American Institute of Mathematical Sciences

Advanced
November  2018, 23(9): 3595-3622. doi: 10.3934/dcdsb.2017216

A space-time discontinuous Galerkin spectral element method for the Stefan problem

Chaoxu Pei , Mark Sussman , and  M. Yousuff Hussaini

Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA

* Corresponding author: Mark Sussman (sussman@math.fsu.edu)

Received  May 2017 Revised  June 2017 Published  November 2018 Early access  September 2017

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.

Keywords: Space-time, discontinuous Galerkin, spectral accuracy, transformations, the Stefan problem.
Mathematics Subject Classification: Primary: 65M70, 35R37.
Citation: Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216
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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.

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

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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.

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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. 

[2]

N. Al-Rawahi and G. Tryggvason, Numerical simulation of dendritic solidification with convection: Three-dimensional flow, J. Comput. Phys., 194 (2004), 677-696. 

[3]

V. Alexiades and A. D. Solomon, Mathematical Modelling of Melting and Freezing Processes Hemisphere Publishing Corporation, Washington, 1981.

[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.

[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.

[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. 

[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. 

[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. 

[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. 

[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.

[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. 

[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.

[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. 

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Figure 1.  The Stefan problem on a Cartesian domain $\Omega \in \mathbb{R}^2$. $\Omega^{s}_t$ and $\Omega^{l}_t$ are time-dependent domains at time $t$, while domain $\Omega$ is fixed. $\Gamma(t)$ is the moving interface at time $t$, and $\boldsymbol{n}=(n_1,n_2)$ is the interface normal pointing in the direction of interface movement. $\phi$ is the level set function introduced in section 2.2
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Figure 2.  A tessellation of the space-time slab ${\mathscr{E}}^n$ of a space-time domain: ${\mathscr{E}}= \Omega \times [t_0,T] \in \mathbb{R}^2$. The coordinates of a point $\bar{\boldsymbol{x}}$ are denoted by $\bar{\boldsymbol{x}}=(x_1,x_{2})$, where $x=x_1$ is the spatial variable and $x_{2}$ denotes the time variable. The square with bold lines is an example of a space-time slab ${\mathscr{E}}^{n}$, such that ${\mathscr{E}}^n={\mathscr{E}} \cap I_n$, where $I_n=[t_n,t_{n+1}]$. We assume that the solid phase is on the left hand side of the interface ${\mathscr{Q}}^n_{\Gamma}$, while the liquid is on the other side. The space-time element ${\mathscr{K}}^{n,s}_j$ is referred to as a regular cell in the solid phase. However, ${\mathscr{K}}^{n,sl}_{j+1}$ and ${\mathscr{K}}^{n,sl}_{j+2}$ are examples of a cut-cell due to the time evolving interface, ${\mathscr{Q}}^n_{\Gamma}$, defined on a nonconforming computational grid. In this example, the curve ${\mathscr{Q}}^n_{\Gamma}$ obtained by 4th order interpolation cuts ${\mathscr{K}}^{n,sl}_{j+1}$ into two subelements, i.e., a pentagon ${\mathscr{K}}^{n,s,\Gamma}_{j+1}$ and a triangle ${\mathscr{K}}^{n,l,\Gamma}_{j+1}$
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Figure 3.  Examples of a cut-cell in a space-time domain ${\mathscr{E}} \in \mathbb{R}^2$. The moving front is evolving from left to right in each case. The shape of the generated subelements could be triangle, quadrilateral or pentagon
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Figure 4.  Examples of a mapping technique applied to two kinds of a cut-cell in a space-time domain ${\mathscr{E}} \in \mathbb{R}^2$. In each case, the curve ${\mathscr{Q}}^n_{\Gamma}$ is mapped to a straight bold line ${\tilde{\mathscr{Q}}}^n_{\Gamma}$, which is either aligned with an inter-element boundary or cutting a regular cell into two subelements
