A conservative nonlocal convection–diffusion model and asymptotically compatible finite difference discretization

Tian, Hao; Ju, Lili; Du, Qiang

doi:10.1016/j.cma.2017.03.020

A conservative nonlocal convection–diffusion model and asymptotically compatible finite difference discretization

Journal Article · Wed Mar 22 00:00:00 EDT 2017 · Computer Methods in Applied Mechanics and Engineering

DOI:https://doi.org/10.1016/j.cma.2017.03.020· OSTI ID:1533652

Tian, Hao ^[1]; Ju, Lili ^[2]; Du, Qiang ^[3]

Ocean University of China, Qingdao (China); DOE/OSTI
University of South Carolina, Columbia, SC (United States)
Columbia University, New York, NY (United States)

Here in this paper, we first propose a nonlocal convection–diffusion model, in which the convection term is constructed in a special upwind manner so that mass conservation and maximum principle are maintained in any space dimension. The well-posedness of the proposed nonlocal model and its convergence to the classical local convection–diffusion model are established. A quadrature-based finite difference discretization is then developed to numerically solve the nonlocal problem and it is shown to be consistent and unconditionally stable. We further demonstrate that the numerical scheme is asymptotically compatible, that is, the approximate solutions converge to the exact solution of the corresponding local problem when δ → 0 and h → 0. Numerical experiments are also performed to complement the theoretical analysis.

View Accepted Manuscript (DOE)

Research Organization:: University of South Carolina, Columbia, SC (United States)

Sponsoring Organization:: Air Force Office of Scientific Research (AFOSR); Army Research Office (ARO); Central Universities; National Science Foundation (NSF); USDOE; USDOE Office of Science (SC)

Grant/Contract Number:: SC0008087; SC0016540

OSTI ID:: 1533652

Alternate ID(s):: OSTI ID: 1550578

Journal Information:: Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering Journal Issue: C Vol. 320; ISSN 0045-7825

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (42)

