Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T09:48:24.996Z Has data issue: false hasContentIssue false
  • English
  • Français

Artificial Boundary Conditions for Nonlocal Heat Equations on Unbounded Domain

Part of: Numerical approximation and computational geometry Integral equations, integral transforms Linear integral equations

Published online by Cambridge University Press:  05 December 2016

Wei Zhang ,
Jiang Yang ,
Jiwei Zhang  and
Qiang Du
Wei Zhang*
Affiliation:
Beijing Computational Science Research Centre, Beijing, P.R. China
Jiang Yang*
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10027, USA
Jiwei Zhang*
Affiliation:
Beijing Computational Science Research Centre, Beijing, P.R. China
Qiang Du*
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10027, USA
*
*Corresponding author. Email addresses:wzhang@csrc.ac.cn (W. Zhang), jyanghkbu@gmail.com (J. Yang), jwzhang@csrc.ac.cn (J. Zhang), qd2125@columbia.edu (Q. Du)
*Corresponding author. Email addresses:wzhang@csrc.ac.cn (W. Zhang), jyanghkbu@gmail.com (J. Yang), jwzhang@csrc.ac.cn (J. Zhang), qd2125@columbia.edu (Q. Du)
*Corresponding author. Email addresses:wzhang@csrc.ac.cn (W. Zhang), jyanghkbu@gmail.com (J. Yang), jwzhang@csrc.ac.cn (J. Zhang), qd2125@columbia.edu (Q. Du)
*Corresponding author. Email addresses:wzhang@csrc.ac.cn (W. Zhang), jyanghkbu@gmail.com (J. Yang), jwzhang@csrc.ac.cn (J. Zhang), qd2125@columbia.edu (Q. Du)
Get access

Abstract

This paper is concerned with numerical approximations of a nonlocal heat equation define on an infinite domain. Two classes of artificial boundary conditions (ABCs) are designed, namely, nonlocal analog Dirichlet-to-Neumann-type ABCs (global in time) and high-order Padé approximate ABCs (local in time). These ABCs reformulate the original problem into an initial-boundary-value (IBV) problem on a bounded domain. For the global ABCs, we adopt a fast evolution to enhance computational efficiency and reduce memory storage. High order fully discrete schemes, both second-order in time and space, are given to discretize two reduced problems. Extensive numerical experiments are carried out to show the accuracy and efficiency of the proposed methods.

Keywords

Artificial boundary conditionsnonlocal modelsPadé approximationnonlocal heat equationsartificial boundary method

MSC classification

Secondary: 45A05: Linear integral equations 65D99: None of the above, but in this section 65R20: Integral equations
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 1 , January 2017 , pp. 16 - 39
Copyright
Copyright © Global-Science Press 2016 

