Richardson Extrapolation Technique for Singularly Perturbed Parabolic Convection-diffusion Problems with a Discontinuous Initial Condition

PDF Review History

Published: 2024-03-27

Page: 132-155

Desta Sodano Sheiso *

Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati - 781 039, India.

*Author to whom correspondence should be addressed.


This article presents the Richardson extrapolation techniques for solving singularly perturbed parabolic convection-diffusion problems with discontinuous initial conditions (DIC). The scheme uses- backward Euler for temporal derivatives on a uniform mesh and classical upwind finite difference method (FDM) for spatial derivatives on a piecewise-uniform (Shishkin) mesh. This scheme provides almost a first-order convergence solution in both space and time variables. The method employs an upwind finite difference operator on a piecewise- uniform mesh to approximate the gap between the analytic function and the parabolic issue solution. The numerical solution's accuracy is improved by using Richardson extrapolation techniques, which raises it from O(N-1 lnN + \(\Delta\) t) to O(N-2 ln2 N +\(\Delta\) t2) in the discrete maximum norm, where N is the number of spatial mesh intervals, and t is the size of the temporal step size. Parameter-uniform error estimates, stability results, and bounds for the truncation errors are all addressed. Finally, numerical experiments are presented to validate our theoretical results.

Keywords: Singularly perturbed parabolic problems, Upwind scheme, discontinuous initial condition, interior layer, Richardson extrapolation technique, Piecewise-uniform Shishkin mesh

How to Cite

Sheiso, D. S. (2024). Richardson Extrapolation Technique for Singularly Perturbed Parabolic Convection-diffusion Problems with a Discontinuous Initial Condition. Asian Journal of Pure and Applied Mathematics, 6(1), 132–155. Retrieved from


Download data is not yet available.


Bobisud L. Parabolic equations with a small parameter and discontinuous data. Journal of Mathematica Analysis and Applications. 1969;26:208-220.

Hemker P, Shishkin G. Discrete approximation of singularly perturbed parabolic pdes with

discontinuou initial condition. in Bail VI Proceedings. 1994;3-4.

O'Riordan E, Shishkin G. Singularly perturbed parabolic problems with non-smooth data. Journal

Computational and Applied Mathematics. 2004;166:233-245.

Sodano D. An upwind nite di erence method to singularly perturbed parabolic convection di usio problems with discontinuous initial conditions on a piecewise-uniform mesh. Mendeley Data. 2024;V2.

Gracia J, O'Riordan E. Numerical approximation of solution derivatives of singularly perturbed parabol problems of convection-di usion type. Mathematics of Computation. 2016;85:581-599.

Gracia JL, O'Riordan E. Parameter-uniform approximations for a singularly perturbed convectio di usio problem with a discontinuous initial condition. Applied Numerical Mathematics. 2021;162:106-123.

Clavero C, Jorge J, Lisbona F. A uniformly convergent scheme on a nonuniform mesh fo convection{di usion parabolic problems. Journal of Computational and Applied Mathematics 2003;154:415-429.

Gracia JL, O'Riordan E. Singularly perturbed reaction{di usion problems with discontinuities in th initia and/or the boundary data. Journal of Computational and Applied Mathematics. 2020;370:112638.

Gracia JL, O'Riordan E. Numerical approximations to a singularly perturbed convection-di usion proble with a discontinuous initial condition. Numerical Algorithms. 2021;88:1851-1873.

Farrell P, Hegarty A, Miller J, O'Riordan E, Shishkin G. Singularly perturbed convection{di usio problem with boundary and weak interior layers. Journal of Computational and Applied Mathematics. 2004;166:133 151.

Farrell P, Hegarty A, Miller JM, O'Riordan E, Shishkin GI. Robust computational techniques for boundar layers. CRC Press; 2000.

Miller J, O'Riordan E, Shishkin G, Kellogg RB. Fitted numerical methods for singular perturbatio problems. SIAM Review. 1997;39:535-537.