Convergence, adaptive refinement, and scaling in 1D peridynamics Bobaru, Florin; Yang, Mijia; Alves, Leonardo Frota International Journal for Numerical Methods in Engineering, Vol. 77, Issue 6 https://doi.org/10.1002/nme.2439	journal	February 2009
Variational solution of fractional advection dispersion equations on bounded domains in ℝd Ervin, Vincent J.; Roop, John Paul Numerical Methods for Partial Differential Equations, Vol. 23, Issue 2 https://doi.org/10.1002/num.20169	journal	January 2007
Peridynamic States and Constitutive Modeling Silling, S. A.; Epton, M.; Weckner, O. Journal of Elasticity, Vol. 88, Issue 2 https://doi.org/10.1007/s10659-007-9125-1	journal	July 2007
Linearized Theory of Peridynamic States Silling, S. A. Journal of Elasticity, Vol. 99, Issue 1 https://doi.org/10.1007/s10659-009-9234-0	journal	January 2010
Cohesive Dynamics and Brittle Fracture Lipton, Robert Journal of Elasticity, Vol. 124, Issue 2 https://doi.org/10.1007/s10659-015-9564-z	journal	December 2015
A Nonlocal Model for Contact Attraction and Repulsion in Heterogeneous Cell Populations Painter, K. J.; Bloomfield, J. M.; Sherratt, J. A. Bulletin of Mathematical Biology, Vol. 77, Issue 6 https://doi.org/10.1007/s11538-015-0080-x	journal	May 2015
M-matrix characterizations.I—nonsingular M-matrices Plemmons, R. J. Linear Algebra and its Applications, Vol. 18, Issue 2 https://doi.org/10.1016/0024-3795(77)90073-8	journal	January 1977
A Petrov-Galerkin finite element method for convection-dominated flows: An accurate upwinding technique for satisfying the maximum principle Mizukami, Akira; Hughes, Thomas J. R. Computer Methods in Applied Mechanics and Engineering, Vol. 50, Issue 2 https://doi.org/10.1016/0045-7825(85)90089-1	journal	August 1985
Reformulation of elasticity theory for discontinuities and long-range forces Silling, S. A. Journal of the Mechanics and Physics of Solids, Vol. 48, Issue 1 https://doi.org/10.1016/S0022-5096(99)00029-0	journal	January 2000
Peridynamic Theory of Solid Mechanics Silling, S. A.; Lehoucq, R. B. Advances in Applied Mechanics https://doi.org/10.1016/S0065-2156(10)44002-8	book	September 2010
Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations Tao, Yunzhe; Tian, Xiaochuan; Du, Qiang Applied Mathematics and Computation, Vol. 305 https://doi.org/10.1016/j.amc.2017.01.061	journal	July 2017
Continuous and discontinuous finite element methods for a peridynamics model of mechanics Chen, X.; Gunzburger, Max Computer Methods in Applied Mechanics and Engineering, Vol. 200, Issue 9-12 https://doi.org/10.1016/j.cma.2010.10.014	journal	February 2011
Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites Hu, Wenke; Ha, Youn Doh; Bobaru, Florin Computer Methods in Applied Mechanics and Engineering, Vol. 217-220 https://doi.org/10.1016/j.cma.2012.01.016	journal	April 2012
Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains Seleson, Pablo; Gunzburger, Max; Parks, Michael L. Computer Methods in Applied Mechanics and Engineering, Vol. 266 https://doi.org/10.1016/j.cma.2013.05.018	journal	November 2013
A fast and faithful collocation method with efficient matrix assembly for a two-dimensional nonlocal diffusion model Wang, Hong; Tian, Hao Computer Methods in Applied Mechanics and Engineering, Vol. 273 https://doi.org/10.1016/j.cma.2014.01.026	journal	May 2014
Nonlocal convection–diffusion problems and finite element approximations Tian, Hao; Ju, Lili; Du, Qiang Computer Methods in Applied Mechanics and Engineering, Vol. 289 https://doi.org/10.1016/j.cma.2015.02.008	journal	June 2015
Robust a posteriori stress analysis for quadrature collocation approximations of nonlocal models via nonlocal gradients Du, Qiang; Tao, Yunzhe; Tian, Xiaochuan Computer Methods in Applied Mechanics and Engineering, Vol. 310 https://doi.org/10.1016/j.cma.2016.07.023	journal	October 2016
A meshfree method based on the peridynamic model of solid mechanics Silling, S. A.; Askari, E. Computers & Structures, Vol. 83, Issue 17-18 https://doi.org/10.1016/j.compstruc.2004.11.026	journal	June 2005
Peridynamics via finite element analysis Macek, Richard W.; Silling, Stewart A. Finite Elements in Analysis and Design, Vol. 43, Issue 15 https://doi.org/10.1016/j.finel.2007.08.012	journal	November 2007
A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities Bobaru, Florin; Duangpanya, Monchai Journal of Computational Physics, Vol. 231, Issue 7 https://doi.org/10.1016/j.jcp.2011.12.017	journal	April 2012
Peridynamic thermal diffusion Oterkus, Selda; Madenci, Erdogan; Agwai, Abigail Journal of Computational Physics, Vol. 265 https://doi.org/10.1016/j.jcp.2014.01.027	journal	May 2014
A nonlocal convection–diffusion equation Ignat, Liviu I.; Rossi, Julio D. Journal of Functional Analysis, Vol. 251, Issue 2 https://doi.org/10.1016/j.jfa.2007.07.013	journal	October 2007
