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Stability and accuracy of time‐stepping schemes and dispersion relations for a nonlocal wave equation
Author(s) -
Guan Qingguang,
Gunzburger Max
Publication year - 2015
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21931
Subject(s) - mathematics , discretization , stability (learning theory) , piecewise , mathematical analysis , finite element method , partial differential equation , wave equation , euler's formula , courant–friedrichs–lewy condition , dispersion (optics) , time derivative , boundary value problem , euler equations , piecewise linear function , backward euler method , physics , machine learning , computer science , optics , thermodynamics
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‐like schemes. For other values of the parameter which includes the forward‐Euler‐like scheme, conditional stability is proved. Dispersion relations for the nonlocal wave equation in one and two dimensions are derived. Comparisons with the analogous results for the classical wave equation are provided as the results of numerical experiments that illustrate the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 500–516, 2015