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Analysis of a higher order uniformly convergent method for singularly perturbed parabolic delay problems

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  • Sharma, Amit
  • Rai, Pratima

Abstract

We develop and analyze a higher-order uniformly convergent method for time dependent parabolic singularly perturbed convection-diffusion (C-D) problems with space dependent delay. Due to the presence of delay, there occurs an interior layer along with a boundary layer in the solution of the considered problem. A Bakhvalov-Shishkin mesh in space direction and a uniform mesh in time direction is constructed to discretize the domain. Numerical approximation is composed of the classical upwind difference scheme for space variable and the implicit-Euler scheme for time variable. The proposed scheme is proved to have ε-uniform convergence of O(K−1+Δt), where K and Δt denote the number of mesh- intervals in space direction and the step size in time direction, respectively. We further apply the Richardson extrapolation and establish that the resulting scheme has ε-uniform convergence of O(K−2+(Δt)2). Numerical results on two test examples are provided to demonstrate the effectiveness of the scheme.

Suggested Citation

  • Sharma, Amit & Rai, Pratima, 2023. "Analysis of a higher order uniformly convergent method for singularly perturbed parabolic delay problems," Applied Mathematics and Computation, Elsevier, vol. 448(C).
    • Handle: RePEc:eee:apmaco:v:448:y:2023:i:c:s0096300323000759
    DOI: 10.1016/j.amc.2023.127906
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    References listed on IDEAS

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    1. M.K. Kadalbajoo & K.K. Sharma, 2002. "Numerical Analysis of Boundary-Value Problems for Singularly-Perturbed Differential-Difference Equations with Small Shifts of Mixed Type," Journal of Optimization Theory and Applications, Springer, vol. 115(1), pages 145-163, October.
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