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] Alpert, B., Greengard, L., and Hagstrom, T., Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation, SIAM J. Numer. Anal., 37 (2000), 1138–1164.CrossRefGoogle Scholar
[2] Alpert, B., Greengard, L. and Hagstrom, T., Nonreflecting Boundary Conditions for the Time-Dependent Wave Equation, Comput. Phy., 180 (2002), 270–296.CrossRefGoogle Scholar
[3] Andreu, F., Mazón, J.M., Rossi, J.D., and Toledo, J., Local and nonlocal weighted p–Laplacian evolution equations with Neumann boundary conditions, Publ. Mat., 55 (2011), 27–66.CrossRefGoogle Scholar
[4] Bakunin, O., Turbulence and Diffusion: Scaling Versus Equations, Springer-Verlag, New York, 2008.Google Scholar
[5] Beylkin, G. and Monzón, L., Approximation by exponential sums revisited, Appl. Comput. Harmon. Anal., 28 (2010), 131–149.CrossRefGoogle Scholar
[6] Berenger, J.-P., A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114:2 (1994), 185–200.CrossRefGoogle Scholar
[7] Bobaru, F. and Duangpanya, M., The peridynamic formulation for transient heat conduction, Int. J. Heat Mass Transfer, 53 (2010), 4047–4059.CrossRefGoogle Scholar
[8] Bobaru, F. and Duangpanya, M., A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities, J. Comput. Phys., 231 (2012), 2764–2785.CrossRefGoogle Scholar
[9] Burch, N. and Lehoucq, R.B., Classical, nonlocal, and fractional diffusion equations on bounded domains, Int. J. Multiscale Comput. Eng., 9 (2011), 661–674.CrossRefGoogle Scholar
[10] Chen, X. and Gunzburger, M., Continuous and discontinuous finite element methods for a peridynamics model of mechanics, Comput. Method Appl. Mech. Eng., 200 (2011), 1237–1250.CrossRefGoogle Scholar
[11] Chew, W. C. and Weedon, W. H., A 3D perfectly matched medium from modified Maxwell equations with stretched coordinates, Microw. Opt. Technol. Lett., 7:13 (1994), 599–604.CrossRefGoogle Scholar
[12] Demmie, P. N. and Silling, S. A., An approach to modeling extreme loading of structures using peridynamics, J. Mech. Mater. Struct., 2:10 (2007), 1921–1945.CrossRefGoogle Scholar
[13] Du, Q., Gunzburger, M., Lehoucq, R. and Zhou, K., Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667–696.CrossRefGoogle Scholar
[14] Du, Q., Gunzburger, M., Lehoucq, R.B. and Zhou, K., A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Mod. Meth. Appl. Sci., 23 (2013), 493–540.CrossRefGoogle Scholar
[15] Du, Q. and Zhou, K., Mathematical analysis for the peridynamic nonlocal continuum theory, ESIAM: Math. Model. Numer. Anal., 45 (2011), 217–234.Google Scholar
[16] Emmrich, E. and Weckner, O., Analysis and numerical approximation of an integro-differential equation modelling non-local effects in linear elasticity, Math. Mech. Solids, 12 (2007), 363–384.CrossRefGoogle Scholar
[17] Emmrich, E. and Weckner, O., On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity, Commun. Math. Sci., 5 (2007), 851–864.CrossRefGoogle Scholar
[18] Emmrich, E. and Weckner, O., The peridynamic equation and its spatial discretisation, Math. Model. Anal., 12:1 (2007), 17–27.CrossRefGoogle Scholar
[19] Engquist, B. and Majda, A., Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., 31 (1977), 629–651.CrossRefGoogle Scholar
[20] Gerstle, W., Silling, S. A., Read, D., Tewary, V., and Lehoucq, R., Peridynamic simulation of electromigration, Comput. Mater. Continua, 8:2 (2008), 75–92.Google Scholar
[21] Givoli, D., Finite element analysis of heat problems in unbounded domains, Numerical Methods in Thermal Problems 6.Part 2 (1989): 1094–1104.Google Scholar
[22] Gunzburger, M. and Lehoucq, R.B., A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 1581–1598.CrossRefGoogle Scholar
[23] Ha, Y. D. and Bobaru, F., Studies of dynamic crack propagation and crack branching with peridynamics, Int. J. Fract., 162:1-2 (2010), 229–244.CrossRefGoogle Scholar
[24] Han, H. and Huang, Z., Exact and approximating boundary conditions for the parabolic problems on unbounded domains, Comput. Math. Appl., 44 (2002), 655–666.CrossRefGoogle Scholar