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Figure 5.  Description of the normal probe method applied in a space-time slab ${\mathscr{E}}^n \in \mathbb{R}^2$ in the fixed frame of reference. The polynomial order, $p=(p^{(x)},p^{(t)})$, in each space-time element is chosen to be $(3,2)$. We assume that the solid phase is on the left hand side of the interface ${\mathscr{Q}}^n_{\Gamma}$ while the liquid is on the other side. The fictitious element in the solid phase at time $t_{n+1}$ is denoted as $e^{n+1,s}_{fic}$, and $e^{n+1,l}_{fic}$ is the fictitious element for the liquid phase at time $t_{n+1}$. The length of both fictitious elements is $h$, which is the same as the length of a regular space-time element, i.e., the length of ${\mathscr{K}}^{n,s}_{j-1}$. The number of Gauss-Lobatto nodes in each fictitious element is $p+1$, where $p=p^{(x)}$. The temperature gradient is evaluated at the node with symbol "$\Box$" in each fictitious element
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Figure 6.  Errors in the temperature (left) and the interface velocity (right) as a function of polynomial order $p^{(x)}$ in the liquid phase. The polynomial order in time direction, $p^{(t)}$, is chosen to be the same as $p^{(x)}$. The comparison is made between the weak form method (46) and the normal probe method (50). The simulation is computed over the time $t=0$ to $t=0.5$ with the time step $\Delta t=3.84 \times 10^{-2}$
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Figure 7.  Errors in the temperature (left) and the interface velocity (right) as a function of polynomial order $p^{(x)}$ in the liquid phase. The simulation is computed over the time $t=0$ to $t=0.5$. The computational domain is divided into $E^{(x)}=5$ in spatial direction, with $E^{(x),l}=4$ in the liquid in the beginning and $E^{(x),l}=2$ in the liquid at the end. The interface velocity is computed by the weak form method (46). The time step is chosen to be $\Delta t=3.84 \times 10^{-2}$
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Figure 8.  Errors in the temperature (left) and the interface velocity (right) as a function of polynomial order $p^{(x)}$ in the liquid phase. The simulation is computed over the time $t=0$ to $t=0.5$. The computational domain is divided into $E^{(x)}=5$ in spatial direction, with $E^{(x),l}=4$ in the liquid in the beginning and $E^{(x),l}=2$ in the liquid at the end. The interface velocity is computed by the normal probe method (50). The time step is chosen to be $\Delta t=3.84 \times 10^{-2}$
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Figure 9.  Errors in the temperature (left) and the interface velocity (right) as a function of polynomial order $p^{(x)}$ in the ice phase. The comparison is made between the weak form method (46) and the normal probe method (50). The interface starts at $x=0.1$ and moves to $x=0.3$. The polynomial order in time direction, $p^{(t)}$, is chosen to be $10$ so that the temporal errors are negligible. The time step is chosen to be $\Delta t=4.9 \times 10^{-2}$. The Stefan number $St$ is set to be $0.02$
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Figure 10.  Errors in the temperature (left) and the interface velocity (right) as a function of polynomial order $p^{(t)}$ in the ice phase. The interface velocity is computed by the weak form method (46). The interface starts at $x=0.1$ and moves to $x=0.3$. The comparison is made among different numbers of the space-time slab, i.e., $E^{(t)}=41$, $E^{(t)}=81$ and $E^{(t)}=162$. The corresponding time step is $\Delta t=4.9 \times 10^{-2}$, $\Delta t=2.5 \times 10^{-2}$ and $\Delta t=1.2 \times 10^{-2}$. The polynomial order in spatial direction, $p^{(x)}$, is chosen to be $10$ so that the spatial errors are negligible. The computational domain is divided into $E^{(x)}=5$ in the spatial direction. The Stefan number $St$ is set to be $0.02$
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Figure 11.  Errors in the temperature (left) and the interface velocity (right) as a function of polynomial order $p^{(t)}$ in the ice phase. The interface velocity is computed by the normal probe method (50). The interface starts at $x=0.1$ and moves to $x=0.3$. The comparison is made among different numbers of the space-time slab, i.e., $E^{(t)}=41$, $E^{(t)}=81$ and $E^{(t)}=162$. The corresponding time step is $\Delta t=4.9 \times 10^{-2}$, $\Delta t=2.5 \times 10^{-2}$ and $\Delta t=1.2 \times 10^{-2}$. The polynomial order in spatial direction, $p^{(x)}$, is chosen to be $10$ so that the spatial errors are negligible. The computational domain is divided into $E^{(x)}=5$ in the spatial direction. The Stefan number $St$ is set to be $0.02$
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Figure 12.  Errors in the temperature (left) and the interface velocity (right) as a function of polynomial order $p^{(x)}$ in the ice phase. The interface velocity is computed by the weak form method (46). The polynomial order in time direction, $p^{(t)}$, is chosen to be the same as $p^{(x)}$. The interface starts at $x=0.1$ and moves to $x=0.4$. The comparison is made among different values of $St$, i.e., $St=0.001$, $St=0.005$, $St=0.01$ and $St=0.05$. The computational domain is divided into $E^{(x)}=6$ in the spatial direction. The time step is chosen to be $\Delta t=0.9 \times 10^{-3}$
