Skip to main content
×
Home
    • Aa
    • Aa

Artificial Boundary Conditions for Nonlocal Heat Equations on Unbounded Domain

  • Wei Zhang (a1), Jiang Yang (a2), Jiwei Zhang (a1) and Qiang Du (a2)
Abstract
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.

Copyright
Corresponding author
*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)
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] B.Alpert , L.Greengard , and T.Hagstrom , Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation, SIAM J. Numer. Anal., 37 (2000), 11381164.

[2] B.Alpert , L.Greengard and T.Hagstrom , Nonreflecting Boundary Conditions for the Time-Dependent Wave Equation, Comput. Phy., 180 (2002), 270296.

[3] F.Andreu , J.M.Mazón , J.D.Rossi , and J.Toledo , Local and nonlocal weighted p–Laplacian evolution equations with Neumann boundary conditions, Publ. Mat., 55 (2011), 2766.

[4] O.Bakunin , Turbulence and Diffusion: Scaling Versus Equations, Springer-Verlag, New York, 2008.

[5] G.Beylkin and L.Monzón , Approximation by exponential sums revisited, Appl. Comput. Harmon. Anal., 28 (2010), 131149.

[6] J.-P.Berenger , A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114:2 (1994), 185200.

[7] F.Bobaru and M.Duangpanya , The peridynamic formulation for transient heat conduction, Int. J. Heat Mass Transfer, 53 (2010), 40474059.

[8] F.Bobaru and M.Duangpanya , A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities, J. Comput. Phys., 231 (2012), 27642785.

[9] N.Burch and R.B.Lehoucq , Classical, nonlocal, and fractional diffusion equations on bounded domains, Int. J. Multiscale Comput. Eng., 9 (2011), 661674.

[10] X.Chen and M.Gunzburger , Continuous and discontinuous finite element methods for a peridynamics model of mechanics, Comput. Method Appl. Mech. Eng., 200 (2011), 12371250.

[11] W. C.Chew and W. H.Weedon , A 3D perfectly matched medium from modified Maxwell equations with stretched coordinates, Microw. Opt. Technol. Lett., 7:13 (1994), 599604.

[12] P. N.Demmie and S. A.Silling , An approach to modeling extreme loading of structures using peridynamics, J. Mech. Mater. Struct., 2:10 (2007), 19211945.

[13] Q.Du , M.Gunzburger , R.Lehoucq and K.Zhou , Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667696.

[14] Q.Du , M.Gunzburger , R.B.Lehoucq and K.Zhou , A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Mod. Meth. Appl. Sci., 23 (2013), 493540.

[15] Q.Du and K.Zhou , Mathematical analysis for the peridynamic nonlocal continuum theory, ESIAM: Math. Model. Numer. Anal., 45 (2011), 217234.

[16] E.Emmrich and O.Weckner , Analysis and numerical approximation of an integro-differential equation modelling non-local effects in linear elasticity, Math. Mech. Solids, 12 (2007), 363384.

[17] E.Emmrich and O.Weckner , On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity, Commun. Math. Sci., 5 (2007), 851864.

[18] E.Emmrich and O.Weckner , The peridynamic equation and its spatial discretisation, Math. Model. Anal., 12:1 (2007), 1727.

[19] B.Engquist and A.Majda , Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., 31 (1977), 629651.

[22] M.Gunzburger and R.B.Lehoucq , A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 15811598.

[23] Y. D.Ha and F.Bobaru , Studies of dynamic crack propagation and crack branching with peridynamics, Int. J. Fract., 162:1-2 (2010), 229244.

[24] H.Han and Z.Huang , Exact and approximating boundary conditions for the parabolic problems on unbounded domains, Comput. Math. Appl., 44 (2002), 655666.

[25] H.Han and Z.Huang , A class of artificial boundary conditions for heat equation in unbounded domains, Comput. Math. Appl., 43 (2002), 889900.

[26] H.Han and X.Wu , Artificial Boundary Method, Spring-Verlag and Tsinghua Unversity Press, Berlin Heidelberg and Beijing, 2013.

[28] S.Jiang and L.Greengard , Fast Evaluation of Nonreflecting Boundary Conditions for the Schrödinger Equation in One Dimension, Comput. Math. Appl., 47 (2004), no. 6-7, 955966.

[29] S.Jiang and L.Greengard , Efficient representation of nonreflecting boundary conditions for the time-dependent Schrödinger equation in two dimensions, Comm. Pure Appl. Math., 61 (2008), 261288.

[31] J.Li , A fast time stepping method for evaluating fractional integrals, SIAM J. Sci. Comput., 31 (2010), 46964714.

[32] Y.Lin and C.Xu , Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 15331552.

[33] R. W.Macek and S.Silling , Peridynamics via finite element analysis, Finite Elem. Anal. Des., 43 (2007), 11691178.

[34] T.Mengesha and Q.Du , Analysis of a scalar peridynamic model with a sign changing kernel, Disc. Cont. Dyn. Systems B, 18 (2013), 14151437.

[35] T.Mengesha and Q.Du , Characterization of function spaces of vector fields via nonlocal derivatives and an application in peridynamics, Nonlinear Anal. A, 140 (2016), 82111.

[36] R.Metzler and J.Klafter , 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.

[37] S. P.Neuman and D. M.Tartakovsky , Perspective on theories of non-Fickian transport in heterogenous media, Adv. Water Resources, 32 (2009), 670680.

[39] S. A.Silling and E.Askari , A meshfree method based on the peridynamic model of solid mechanics, Comput. Struct., 83:17-18 (2005), 15261535.

[40] S.Silling and R.B.Lehoucq , Peridynamic theory of solid mechanics, Adv. Appl. Mech., 44 (2010), 73166.

[41] S.Silling , O.Weckner , E.Askari and F.Bobaru , Crack nucleation in a peridynamic solid, Int. J. Fracture, 162 (2010), 219227.

[42] Z.Sun and X.Wu , A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), 193209.

[43] X.Tian and Q.Du , Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations, SIAM J. Numer. Anal., 51 (2013), 34583482.

[44] X.Tian and Q.Du , Asymptotically compatible schemes and applications to robust discretization of nonlocal models, SIAM J. Numer. Anal., 52 (2014), 16411665.

[45] R.A.Wildman and G.A.Gazonas , A perfectly matched layer for peridynamics in two dimensions, J. Mech. Mater. Struct., 7:8-9 (2012), 765781.

[49] K.Zhou and Q.Du , Mathematical and Numerical Analysis of Linear Peridynamic Models with Nonlocal Boundary Conditions, SIAM J. Numer. Anal., 48 (2010), 17591780.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords: