-tracking shock-capturing method for two gases, Communications in Applied Mathematics and Computational Science, 11 (2016), 1-35.
doi: 10.2140/camcos.2016.11.1. |
[60] |
M. Vahab, C. Pei, M. Y. Hussaini, M. Sussman and Y. Lian, An adaptive coupled level set and moment-of-fluid method for simulating droplet impact and solidification on solid surfaces with application to aircraft icing, in 54th AIAA Aerospace Sciences Meeting 2016, p1340. Google Scholar |
[61] |
J. J. W. van der Vegt and J. J. Sudirham,
A space-time discontinuous Galerkin method for the time-dependent Oseen equations,, Appl. Numer. Math., 58 (2008), 1892-1917.
doi: 10.1016/j.apnum.2007.11.010. |
[62] |
S. W. Welch and J. Wilson, A volume of fluid based method for fluid flows with phase change, J. Comput. Phys., 160 (2000), 662-682. Google Scholar |













iter | ||||
(2, 1) | 1.24E-003 | 1.75E-003 | 6.12E-003 | 11 |
(3, 1) | 5.16E-005 | 6.40E-005 | 2.68E-004 | 11 |
(4, 1) | 1.28E-006 | 1.47E-006 | 7.20E-006 | 9 |
(5, 1) | 2.40E-008 | 2.63E-008 | 1.44E-007 | 8 |
(6, 1) | 3.77E-010 | 4.02E-010 | 2.31E-009 | 5 |
iter | ||||
(2, 1) | 1.24E-003 | 1.75E-003 | 6.12E-003 | 11 |
(3, 1) | 5.16E-005 | 6.40E-005 | 2.68E-004 | 11 |
(4, 1) | 1.28E-006 | 1.47E-006 | 7.20E-006 | 9 |
(5, 1) | 2.40E-008 | 2.63E-008 | 1.44E-007 | 8 |
(6, 1) | 3.77E-010 | 4.02E-010 | 2.31E-009 | 5 |
iter | ||||
(2, 1) | 3.89E-003 | 5.29E-003 | 1.34E-002 | 12 |
(3, 1) | 1.55E-004 | 1.85E-004 | 5.08E-004 | 12 |
(4, 1) | 3.73E-006 | 4.16E-006 | 1.25E-005 | 10 |
(5, 1) | 6.80E-008 | 7.32E-008 | 2.29E-007 | 6 |
(6, 1) | 9.97E-010 | 1.05E-009 | 3.47E-009 | 5 |
iter | ||||
(2, 1) | 3.89E-003 | 5.29E-003 | 1.34E-002 | 12 |
(3, 1) | 1.55E-004 | 1.85E-004 | 5.08E-004 | 12 |
(4, 1) | 3.73E-006 | 4.16E-006 | 1.25E-005 | 10 |
(5, 1) | 6.80E-008 | 7.32E-008 | 2.29E-007 | 6 |
(6, 1) | 9.97E-010 | 1.05E-009 | 3.47E-009 | 5 |
Order of convergence | |||
Weak form (46) | Normal prob (50) | ||
(4, 4) | 5 | — | — |
10 | 3.58 | 3.79 | |
20 | 4.04 | 3.85 | |
(5, 5) | 5 | — | — |
10 | 4.65 | 4.82 | |
20 | 5.38 | 4.89 |
Order of convergence | |||
Weak form (46) | Normal prob (50) | ||
(4, 4) | 5 | — | — |
10 | 3.58 | 3.79 | |
20 | 4.04 | 3.85 | |
(5, 5) | 5 | — | — |
10 | 4.65 | 4.82 | |
20 | 5.38 | 4.89 |
iter | ||||
(3, 3) | 3.86E-006 | 8.44E-007 | 1.87E-007 | 7 |
(3, 10) | 2.46E-006 | 7.67E-008 | 1.67E-008 | 7 |
(4, 4) | 3.82E-009 | 9.46E-010 | 2.11E-010 | 7 |
(4, 10) | 2.41E-009 | 1.74E-010 | 3.98E-011 | 7 |
(5, 5) | 1.48E-009 | 4.72E-010 | 1.05E-010 | 7 |
(5, 10) | 4.18E-010 | 1.28E-010 | 2.84E-011 | 7 |
(6, 6) | 3.07E-012 | 9.52E-013 | 2.12E-013 | 7 |
(6, 10) | 1.14E-012 | 3.47E-013 | 7.79E-014 | 7 |
(7, 7) | 3.19E-013 | 9.91E-014 | 2.19E-014 | 7 |
(7, 10) | 1.46E-013 | 4.55E-014 | 1.09E-014 | 7 |
iter | ||||
(3, 3) | 3.86E-006 | 8.44E-007 | 1.87E-007 | 7 |
(3, 10) | 2.46E-006 | 7.67E-008 | 1.67E-008 | 7 |
(4, 4) | 3.82E-009 | 9.46E-010 | 2.11E-010 | 7 |
(4, 10) | 2.41E-009 | 1.74E-010 | 3.98E-011 | 7 |
(5, 5) | 1.48E-009 | 4.72E-010 | 1.05E-010 | 7 |
(5, 10) | 4.18E-010 | 1.28E-010 | 2.84E-011 | 7 |
(6, 6) | 3.07E-012 | 9.52E-013 | 2.12E-013 | 7 |
(6, 10) | 1.14E-012 | 3.47E-013 | 7.79E-014 | 7 |
(7, 7) | 3.19E-013 | 9.91E-014 | 2.19E-014 | 7 |
(7, 10) | 1.46E-013 | 4.55E-014 | 1.09E-014 | 7 |
iter | ||||
(3, 3) | 1.36E-005 | 4.65E-006 | 2.34E-007 | 7 |
(3, 10) | 1.29E-005 | 4.42E-006 | 1.84E-007 | 8 |
(4, 4) | 8.27E-008 | 2.67E-008 | 1.26E-009 | 8 |
(4, 10) | 8.50E-008 | 2.75E-008 | 1.43E-009 | 8 |
(5, 5) | 1.44E-008 | 4.58E-009 | 1.02E-009 | 7 |
(5, 10) | 1.42E-008 | 4.52E-009 | 1.01E-009 | 7 |
(6, 6) | 1.43E-012 | 1.62E-014 | 4.03E-013 | 10 |
(6, 10) | 1.20E-012 | 3.86E-013 | 4.81E-013 | 8 |
(7, 7) | 4.08E-012 | 1.27E-012 | 2.80E-013 | 10 |
(7, 10) | 4.20E-012 | 1.31E-012 | 2.87E-013 | 9 |
iter | ||||
(3, 3) | 1.36E-005 | 4.65E-006 | 2.34E-007 | 7 |
(3, 10) | 1.29E-005 | 4.42E-006 | 1.84E-007 | 8 |
(4, 4) | 8.27E-008 | 2.67E-008 | 1.26E-009 | 8 |
(4, 10) | 8.50E-008 | 2.75E-008 | 1.43E-009 | 8 |
(5, 5) | 1.44E-008 | 4.58E-009 | 1.02E-009 | 7 |
(5, 10) | 1.42E-008 | 4.52E-009 | 1.01E-009 | 7 |
(6, 6) | 1.43E-012 | 1.62E-014 | 4.03E-013 | 10 |
(6, 10) | 1.20E-012 | 3.86E-013 | 4.81E-013 | 8 |
(7, 7) | 4.08E-012 | 1.27E-012 | 2.80E-013 | 10 |
(7, 10) | 4.20E-012 | 1.31E-012 | 2.87E-013 | 9 |
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