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Figure 13.  Errors in the temperature (left) and the interface velocity (right) as a function of polynomial order $p^{(x)}$ in the ice phase. The interface velocity is computed by the normal probe method (50). The polynomial order in time direction, $p^{(t)}$, is chosen to be the same as $p^{(x)}$. The interface starts at $x=0.1$ and moves to $x=0.4$. The comparison is made among different values of $St$, i.e., $St=0.001$, $St=0.02$, $St=0.05$ and $St=0.08$. The computational domain is divided into $E^{(x)}=6$ in the spatial direction. The time step is chosen to be $\Delta t=0.9 \times 10^{-3}$
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Table 1.  Errors of the temperature ($\|Err_{\theta}\|_{\infty}$), interface position ($\|Err_{\Gamma}\|_{\infty}$) and interface velocity ($\|Err_{V_n}\|_{\infty}$) in the last space-time slab. The interface velocity is computed by applying the weak form method (46). The number of the Picard iterations is listed in the last column with $tol=10^{-13}$
$p=(p^{(x)},p^{(t)})$ $\|Err_{\theta}\|_{\infty}$ $\|Err_{\Gamma}\|_{\infty}$ $\|Err_{V_n}\|_{\infty}$ 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
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$p=(p^{(x)},p^{(t)})$ $\|Err_{\theta}\|_{\infty}$ $\|Err_{\Gamma}\|_{\infty}$ $\|Err_{V_n}\|_{\infty}$ 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
Table 2.  Errors of the temperature ($\|Err_{\theta}\|_{\infty}$), interface position ($\|Err_{\Gamma}\|_{\infty}$) and interface velocity ($\|Err_{V_n}\|_{\infty}$) in the last space-time slab. The interface velocity is computed by applying the normal probe method (50). The number of the Picard iterations is listed in the last column with $tol=10^{-13}$
$p=(p^{(x)},p^{(t)})$ $\|Err_{\theta}\|_{\infty}$ $\|Err_{\Gamma}\|_{\infty}$ $\|Err_{V_n}\|_{\infty}$ 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
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$p=(p^{(x)},p^{(t)})$ $\|Err_{\theta}\|_{\infty}$ $\|Err_{\Gamma}\|_{\infty}$ $\|Err_{V_n}\|_{\infty}$ 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
Table 3.  Order of convergence of the temperature ($\|Err_{\theta}\|_{\infty}$) for both options in the last space-time slab
Order of convergence
$p=(p^{(x)},p^{(t)})$ $E^{(x)}$ 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
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Order of convergence
$p=(p^{(x)},p^{(t)})$ $E^{(x)}$ 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
Table 4.  Errors of the temperature ($\|Err_{\theta}\|_{\infty}$), interface position ($\|Err_{\Gamma}\|_{\infty}$) and interface velocity ($\|Err_{V_n}\|_{\infty}$) in the last space-time slab. The interface velocity is computed by applying the weak form method (46). The computational domain is divided into $E^{(x)}=6$ in the spatial direction. The interface starts at $x=0.1$ and moves to $x=0.3$. The time step is chosen to be $\Delta t=4.9 \times 10^{-2}$. The number of the Picard iteration is listed in the last column with $tol=10^{-15}$
$p=(p^{(x)},p^{(t)})$ $\|Err_{\theta}\|_{\infty}$ $\|Err_{\Gamma}\|_{\infty}$ $\|Err_{V_n}\|_{\infty}$ 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
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$p=(p^{(x)},p^{(t)})$ $\|Err_{\theta}\|_{\infty}$ $\|Err_{\Gamma}\|_{\infty}$ $\|Err_{V_n}\|_{\infty}$ 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
Table 5.  Errors of the temperature ($\|Err_{\theta}\|_{\infty}$), interface position ($\|Err_{\Gamma}\|_{\infty}$) and interface velocity ($\|Err_{V_n}\|_{\infty}$) in the last space-time slab. The interface velocity is computed by applying the normal probe method (50). The computational domain is divided into $E^{(x)}=6$ in the spatial direction. The interface starts at $x=0.1$ and moves to $x=0.3$. The time step is chosen to be $\Delta t=4.9 \times 10^{-2}$. The number of the Picard iteration is listed in the last column with $tol=10^{-15}$
$p=(p^{(x)},p^{(t)})$ $\|Err_{\theta}\|_{\infty}$ $\|Err_{\Gamma}\|_{\infty}$ $\|Err_{V_n}\|_{\infty}$ 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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$p=(p^{(x)},p^{(t)})$ $\|Err_{\theta}\|_{\infty}$ $\|Err_{\Gamma}\|_{\infty}$ $\|Err_{V_n}\|_{\infty}$ 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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