A second order fitted operator finite difference scheme for a singularly perturbed degenerate parabolic problem

Document Type : Research Paper

Authors

Pure and Applied Analytics Focus Area, North West University, Mafikeng Campus, Private Bag X2046, Mmabatho, 2735, South Africa

Abstract

A second order finite difference scheme is constructed to solve a singularly perturbed degenerate parabolic convection diffusion problem via Rothe's method. The solution of the problem exhibits a boundary layer on the left side of the spatial domain.  By means of the Crank Nicolson finite difference scheme, the time derivative is discretised to obtain a set of semi-discrete boundary value problems. Using a fitted operator finite difference scheme based on the midpoint downwind scheme,  the system of boundary value problems are discretized and analysed for convergence. Second order accuracy is established for each discretisation process. Numerical simulations are carried out to validate the theoretical error estimate.

Keywords

[1] C. Clavero, J. L. Gracia, G. I. Shishkin and L. P. Shishkina, Convergent ε-uniformly for parabolic singularly
perturbed problems with a degenerating convective term and a discontinuous source, Math. Model. Anal. 20(5)
(2015) 641–657.
[2] R. K. Dunne, E. O’Riordan and G. I Shishkin, A fitted mesh method for a class of singularly perturbed parabolic
problems with a boundary turning point, Comput. Meth. Appl. Math. 3(3) (2003) 361–372.
[3] V. Gupta, S.K. Sahoo and R.K. Dubey, Parameter-uniform fitted mesh higher order finite difference scheme for
singularly perturbed problem with an interior turning point, arXiv preprint arXiv:1909.07128.
[4] M.K. Kadalbajoo and K.C. Patidar, A survey of numerical techniques for solving singularly perturbed ordinary
differential equations, Appl. Math. Comput. 130(2) (2002) 457–510.
[5] M.K. Kadalbajoo and K.C. Patidar, Singularly perturbed problems in partial differential equations, Appl. Math.
Comput. 134(2) (2003) 371–429.[6] M.K. Kadalbajoo and V. Gupta, A brief survey on numerical methods for solving singularly perturbed problems,
Appl. Math. Comput. 217 (2010) 3641–3716.
[7] M.M. Khalsaraei, Nonstandard explicit third-order Runge-Kutta method with positivity property, Int. J. Nonlinear
Anal. Appl. 8(2) (2017) 37–46.
[8] J.P. Kauthen and V. Gupta, A survey of singularly perturbed Volterra equations, Appl. Numerical Math. 24(23)
(1997) 95–114.
[9] R.B. Kellog and A. Tsan, Analysis of some difference approximations for a singular perturbation problem without
turning points, Math. Comput. 31(144) (1978) 1025–1039.
[10] A. Majumdar and S. Natesan, Second-order uniformly convergent Richardson extrapolation method for singularly
perturbed degenerate parabolic PDEs, Int. J. Appl. Comput. Math. 3(1) (2017) 31–53.
[11] A. Majumdar and S. Natesan, Alternating direction numerical scheme for singularly perturbed 2D degenerate
parabolic convection-diffusion problems, Appl. Math. Comput. 313 (2017) 453–473.
[12] A. Majumdar and S. Natesan, An ε-uniform hybrid numerical scheme for a singularly perturbed degenerate
parabolic convection-diffusion problem, Int. J. Comput. Math. 96(7) (2019) 1313–1334.
[13] N.A. Mbroh, S.C. Oukouomi Noutchie and R.Y. Mpika Massoukou, A uniformly convergent finite difference
scheme for Robin type singularly perturbed parabolic convection diffusion problem, Math. Comput. Simul. 174
(2020) 218–232.
[14] J.J.H. Miller, E. O’Riordan and G. I. Shishkin, Fitted Numerical Methods for Singularly Perturbed Problems:
Error Estimates in The Maximum Norm for Linear Problems in One and Two Dimension, World Scientific
Publications, Singapore, 2012.
[15] M.J. Ng-Stynes, E. O’Riordan and M. Stynes, Numerical methods for time-dependent convection-diffusion equations, J. Comput. Appl. Math. 21(3) (1988) 289–310.
[16] K. C. Patidar, High order fitted operator numerical method for self-adjoint singular perturbation problems, Appl.
Math. Comput., 171(1) (2005) 547–566.
[17] P. Rai and S. Yadav, Robust numerical schemes for singularly perturbed delay parabolic convection-diffusion
problems with degenerate coefficient, Int. J. Comput. Math. 98(1) (2020) 195–221.
[18] H. Roos, M. Stynes and L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations:
Convection-Diffusion-Reaction and Flow Problems, Springer Science & Business Media, 2008.
[19] M. Viscor and M. Stynes, A robust finite difference method for a singularly perturbed degenerate parabolic problem
Part I,Int. J. Numerical Anal. Model. 7(3) (2010).
[20] R. Vulanovi´c and P. A. Farrell, Continuous and numerical analysis of a multiple boundary turning point problem,
SIAM J. Numerical Anal. 30(5) (1993) 1400–1418.
Volume 12, Special Issue
December 2021
Pages 677-687
  • Receive Date: 23 July 2020
  • Revise Date: 25 September 2020
  • Accept Date: 19 October 2020