2014
DOI: 10.1002/num.21931
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Stability and accuracy of time‐stepping schemes and dispersion relations for a nonlocal wave equation

Abstract: A time‐dependent nonlocal wave equation is considered. A feature of the model is that instead of boundary conditions, constraints over regions having finite measures are imposed. The Newmark scheme is considered for discretizing the time derivative and piecewise‐linear finite element methods are used for spatial discretization. For certain ranges of a parameter appearing in the Newmark scheme, unconditional stability is proved; in particular, this result applies to the backward‐Euler‐like and Crank‐Nicolson‐li… Show more

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Cited by 19 publications
(21 citation statements)
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References 19 publications
(28 reference statements)
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“…Let θ ∈ [0, 1] be the parameter which controls the contribution of the implicit and explicit scheme. Let (û k ,v k ) be the solution of Equation 18 and Equation 19 for given fixed θ.…”
Section: Extension To the Implicit Schemesmentioning
confidence: 99%
“…Let θ ∈ [0, 1] be the parameter which controls the contribution of the implicit and explicit scheme. Let (û k ,v k ) be the solution of Equation 18 and Equation 19 for given fixed θ.…”
Section: Extension To the Implicit Schemesmentioning
confidence: 99%
“…Numerical analysis of linear peridynamic models for one‐dimensional (1D) bars have been given in the work of Bobaru et al and Weckner and Emmrich . Related approximations of nonlocal diffusion models are discussed in the works of Tian et al, Chen and Gunzburger, and Du et al A stability analysis of the numerical approximation to solutions of linear nonlocal wave equations is given in the work of Guan and Gunzburger …”
Section: Introductionmentioning
confidence: 99%
“…16 Related approximations of nonlocal diffusion models are discussed in the works of Tian et al, 19 Chen and Gunzburger, 20 and Du et al 21 A stability analysis of the numerical approximation to solutions of linear nonlocal wave equations is given in the work of Guan and Gunzburger. 22 In this work, we analyze the discrete approximations to the nonlinear nonlocal model developed in the works of Lipton. 23,24 This model is a smooth version of the prototypical micro-elastic model introduced in the work of Silling, 1 see Section 2.…”
Section: Introductionmentioning
confidence: 99%
“…Also, in what concerns the numerical analysis, there are multiple contributions to facilitate the comprehension of the equations and the efficiency of different numerical schemes. [23][24][25][26][27][28][29] However, there is not much literature concerning nonlocal optimal design models. We refer, among others, Andrés and Muñoz, 30,31 Bonder and Spedaletti, 32 D'Elia and Gunzburger, 33,34 and Zhou and Du 35,36 mainly in the context of elliptic equations.…”
Section: Introductionmentioning
confidence: 99%