Convergence of two implicit numerical schemes for diffusion mathematical models with delay

García, Piqueras; Castro, María Ángeles; Martín, J.A.; Sirvent, Antonio

doi:10.1016/j.mcm.2010.02.016

Cited by 12 publications

(5 citation statements)

References 5 publications

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“…Bounds on the local truncation error of the scheme have been provided, its properties of consistency and stability have been analysed, and conditions for the convergence of the method have been established and illustrated with numerical examples. Open problems for future work would include to formally derive the mixed delayed model from physical foundations, as was done in [29] for the pure delayed model, and the development of higher order implicit schemes, in a similar way to those proposed in [9] for the constant coefficients case.…”

Section: Discussionmentioning

confidence: 99%

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Difference schemes for time-dependent heat conduction models with delay

Castro¹,

Rodríguez²,

Cabrera³

et al. 2013

International Journal of Computer Mathematics

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Different non-Fourier models of heat conduction, that incorporate time lags in the heat flux and/or the temperature gradient, have been increasingly considered in the last years to model microscale heat transfer problems in engineering. Numerical schemes to obtain approximate solutions of constant coefficients lagging models of heat conduction have already been proposed. In this work, an explicit finite difference scheme for a model with coefficients variable in time is developed, and their properties of convergence and stability are studied. Numerical computations showing examples of applications of the scheme are presented.

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Section: Discussionmentioning

confidence: 99%

“…Difference schemes for parabolic problems with delay have also been proposed (see, e.g. [1,2,19,31]), including problem (2) in the constant coefficients case [8,9]. The aim of this work is the construction of an explicit difference scheme for the time-dependent problem (2)-(4), characterizing its convergence properties.…”

Section: Introductionmentioning

confidence: 99%

Difference schemes for time-dependent heat conduction models with delay

Castro¹,

Rodríguez²,

Cabrera³

et al. 2013

International Journal of Computer Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…There are plenty of phenomena where the presence of delays is to necessarily considered [3,4,5,6,7]. Equation of the form (1.1) has been studied extensively in the case where r 3 = r 4 = 0 (see,e.g., [8,9,10,11,12,13]. The accessibility of exact solutions as well as efficient methods to obtain numerical approximations of required precision are very important.…”

Section: Introductionmentioning

confidence: 99%

Cubic B-spline Galerkin Finite Element Method for the Generalized Reaction Diffusion Equation with Delay

Lubo,

Duressa

2024

Preprint

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In this paper, a generalized reaction diffusion equation with delay has been numerically solved by a cubic Bspline Galerkin finite element method. The full discrete numerical process is based on trapezoidal rule. To demonstrate the efficiency of the method, numerical illustrations have been given. Graphs are also displayed in support of the numerical results. Both the theoretical and computational rate of convergence of the numerical method have been examined and found to be in agreement. As it can be observed from the numerical results given in tables and graphs, the proposed method approximates the exact solution very well. The accuracy of numerical scheme is confirmed by computing L2 and L∞ error norms

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“…Later on, several researchers have discussed the stability and a priori error analysis of numerical methods for (1.1)-(1.3) with a constant delay, i.e., τ (t) = t − η(t) = τ with τ > 0 being a real constant. The authors of [13,14] considered the consistence, stability and convergence of an explicit difference scheme and two implicit difference schemes for the Eqs. (1.1)-(1.3) with a constant delay, respectively.…”

Section: Introductionmentioning

confidence: 99%

A Posteriori Error Estimates for Fully Discrete Finite Element Method for Generalized Diffusion Equation with Delay

Wang

Xiao

2020

J Sci Comput

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In this paper, we derive several a posteriori error estimators for generalized diffusion equation with delay in a convex polygonal domain. The Crank-Nicolson method for time discretization is used and a continuous, piecewise linear finite element space is employed for the space discretization. The a posteriori error estimators corresponding to space discretization are derived by using the interpolation estimates. Two different continuous, piecewise quadratic reconstructions are used to obtain the error due to the time discretization. To estimate the error in the approximation of the delay term, linear approximations of the delay term are used in a crucial way. As a consequence, a posteriori upper and lower error bounds for fully discrete approximation are derived for the first time. In particular, long-time a posteriori error estimates are obtained for stable systems. Numerical experiments are presented which confirm our theoretical results.

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Convergence of two implicit numerical schemes for diffusion mathematical models with delay

Cited by 12 publications

References 5 publications

Difference schemes for time-dependent heat conduction models with delay

Difference schemes for time-dependent heat conduction models with delay

Cubic B-spline Galerkin Finite Element Method for the Generalized Reaction Diffusion Equation with Delay

A Posteriori Error Estimates for Fully Discrete Finite Element Method for Generalized Diffusion Equation with Delay

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