Asymptotically Compatible Schemes for Peridynamics Based on Numerical Quadratures

Volume 1: Advances in Aerospace Technology ◽

10.1115/imece2014-39620 ◽

2014 ◽

Cited By ~ 7

Author(s):

Qiang Du ◽

Xiaochuan Tian

Keyword(s):

Diffusion Models ◽

Nonlocal Diffusion ◽

Local Models ◽

Numerical Schemes ◽

Nonlocal Models ◽

Numerical Quadratures

We present some studies of numerical schemes for nonlocal peridynamic and nonlocal diffusion models. We describe asymptotically compatible (AC) schemes recently developed for robust discretizations of nonlocal models. The AC schemes for peridynamic models provide convergent approximations to nonlocal models associated with fixed horizon parameter as well as their limiting local models. We illustrate what quadrature based discretizations can be AC schemes and what may fail to be AC.

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Numerical Schemes for Pricing Asian Options Under State-Dependent Regime-Switching Jump-Diffusion Models

SSRN Electronic Journal ◽

10.2139/ssrn.2571507 ◽

2015 ◽

Author(s):

Duy-Minh Dang ◽

Duy Nguyen ◽

Granville Sewell

Keyword(s):

Regime Switching ◽

Jump Diffusion ◽

Diffusion Models ◽

Asian Options ◽

Numerical Schemes ◽

State Dependent

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Peridynamics and Nonlocal Diffusion Models: Fast Numerical Methods

Handbook of Nonlocal Continuum Mechanics for Materials and Structures ◽

10.1007/978-3-319-22977-5_35-1 ◽

2017 ◽

pp. 1-23 ◽

Cited By ~ 1

Author(s):

Hong Wang

Keyword(s):

Numerical Methods ◽

Diffusion Models ◽

Nonlocal Diffusion

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Asymptotically compatible discretization of multidimensional nonlocal diffusion models and approximation of nonlocal Green’s functions

IMA Journal of Numerical Analysis ◽

10.1093/imanum/dry011 ◽

2018 ◽

Vol 39 (2) ◽

pp. 607-625 ◽

Cited By ~ 4

Author(s):

Qiang Du ◽

Yunzhe Tao ◽

Xiaochuan Tian ◽

Jiang Yang

Keyword(s):

Green's Functions ◽

Numerical Solutions ◽

Green’S Functions ◽

Diffusion Equations ◽

Nonlocal Diffusion ◽

Optimal Order ◽

Numerical Approximations ◽

Splitting Technique ◽

Singular Measures ◽

Nonlocal Models

AbstractNonlocal diffusion equations and their numerical approximations have attracted much attention in the literature as nonlocal modeling becomes popular in various applications. This paper continues the study of robust discretization schemes for the numerical solution of nonlocal models. In particular, we present quadrature-based finite difference approximations of some linear nonlocal diffusion equations in multidimensions. These approximations are able to preserve various nice properties of the nonlocal continuum models such as the maximum principle and they are shown to be asymptotically compatible in the sense that as the nonlocality vanishes, the numerical solutions can give consistent local limits. The approximation errors are proved to be of optimal order in both nonlocal and asymptotically local settings. The numerical schemes involve a unique design of quadrature weights that reflect the multidimensional nature and require technical estimates on nonconventional divided differences for their numerical analysis. We also study numerical approximations of nonlocal Green’s functions associated with nonlocal models. Unlike their local counterparts, nonlocal Green’s functions might become singular measures that are not well defined pointwise. We demonstrate how to combine a splitting technique with the asymptotically compatible schemes to provide effective numerical approximations of these singular measures.

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A collocation method based on localized radial basis functions with reproducibility for nonlocal diffusion models

Computational and Applied Mathematics ◽

10.1007/s40314-021-01665-6 ◽

2021 ◽

Vol 40 (8) ◽

Author(s):

Jiashu Lu ◽

Yufeng Nie

Keyword(s):

Collocation Method ◽

Radial Basis Functions ◽

Diffusion Models ◽

Basis Functions ◽

Nonlocal Diffusion ◽

Radial Basis

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Nonlocal and local models for taxis in cell migration: a rigorous limit procedure

Journal of Mathematical Biology ◽

10.1007/s00285-020-01536-4 ◽

2020 ◽

Vol 81 (6-7) ◽

pp. 1251-1298 ◽

Cited By ~ 1

Author(s):

Maria Eckardt ◽

Kevin J. Painter ◽

Christina Surulescu ◽

Anna Zhigun

Keyword(s):

Cell Migration ◽

General Solution ◽

Numerical Simulations ◽

Integral Operators ◽

Mathematical Framework ◽

Local Models ◽

Nonlocal Operators ◽

Nonlocal Models ◽

Limit Procedure

AbstractA rigorous limit procedure is presented which links nonlocal models involving adhesion or nonlocal chemotaxis to their local counterparts featuring haptotaxis and classical chemotaxis, respectively. It relies on a novel reformulation of the involved nonlocalities in terms of integral operators applied directly to the gradients of signal-dependent quantities. The proposed approach handles both model types in a unified way and extends the previous mathematical framework to settings that allow for general solution-dependent coefficient functions. The previous forms of nonlocal operators are compared with the new ones introduced in this paper and the advantages of the latter are highlighted by concrete examples. Numerical simulations in 1D provide an illustration of some of the theoretical findings.

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