Volume 1, Issue 1
Analysis of Coarse-Grained Lattice Models and Connections to Nonlocal Interactions

Qiang Du, Xiantao Li & Liming Yuan

CSIAM Trans. Appl. Math., 1 (2020), pp. 155-185.

Published online: 2020-03

[An open-access article; the PDF is free to any online user.]

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  • Abstract

We study coarse-grained models of some linear static lattice models with interactions up to second nearest neighbors. It will be demonstrated how nonlocal interactions, as described by a nonlocal kernel function, arise from a coarse-graining procedure. Some important properties of the nonlocal kernels will be established such as its decay rate and positivity. We also study the scaling behavior of the kernel functions as the level of coarse-graining changes. In addition, we suggest closure approximations of the nonlocal interactions that can be expressed in local PDE forms by introducing auxiliary variables.

  • Keywords

Linear static models, coarse-graining, next nearest neighbor interactions, nonlocal models.

  • AMS Subject Headings

65N15, 74G15, 70E55

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

qd2125@columbia.edu (Qiang Du)

xli@math.psu.edu (Xiantao Li)

lzy104@psu.edu (Liming Yuan)

  • BibTex
  • RIS
  • TXT
@Article{CSIAM-AM-1-155, author = {Du , Qiang and Li , Xiantao and Yuan , Liming }, title = {Analysis of Coarse-Grained Lattice Models and Connections to Nonlocal Interactions}, journal = {CSIAM Transactions on Applied Mathematics}, year = {2020}, volume = {1}, number = {1}, pages = {155--185}, abstract = {

We study coarse-grained models of some linear static lattice models with interactions up to second nearest neighbors. It will be demonstrated how nonlocal interactions, as described by a nonlocal kernel function, arise from a coarse-graining procedure. Some important properties of the nonlocal kernels will be established such as its decay rate and positivity. We also study the scaling behavior of the kernel functions as the level of coarse-graining changes. In addition, we suggest closure approximations of the nonlocal interactions that can be expressed in local PDE forms by introducing auxiliary variables.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.2020-0009}, url = {http://global-sci.org/intro/article_detail/csiam-am/16798.html} }
TY - JOUR T1 - Analysis of Coarse-Grained Lattice Models and Connections to Nonlocal Interactions AU - Du , Qiang AU - Li , Xiantao AU - Yuan , Liming JO - CSIAM Transactions on Applied Mathematics VL - 1 SP - 155 EP - 185 PY - 2020 DA - 2020/03 SN - 1 DO - http://doi.org/10.4208/csiam-am.2020-0009 UR - https://global-sci.org/intro/article_detail/csiam-am/16798.html KW - Linear static models, coarse-graining, next nearest neighbor interactions, nonlocal models. AB -

We study coarse-grained models of some linear static lattice models with interactions up to second nearest neighbors. It will be demonstrated how nonlocal interactions, as described by a nonlocal kernel function, arise from a coarse-graining procedure. Some important properties of the nonlocal kernels will be established such as its decay rate and positivity. We also study the scaling behavior of the kernel functions as the level of coarse-graining changes. In addition, we suggest closure approximations of the nonlocal interactions that can be expressed in local PDE forms by introducing auxiliary variables.

Qiang Du, Xiantao Li & Liming Yuan. (2020). Analysis of Coarse-Grained Lattice Models and Connections to Nonlocal Interactions. CSIAM Transactions on Applied Mathematics. 1 (1). 155-185. doi:10.4208/csiam-am.2020-0009
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