Roos HG, Stynes M, Tobiska L. Robust numerical methods for singularly perturbed di erentia equations Convection-di usion-reaction and ow problems. Springer. Science Business Media. 2008;24.

Miller J, O'Riordan E, Shishkin G, Shishkina L. Fitted mesh methods for problems with paraboli boundar layers. in Mathematical Proceedings of the Royal Irish Academy. JSTOR. 1998;173-190.

Clavero C, Gracia JL, Shishkin GI, Shishkina LP. An ecient numerical scheme for 1d paraboli singularly perturbed problems with an interior and boundary layers. Journal of Computational and Appli Mathematics. 2017;318:634-645.

Miller JJ, O'riordan E, Shishkin GI. Fitted numerical methods for singular perturbation problems: Erro estimates in the maximum norm for linear problems in one and two dimensions. World Scienti c; 1996.

Shishkin GI. Grid approximation of singularly perturbed parabolic convection-di usion equations with piecewise-smooth initial condition. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki. 2006;46:52- 76.

Shishkin GI. The Richardson scheme for the singularly perturbed parabolic reaction-di usion equatio in the case of a discontinuous initial condition. Computational Mathematics and Mathematical Physics 2009;49:1348-1368.

Gracia J, O'Riordan E. A singularly perturbed convection{di usion problem with a moving interio layer Int. J. Numer. Anal. Model. 2012;9:823-843.

O'Riordan E, Pickett M, Shishkin G. Parameter-uniform nite di erence schemes for singularly perturb parabolic di usion-convection-reaction problems. Mathematics of Computation. 2006;75:1135-1154.

Doolan EP, Miller JJ, Schilders WH. Uniform numerical methods for problems with initial and boundar layers. Boole Press; 1980.

Lozano JLG, Gracia CC. Richardson extrapolation on generalized shishkin meshes for singularl perturbe problems. in VIII Journ´ees Zaragoza-Pau de Math´ematiques Appliqu´ees et de Statistiques, Prensas d la Universidad de Zaragoza. 2003;169-178.

Natividad MC, Stynes M. Richardson extrapolation for a convection{di usion problem using a shishki mesh. Applied Numerical Mathematics. 2003;45:315-329.

Shishkin G. Robust novel high-order accurate numerical methods for singularly perturbe convectiond usion problems. Mathematical Modelling and Analysis. 2005;10:393-412.

Shishkin G, Shishkina L. The richardson extrapolation technique for quasilinear parabolic singularl perturbed convection-di usion equations. in Journal of Physics: Conference Series, IOP Publishing 2006;55:203.

Clavero C, Gracia J, Jorge J. High-order numerical methods for one-dimensional parabolic singularl perturbed problems with regular layers. Numerical Methods for Partial Di erential Equations: A International Journal. 2005;21:149-169.

Munyakazi JB, Patidar KC. On richardson extrapolation for tted operator nite di erence methods Applied Mathematics and Computation. 2008;201:465-480.

Linss. Layer-adapted meshes for reaction-convection-di usion problems. Springer; 2009.

Yadav NS, Mukherjee K. On -uniform higher order accuracy of new ecient numerical method and it extrapolation for singularly perturbed parabolic problems with boundary layer. International Journal o Applied and Computational Mathematics. 2021;7:72.

Dunne RK, O'Riordan E, Shishkin GI. A tted mesh method for a class of singularly perturbed paraboli problems with a boundary turning point. Computational Methods in Applied Mathematics. 2003;3:361-372.

Keller HB. Numerical methods for two-point boundary-value problems. Courier Dover Publications 2018.

Lin T. An upwind di erence scheme on a novel shishkin-type mesh for a linear convection{ di usio problem. Journal of Computational and Applied Mathematics. 1999;110:93-104.

Stynes M, Roos HG. The midpoint upwind scheme. Applied Numerical Mathematics. 1997;23:361-374.