Richardson extrapolation technique for singularly perturbed delay parabolic partial differential eqn

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L. Govindarao, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 44-48 ISSN: 2395-3519

International Journal of Advanced Trends in Computer Applications www.ijatca.com

Richardson extrapolation technique for singularly perturbed delay parabolic partial differential equation 1

L. Govindarao, 2J. Mohapatra 1,2 Department of Mathematics, National Institute of Technology, Rourkela, India-769008 1

jugal@nitrkl.ac.in, 2lolugu.govindarao@gmail.com

Abstract: This article presents a higher-order parameter uniformly convergent method for a singularly perturbed delay parabolic reaction-diffusion initial-boundary-value problem. For the discretization of the time derivative, we use the Crank-Nicolson scheme on the uniform mesh and for the spatial discretization, we use the central difference scheme on the Shishkin mesh, which provides a second order convergence rate. To enhance the order of convergence, we apply the Richardson extrapolation technique. We prove that the proposed method converges uniformly with respect to the perturbation parameter and also attains almost fourth order convergence rate. Finally, to support the theoretical results, we present some numerical experiments by using the proposed method.

Keywords: Singular perturbation, Delay parabolic problems, Crank-Nicolson scheme, Richardson extrapolation.

I.

and

INTRODUCTION

In this article, we consider the following singularly perturbed delay parabolic reaction-diffusion InitialBoundary-Value Problems (IBVP): đ?œ• đ?œ•đ?‘Ą

+ đ??żđ?œ€,đ?‘Ľ đ?‘˘ đ?‘Ľ, đ?‘Ą = −đ?‘? đ?‘Ľ, đ?‘Ą đ?‘˘ đ?‘Ľ, đ?‘Ą − đ?œ? + đ?‘“ đ?‘Ľ, đ?‘Ą , đ?‘Ľ, đ?‘Ą ∈ đ??ˇ,

đ?‘˘ đ?‘Ľ, đ?‘Ą = đ?œƒđ?‘? đ?‘Ľ, đ?‘Ą , đ?‘Ľ, đ?‘Ą ∈ Γđ?‘? , đ?‘˘ 0, đ?‘Ą = đ?œƒđ?‘™ đ?‘Ą đ?‘œđ?‘› Γđ?‘™ = 0, đ?‘Ą : 0 ≤ đ?‘Ą ≤ đ?‘‡ , đ?‘˘ 0, đ?‘Ą = đ?œƒđ?‘™ đ?‘Ą đ?‘œđ?‘› Γđ?‘™ = 0, đ?‘Ą : 0 ≤ đ?‘Ą ≤ đ?‘‡ ,

(1.1)

where, đ??żđ?œ€,đ?‘Ľ đ?‘˘ đ?‘Ľ, đ?‘Ą = −đ?œ€đ?‘˘đ?‘Ľđ?‘Ľ đ?‘Ľ, đ?‘Ą + đ?‘Ž đ?‘Ľ đ?‘˘ đ?‘Ľ, đ?‘Ą . Here Ί = 0, 1 , đ??ˇ = Ί Ă— 0, đ?‘‡ , Γ = Γđ?‘™ âˆŞ Γđ?‘? âˆŞ Γđ?‘&#x; . Γđ?‘? and Γđ?‘&#x; are the left and the right sides of the rectangular domain đ??ˇcorresponding to đ?‘Ľ = 0 and đ?‘Ľ = 1, respectively. Γđ?‘? = [0, 1] Ă— [−đ?œ?, 0]. Also, 0 < đ?œ€ ≪ 1 and đ?œ? > 0 are given constants. The functions đ?‘Ž(đ?‘Ľ), đ?‘?(đ?‘Ľ, đ?‘Ą), đ?‘“(đ?‘Ľ, đ?‘Ą) on đ??ˇand đ?œƒđ?‘? đ?‘Ľ, đ?‘Ą , đ?œƒđ?‘™ (đ?‘Ą), đ?œƒđ?‘&#x;(đ?‘Ą) onΓ, are sufficiently smooth, bounded functions that satisfy, đ?‘Ž đ?‘Ľ ≼ đ?›˝ ≼ 0, đ?‘?(đ?‘Ľ, đ?‘Ą) > 0 onđ??ˇ. The terminal time đ?‘‡is assumed to satisfy the condition đ?‘‡ = đ?‘˜đ?œ? for some positive integer k. The required compatibility conditions at the corner points and the delay terms are đ?œƒđ?‘? (0, 0) = đ?œƒđ?‘™ (0), đ?œƒđ?‘? (1, 0) = đ?œƒđ?‘&#x;(0),

đ?‘‘đ?œƒđ?‘™ 0 đ?œ• 2 đ?œƒđ?‘? 0, 0 −đ?œ€ + đ?‘Ž 0 đ?œƒđ?‘? 0, 0 đ?‘‘đ?‘Ą đ?œ•đ?‘Ľ 2 = −đ?‘? 0, 0 đ?œƒđ?‘? 0, −đ?œ? + đ?‘“ 0, 0 , đ?‘‘đ?œƒđ?‘&#x; 0 đ?œ• 2 đ?œƒđ?‘? 1, 0 −đ?œ€ + đ?‘Ž 1 đ?œƒđ?‘? 1, 0 đ?‘‘đ?‘Ą đ?œ•đ?‘Ľ 2 = −đ?‘? 1, 0 đ?œƒđ?‘? 1, −đ?œ? + đ?‘“ 1, 0 . Under the above assumptions and compatibility conditions, problem (1.1) admits a unique solution and the solution exhibits boundary layers along x=0, x=1(Ansari et. al.(2007); Kumar and Sekhara Rao (2010)). One can refer (Farrel et. al.(2000); Bansal et. al.(2015); Salama and Al-Amerya (2018); Govindarao and Mohapatra (2018)) reference therein for more details of singular perturbation. There are few articles dealing with the theory and the numerical methods for equation (1.1). Ansari et. al.(Ansari et. al. (2007)) solved the singularly perturbed delay parabolic reaction diffusion problem on piecewise uniform Shishkin mesh. Das and Natesan. (Das and Natesan (2018)) solved the delay parabolic convection diffusion problem. But most of the methods discussed above using finite difference schemes are of first order or second order accurate. So there is a need

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