The effect of long-range forces on the dynamics of a bar Weckner, Olaf; Abeyaratne, Rohan Journal of the Mechanics and Physics of Solids, Vol. 53, Issue 3 https://doi.org/10.1016/j.jmps.2004.08.006	journal	March 2005
Cauchy problem of a nonlocal -Laplacian evolution equation with nonlocal convection Sun, Jiebao; Li, Jing; Liu, Qiang Nonlinear Analysis: Theory, Methods & Applications, Vol. 95 https://doi.org/10.1016/j.na.2013.09.023	journal	January 2014
Deformation of a Peridynamic Bar Silling, S. A.; Zimmermann, M.; Abeyaratne, R. Journal of Elasticity, Vol. 73, Issue 1-3 https://doi.org/10.1023/B:ELAS.0000029931.03844.4f	journal	December 2003
Mathematical analysis for the peridynamic nonlocal continuum theory Du, Qiang; Zhou, Kun ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 45, Issue 2 https://doi.org/10.1051/m2an/2010040	journal	August 2010
A posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models Du, Qiang; Ju, Lili; Tian, Li Mathematics of Computation, Vol. 82, Issue 284 https://doi.org/10.1090/S0025-5718-2013-02708-1	journal	May 2013
Asymptotically compatible discretization of multidimensional nonlocal diffusion models and approximation of nonlocal Green’s functions Du, Qiang; Tao, Yunzhe; Tian, Xiaochuan IMA Journal of Numerical Analysis, Vol. 39, Issue 2 https://doi.org/10.1093/imanum/dry011	journal	April 2018
Multidimensional advection and fractional dispersion Meerschaert, Mark M.; Benson, David A.; Bäumer, Boris Physical Review E, Vol. 59, Issue 5 https://doi.org/10.1103/PhysRevE.59.5026	journal	May 1999
Nonlocal Linear Image Regularization and Supervised Segmentation Gilboa, Guy; Osher, Stanley Multiscale Modeling & Simulation, Vol. 6, Issue 2 https://doi.org/10.1137/060669358	journal	January 2007
Nonlocal Operators with Applications to Image Processing Gilboa, Guy; Osher, Stanley Multiscale Modeling & Simulation, Vol. 7, Issue 3 https://doi.org/10.1137/070698592	journal	January 2009
A Nonlocal Vector Calculus with Application to Nonlocal Boundary Value Problems Gunzburger, Max; Lehoucq, R. B. Multiscale Modeling & Simulation, Vol. 8, Issue 5 https://doi.org/10.1137/090766607	journal	January 2010
Image Denoising Methods. A New Nonlocal Principle Buades, A.; Coll, B.; Morel, J. M. SIAM Review, Vol. 52, Issue 1 https://doi.org/10.1137/090773908	journal	January 2010
Mathematical and Numerical Analysis of Linear Peridynamic Models with Nonlocal Boundary Conditions Zhou, Kun; Du, Qiang SIAM Journal on Numerical Analysis, Vol. 48, Issue 5 https://doi.org/10.1137/090781267	journal	January 2010
A New Approach for a Nonlocal, Nonlinear Conservation Law Du, Qiang; Kamm, James R.; Lehoucq, R. B. SIAM Journal on Applied Mathematics, Vol. 72, Issue 1 https://doi.org/10.1137/110833233	journal	January 2012
Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints Du, Qiang; Gunzburger, Max; Lehoucq, R. B. SIAM Review, Vol. 54, Issue 4 https://doi.org/10.1137/110833294	journal	January 2012
A Convergent Adaptive Finite Element Algorithm for Nonlocal Diffusion and Peridynamic Models Du, Qiang; Tian, Li; Zhao, Xuying SIAM Journal on Numerical Analysis, Vol. 51, Issue 2 https://doi.org/10.1137/120871638	journal	January 2013
Analysis and Comparison of Different Approximations to Nonlocal Diffusion and Linear Peridynamic Equations Tian, Xiaochuan; Du, Qiang SIAM Journal on Numerical Analysis, Vol. 51, Issue 6 https://doi.org/10.1137/13091631X	journal	January 2013
Asymptotically Compatible Schemes and Applications to Robust Discretization of Nonlocal Models Tian, Xiaochuan; Du, Qiang SIAM Journal on Numerical Analysis, Vol. 52, Issue 4 https://doi.org/10.1137/130942644	journal	January 2014
A Nonlocal Vector Calculus, Nonlocal Volume-Constrained Problems, and Nonlocal Balance laws Du, Qiang; Gunzburger, Max; Lehoucq, R. B. Mathematical Models and Methods in Applied Sciences, Vol. 23, Issue 03 https://doi.org/10.1142/S0218202512500546	journal	January 2013
Fractional Diffusion on Bounded Domains Defterli, Ozlem; D’Elia, Marta; Du, Qiang Fractional Calculus and Applied Analysis, Vol. 18, Issue 2 https://doi.org/10.1515/fca-2015-0023	journal	January 2015
Non-local kinetic and macroscopic models for self-organised animal aggregations A. Carrillo, José; Eftimie, Raluca Kinetic & Related Models, Vol. 8, Issue 3 https://doi.org/10.3934/krm.2015.8.413	journal	January 2015

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42 ENGINEERING
97 MATHEMATICS AND COMPUTING
asymptotically compatible
mass conservation
maximum principle
nonlocal convection–diffusion
quadrature-based finite difference

A conservative nonlocal convection–diffusion model and asymptotically compatible finite difference discretization

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