[25] Han, H. and Huang, Z., A class of artificial boundary conditions for heat equation in unbounded domains, Comput. Math. Appl., 43 (2002), 889–900.CrossRefGoogle Scholar
[26] Han, H. and Wu, X., Artificial Boundary Method, Spring-Verlag and Tsinghua Unversity Press, Berlin Heidelberg and Beijing, 2013.CrossRefGoogle Scholar
[27] Jiang, S., Fast Evaluation of the Nonreflecting Boundary Conditions for the Schrödinger Equation, Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University, New York, 2001.Google Scholar
[28] Jiang, S. and Greengard, L., Fast Evaluation of Nonreflecting Boundary Conditions for the Schrödinger Equation in One Dimension, Comput. Math. Appl., 47 (2004), no. 6-7, 955–966.CrossRefGoogle Scholar
[29] Jiang, S. and Greengard, L., Efficient representation of nonreflecting boundary conditions for the time-dependent Schrödinger equation in two dimensions, Comm. Pure Appl. Math., 61 (2008), 261–288.CrossRefGoogle Scholar
[30] Jiang, S.D. et al., Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, preprint.Google Scholar
[31] Li, J., A fast time stepping method for evaluating fractional integrals, SIAM J. Sci. Comput., 31 (2010), 4696–4714.CrossRefGoogle Scholar
[32] Lin, Y. and Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533–1552.CrossRefGoogle Scholar
[33] Macek, R. W. and Silling, S., Peridynamics via finite element analysis, Finite Elem. Anal. Des., 43 (2007), 1169–1178.CrossRefGoogle Scholar
[34] Mengesha, T. and Du, Q., Analysis of a scalar peridynamic model with a sign changing kernel, Disc. Cont. Dyn. Systems B, 18 (2013), 1415–1437.Google Scholar
[35] Mengesha, T. and Du, Q., Characterization of function spaces of vector fields via nonlocal derivatives and an application in peridynamics, Nonlinear Anal. A, 140 (2016), 82–111.CrossRefGoogle Scholar
[36] Metzler, R. and Klafter, J., The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), R161.CrossRefGoogle Scholar
[37] Neuman, S. P. and Tartakovsky, D. M., Perspective on theories of non-Fickian transport in heterogenous media, Adv. Water Resources, 32 (2009), 670–680.CrossRefGoogle Scholar
[38] Oldham, K.B. and Spanier, J., The Fractional Calculus, Academic Press, New York, 1974.Google Scholar
[39] Silling, S. A. and Askari, E., A meshfree method based on the peridynamic model of solid mechanics, Comput. Struct., 83:17-18 (2005), 1526–1535.CrossRefGoogle Scholar
[40] Silling, S. and Lehoucq, R.B., Peridynamic theory of solid mechanics, Adv. Appl. Mech., 44 (2010), 73–166.CrossRefGoogle Scholar
[41] Silling, S., Weckner, O., Askari, E. and Bobaru, F., Crack nucleation in a peridynamic solid, Int. J. Fracture, 162 (2010), 219–227.CrossRefGoogle Scholar
[42] Sun, Z. and Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193–209.CrossRefGoogle Scholar
[43] Tian, X. and Du, Q., Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations, SIAM J. Numer. Anal., 51 (2013), 3458–3482.CrossRefGoogle Scholar
[44] Tian, X. and Du, Q., Asymptotically compatible schemes and applications to robust discretization of nonlocal models, SIAM J. Numer. Anal., 52 (2014), 1641–1665.CrossRefGoogle Scholar
[45] Wildman, R.A. and Gazonas, G.A., A perfectly matched layer for peridynamics in two dimensions, J. Mech. Mater. Struct., 7:8-9 (2012), 765–781.CrossRefGoogle Scholar
[46] Wildman, R.A. and Gazonas, G.A., A perfectly matched layer for peridynamics in one dimension, Technical report ARL-TR-5626, U.S. Army Research Laboratory, Aberdeen, MD, 2011.Google Scholar
[47] Wu, X. and Zhang, J., High order local absorbing boundary conditions for heat equation in unbounded domains, J. Comput. Math., 29 (2011), 74–90.Google Scholar
[48] Zheng, C., Approximation, stability and fast evaluation of exact artificial boundary condition for one-dimensional heat equation, J. Comput. Math., 25 (2007), 730–745.Google Scholar
[49] Zhou, K. and Du, Q., Mathematical and Numerical Analysis of Linear Peridynamic Models with Nonlocal Boundary Conditions, SIAM J. Numer. Anal., 48 (2010), 1759–1780.CrossRefGoogle Scholar