Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T14:44:18.095Z Has data issue: false hasContentIssue false
  • English
  • Français

A Numerical Method for Solving Matrix Coefficient Heat Equations with Interfaces

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  10 November 2015

Liqun Wang  and
Liwei Shi
Liqun Wang
Affiliation:
Department of Mathematics, China University of Petroleum (Beijing), Beijing, 102249, China
Liwei Shi*
Affiliation:
Department of Science and Technology Teaching, China University of Political Science and Law, Beijing, 102249, China
*
*Corresponding author. Email addresses: wliqunhmily@gmail.com (L.-Q. Wang), sliweihmily@gmail.com (L.-W. Shi)
Get access

Abstract

In this paper, we propose a numerical method for solving the heat equations with interfaces. This method uses the non-traditional finite element method together with finite difference method to get solutions with second-order accuracy. It is capable of dealing with matrix coefficient involving time, and the interfaces under consideration are sharp-edged interfaces instead of smooth interfaces. Modified Euler Method is employed to ensure the accuracy in time. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up) on the sharp-edged interface corner. Extensive numerical experiments illustrate the feasibility of the method.

Keywords

Heat equationSharp-edged interfaceJump conditionMatrix coefficient

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 4 , November 2015 , pp. 475 - 495
Copyright
Copyright © Global-Science Press 2015 

