Robust numerical schemes for time delayed singularly perturbed parabolic problems with discontinuous convection and source terms
Abstract
This article deals with two different numerical approaches for solving singularly perturbed parabolic problems with time delay and interior layers. In both approaches, the implicit Euler scheme is used for the time scale. In the first approach, the upwind scheme is used to deal with the spatial derivatives whereas in the second approach a hybrid scheme is used, comprising the midpoint upwind scheme and the central difference scheme at appropriate domains. Both schemes are applied on two different layer resolving meshes, namely a Shishkin mesh and a Bakhvalov–Shishkin mesh. Stability and error analysis are provided for both schemes. The comparison is made in terms of the maximum absolute errors, rates of convergence, and the computational time required. Numerical outputs are presented in the form of tables and graphs to illustrate the theoretical findings.
References
[1]
Kyrycho YN and Hogan SJ On the use of delay equations in engineering applications J. Vib. Control 2010 16 7–8 943-960
[2]
Rihan FA Delay Differential Equations and Applications to Biology 2021 Singapore Springer
[3]
Ansari AR, Bakr SA, and Shishkin GI A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations J. Comput. Appl. Math. 2007 205 1 552-566
[4]
Das A and Natesan S Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection-diffusion problems on Shishkin mesh Appl. Math. Comput. 2015 271 168-186
[5]
Govindarao L, Mohapatra J, and Das A A fourth-order numerical scheme for singularly perturbed delay parabolic problem arising in population dynamics J. Appl. Math. Comput. 2020 63 1 171-195
[6]
Kaushik A and Sharma M A robust numerical approach for singularly perturbed time delayed parabolic partial differential equations Comput. Math. Model. 2012 23 1 96-106
[7]
Govindarao L, Sahu SR, and Mohapatra J Uniformly convergent numerical method for singularly perturbed time delay parabolic problem with two small parameters Iran. J. Sci. Technol. Trans. A Sci. 2019 43 2373-2383
[8]
Sahu S and Mohapatra J Numerical investigation of time delay parabolic differential equation involving two small parameters Eng. Comput. 2021 38 6 2882-2899
[9]
Priyadarshana S, Mohapatra J, and Pattanaik SR Parameter uniform optimal order numerical approximations for time-delayed parabolic convection diffusion problems involving two small parameters Comput. Appl. Math. 2022
[10]
Priyadarshana S, Mohapatra J, and Govindrao L An efficient uniformly convergent numerical scheme for singularly perturbed semilinear parabolic problems with large delay in time J. Appl. Math. Comput. 2022 68 2617-2639
[11]
Priyadarshana S and Mohapatra J Weighted variable based numerical scheme for time-lagged semilinear parabolic problems including small parameter J. Appl. Math. Comput. 2023 69 2439-2463
[12]
Priyadarshana S, Mohapatra J, and Pattanaik SR A second order fractional step hybrid numerical algorithm for time delayed singularly perturbed 2D convection-diffusion problems Appl. Numer. Math. 2023 189 107-129
[13]
Mohapatra J, Priyadarshana S, and Reddy NR Uniformly convergent computational method for singularly perturbed time delayed parabolic differential-difference equations Eng. Comput. 2023 40 3 694-717
[14]
Priyadarshana S and Mohapatra J An efficient numerical approximation for mixed singularly perturbed parabolic problems involving large time-lag Indian J. Pure Appl. Math. 2023
[15]
Mukherjee K and Natesan S -uniform error estimate of hybrid numerical scheme for singularly perturbed parabolic problems with interior layers Numer. Algorithms 2011 58 1 103-141
[16]
Farrell PA, Hegarty AF, Miller JJH, O’Riordan E, and Shishkin GI Global maximum norm parameter-uniform numerical method for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient Math. Comput. Model. 2004 40 11–12 1375-1392
[17]
Farrell PA, Hegarty AF, Miller JJH, O’Riordan E, and Shishkin GI Singularly perturbed convection diffusion problems with boundary and weak interior layers J. Comput. Appl. Math. 2004 166 1 133-151
[18]
Shanthi V, Ramanujam N, and Natesan S Fitted mesh method for singularly perturbed reaction convection-diffusion problems with boundary and interior layers J. Appl. Math. Comput. 2006 22 1–2 49-65
[19]
O’Riordan E and Shishkin GI Singularly perturbed parabolic problems with non-smooth data J. Comput. Appl. Math. 2004 166 233-245
[20]
Shishkin GI A difference scheme for a singularly perturbed parabolic equation with discontinuous coefficients and concentrated factors USSR Comput. Math. Math. Phys. 1989 29 5 9-15
[21]
Mukherjee K and Natesan S Optimal error estimate of upwind scheme on Shishkin-type meshes for singularly perturbed parabolic problems with discontinuous convection coefficients BIT Numer. Math. 2011 51 289-315
[22]
Yadav NS and Mukherjee K Uniformly convergent new hybrid numerical method for singularly perturbed parabolic problems with interior layers Int. J. Appl. Comput. Math. 2020
[23]
Ladyženskaja OA, Solonnikov VA, and Ural’ceva NN Linear and Quasi-Linear Equations of Parabolic Type 1968 Providence American Mathematical Society
[24]
Roos HG, Stynes M, and Tobiska L Robust Numerical Methods for Singularly Perturbed Differential Equations: Convection-Diffusion-Reaction and Flow Problems 2008 Heidelberg Springer
[25]
Pao CV Nonlinear Parabolic and Elliptic Equations 1992 New York Plenum Press
[26]
Kellogg RB and Tsan A Analysis of some differences approximations for a singular perturbation problem without turning point Math. Comput. 1978 32 144 1025-1039
Recommendations
ε-Uniform error estimate of hybrid numerical scheme for singularly perturbed parabolic problems with nterior layers
This article is devoted to the study of a hybrid numerical scheme for a class of singularly perturbed parabolic convection-diffusion problems with discontinuous convection coefficients. In general, the solutions of this class of problems possess strong ...
Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments
In this work, a parameter uniform numerical method is developed to find the approximate solution of time-dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments in the space variable. The earlier work on such ...
Uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection-diffusion problems on Shishkin mesh
This article studies the numerical solution of singularly perturbed delay parabolic convection-diffusion initial-boundary-value problems. Since the solution of these problems exhibit regular boundary layers in the spatial variable, we use the piecewise-...
Comments
Please enable JavaScript to view thecomments powered by Disqus.Information & Contributors
Information
Published In
© The Author(s) 2023.
Publisher
Springer-Verlag
Berlin, Heidelberg
Publication History
Published: 29 November 2023
Accepted: 21 October 2023
Revision received: 08 July 2023
Received: 04 November 2022
Author Tags
Author Tags
Qualifiers
- Research-article
Funding Sources
- Universidad de Salamanca
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 28 Nov 2024
Other Metrics
Citations
View Options
View options
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in