Access options

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

References

[1]Chadam, S. M. and Yin, H., A diffusion equation with localized chemical reactions, Proc. of Edinburgh Math., vol. 37 (1993), pp. 101–118.CrossRefGoogle Scholar
[2]Kandilarov, J. D., A Rothe-immersed interface method for a class of parabolic interface problems, Numer. Anal. Appl., vol. 3401 (2005), pp. 328–336.Google Scholar
[3]Peskin, C., Numerical analysis of blood flow in the heart, J. Comput. Phys., vol. 25 (1977), pp. 220–252.CrossRefGoogle Scholar
[4]Peskin, C. and Printz, B., Improved volume conservation in the computation of flows with immersed elastic boundaries, J. Comput. Phys., vol. 105 (1993), pp. 33–46.CrossRefGoogle Scholar
[5]Sussman, M., Smereka, P., and Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., vol. 114 (1994), pp. 146–154.CrossRefGoogle Scholar
[6]Li, Z., An overview of the immersed interface method and its applications, Taiwanese J. Math., vol. 7 (2003), pp. 1–49.CrossRefGoogle Scholar
[7]Leveque, R. J. and Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., vol. 31 (1994), pp. 1019–1044.CrossRefGoogle Scholar
[8]Liu, X. D., Fedkiw, R. P., and Kang, M., A boundary condition capturing method for Pois-son's equation on irregular domains, J. Comput. Phys., vol. 160 (2000), 151–178.CrossRefGoogle Scholar
[9]Fedkiw, R., Aslam, T., Merriman, B., and Osher, S., A non-oscillatory eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys., vol. 152 (1999), 457–492.CrossRefGoogle Scholar
[10]Wan, J. W. L., and Liu, X. D., A boundary condition capturing multigrid approach to irregular boundary problems, SIAM J. Sci. Comput., vol. 25, no. 6 (2004), 1982–2003.CrossRefGoogle Scholar
[11]Macklin, P. and Lowengrub, J. S., A new ghost cell / level set method for moving boundary problems: application to tumor growth, J. Sci. Comput., vol. 35 (2008), pp. 266–299.CrossRefGoogle ScholarPubMed
[12]Yu, S., Zhou, Y., and Wei, G. W., Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces, J. Comput. Phys., vol. 224 (2007), pp. 729–756.CrossRefGoogle Scholar
[13]Zhou, Y. C., Zhao, S., Feig, M., and Wei, G. W., High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources, J. Comput. Phys., vol. 213 (2006), pp. 1–30.CrossRefGoogle Scholar
[14]Moes, N., Dolbow, J., and Belytschko, T., A finite element method for crack growth without remeshing, Int. J. Numer. Meth. Engng., vol. 46, no. 1 (1999), pp. 131–150.3.0.CO;2-J>CrossRefGoogle Scholar
[15]Wagner, G. J., Moes, N., Liu, W. K., and Belytschko, T., The extended finite element method for rigid particles in Stokes flow, Int. J. Numer. Meth. Engng., vol. 51, no. 3 (2001), pp. 293–313.CrossRefGoogle Scholar
[16]Chessa, J. and Belytschko, T., An extended finite element method for twophase fluids: flow simulation and modeling, J. Appl. Mech., vol. 70, no. 1 (2003), pp. 10–17.CrossRefGoogle Scholar
[17]Gong, Y., Li, B., and Li, Z., Immersed-interface finite-element methods for elliptic interface problems with non-homogeneous jump conditions, SIAM J. Numer. Anal., vol. 46 (2008), pp. 472–495.CrossRefGoogle Scholar
[18]Li, Z., Lin, T., and Wu, X., New Cartesian grid methods for interface problems using the finite element formulation, Numerische Mathematik, vol. 96, no. 1 (2003), pp. 61–98.CrossRefGoogle Scholar
[19]Babuska, I., The finite element methodfor elliptic equations with discontinuous coefficients, Computing, vol. 5 (1970), pp. 207–218.CrossRefGoogle Scholar
[20]Hansbo, A. and Hansbo, P., An unfitted finite element method, based on Nitsches method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., vol. 191 (2002), pp. 5537–5552.CrossRefGoogle Scholar
[21]Chen, Z. and Zou, J., Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., vol. 79 (1996), pp. 175–202.CrossRefGoogle Scholar
[22]Attanayake, C. and Senaratne, D., Convergence of an immersed finite element method for semilinear parabolic interface problems, Appl. Math. Sci., vol. 5, no. 3 (2011), pp. 135–147.Google Scholar
[23]Hou, S. and Liu, X. D., A numerical methodfor solving variable coefficient elliptic equations with interfaces, J. Comput. Phys., vol. 202 (2005), pp. 411–445.CrossRefGoogle Scholar
[24]Hou, S., Wang, W., and Wang, L., Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces, J. Comput. Phys., vol. 229 (2010), pp. 7162–7179.CrossRefGoogle Scholar
[25] S. Hou, Song, P., Wang, L., and Zhao, H., A weak formulation for solving elliptic interface problems without body fitted grid, J. Comput. Phys., vol. 249 (2013), pp. 80–95.Google Scholar
[26]Johansen, H. and Colella, P., A cartesian grid embedded boundary method for poisson's equation on irregular domains, J. Comput. Phys., vol. 147 (1998), pp. 60–85.CrossRefGoogle Scholar
[27]Mccorquodale, P., Colella, P., and Johansen, H., A cartesian grid embedded boundary method for the heat equation on irregular domains, J. Comput. Phys., vol. 173 (2001), pp. 620–635.CrossRefGoogle Scholar
[28]Schwartz, P., Barad, M., Colella, P., and Ligocki, T., A cartesian grid embedded boundary method for the heat equation and poisson's equation in three dimensions, J. Comput. Phys., vol. 211 (2006), pp. 531–550.CrossRefGoogle Scholar
[29]Wang, S., Samulyak, R., and Guo, T., An embedded boundary method for elliptic and parabolic problems with interfaces and application to multi-material systems with phase transitions, Acta Mathematica Scientia, vol. 30, no.2 (2010), pp. 499–521.CrossRefGoogle Scholar
[30]Hou, S., Li, Z., Wang, L., and Wang, W., A numerical method for solving elasticity equations with interfaces, Comm. Comput. Phys., vol. 12 (2012), pp. 595–612.CrossRefGoogle ScholarPubMed
[31]Hou, S., Wang, L., and Wang, W., A numerical method for solving the elliptic interface problems with multi-domains and triple junction points, J. Comput. Math., vol. 30 (2012), pp. 504–516.CrossRefGoogle Scholar
[32]Wang, L., Hou, S., and Shi, L., A numerical method for solving 3D elasticity equations with sharp-edged interfaces, Int. J. PDE., (2013) Article ID 476873.CrossRefGoogle Scholar
[33]Wang, L. and Shi, L., Numerical method for solving matrix coefficient elliptic equations in irregular domain with sharp-edged boundary, Adv. Numer. Anal., (2013) Article ID 967342.CrossRefGoogle Scholar
[34]Wang, L., Hou, S., and Shi, L., An improved non-traditional finite element formulation for solving the elliptic interface problems, J. Comput. Math., vol. 32 (2014), pp. 39–57.CrossRefGoogle Scholar
[35]Wang, L. and Shi, L., A simple method for matrix-valued coefficient elliptic equations with sharp-edged interfaces, Appl. Math. Comput., vol. 242 (2014), pp. 917–930Google Scholar
[36]Li, Z. and Shen, Y., A numerical method for solving heat equations involving interfaces, Electron. J. Diff. Eqns., Conf. 03 (1999), pp. 100–108.Google Scholar
[37]Huang, J. and Zou, J., Some New A Priori Estimates for Second-Order Elliptic and Parabolic Interface Problems, J. Diff. Eqns., vol. 184 (2002), pp. 570–586.CrossRefGoogle Scholar
[38]Yang, X., Li, B., and Li, Z., The Immersed Interface Method for Elasticity Problems with Interfaces, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10/2003.Google Scholar
[39]Dai, W., Khaliq, A., and Nassar, R., An Introduction to Finite Element Mtehod, Louisiana Tech University Text Book, Ruston, Louisiana, USA.Google Scholar
[40]Necas, J., Introduction to the theory of nonlinear elliptic equations, Teubner-Texte zur Mathematik, Band 52 (1983), ISSN 0138-502X.Google Scholar