Journal of Applied and Computational Mechanics

Uniformly Convergent Numerical Method for Two-parametric ‎Singularly Perturbed Parabolic Convection-diffusion Problems

Document Type : Research Paper

Authors

1 Department of Mathematics, Wollega University, Nekemte, 395, Ethiopia‎

2 Department of Mathematics, Jimma University, Jimma, 378, Ethiopia‎

Abstract

This paper deals with the numerical treatment of two-parametric singularly perturbed parabolic convection-diffusion problems. The scheme is developed through the Crank-Nicholson discretization method in the temporal dimension followed by fitting the B-spline collocation method in the spatial direction. The effect of the perturbation parameters on the solution profile of the problem is controlled by fitting a parameter. As a result, it has been observed that the method is a parameter-uniform convergent and its order of convergence is two. This is shown by the boundedness of the solution, its derivatives, and error estimation. The effectiveness of the proposed method is demonstrated by model numerical examples, and more accurate solutions are obtained as compared to previous findings available in the literature.

Keywords

Main Subjects

[1] Bobisud L., Second-order linear parabolic equations with a small parameter, Archive for Rational Mechanics and Analysis, 27(5), 1968, 385–397.
[2] Kadalbajoo M.K., Arora P., B-spline collocation method for the singular-perturbation problem using artificial viscosity, Computers and Mathematics with Applications, 57(4), 2009, 650-663.
[3] Kadalbajoo M.K., Awasthi A., Crank-Nicolson finite difference method based on a midpoint upwind scheme on a non-uniform mesh for time-dependent singularly perturbed convection-diffusion equations, International Journal of Computer Mathematics, 85(5), 2008, 771-790.
[4] Kadalbajoo M.K., Yadaw A.S., Parameter-uniform finite element method for two-parameter singularly perturbed parabolic reaction-diffusion problems, International Journal of Computational Methods, 9(4), 2012, 1250047-16.
[5] Jha A., Kadalbajoo M.K., A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems, International Journal of Computer Mathematics, 92(6), 2015, 1204–1221.
[6] Munyakazi J.B., A robust finite difference method for two-parameter parabolic convection-diffusion problems, Applied Mathematics and Information Sciences, 9(6), 2015, 2877–2883.
[7] Gupta V., Kadalbajoo M.K., A parameter-uniform higher order finite difference scheme for singularly perturbed time-dependent parabolic problem with two small parameters, International Journal of Computer Mathematics, 96(6), 2019, 474-499.
[8] Bullo T.A., Duressa G.F., Degla G.A., Higher Order Fitted Operator Finite Difference Method for Two-Parameter Parabolic Convection-Diffusion Problems, International Journal of Engineering & Applied Sciences, 11(4), 2019, 455-467.
[9] Bullo T.A., Duressa G.F., Degla G.A., Fitted Operator Average Finite Difference Method for Solving Singularly Perturbed Parabolic Convection-Diffusion Problems, International Journal of Engineering & Applied Sciences, 11(3), 2019, 414-427.
[10] Rao S.C.S., Kumar M., Exponential B-spline collocation method for self-adjoint singularly perturbed boundary value problems, Applied Numerical Mathematics, 58, 2008, 572–1581.
[11] Kuerten J.G.M, van der Geld C.W.M, Geurts B.J., Turbulence modification and heat transfer enhancement by inertial particles in turbulent channel flow, Physics of Fluids, 23, 2011, 12331–8.
[12] Ma H., Duan Z., Similarities of Flow and Heat Transfer around a Circular Cylinder, Symmetry, 12, 2020, 1–15.
[13] Van Dyke M., Perturbation Methods in Fluid Mechanics, Nature, 206, 1965, 226–227.
[14] Mo J., Ni M., Recent progress in study of singular perturbation problems, Journal of Shanghai University, 13(1), 2009, 1–5.
[15] Aminikhah H., Alavi J., Numerical Solution of Convection-Diffusion Equation Using Cubic B-Spline Quasi-Interpolation, Thai Journal of Mathematics, 14(3), 2016, 599–613.
[16] Mohanty R.K., Dahiya V., Khosla N., Spline in Compression Methods for Singularly Perturbed 1D Parabolic Equations with Singular Coefficients, Open Journal of Discrete Mathematics, 2, 2012, 70-77.
[17] Mamatha K., Phaneendra K., Solution of convection-diffusion problems using fourth order adaptive cubic spline method, Mathematics in Computer Science, 10(4), 2020, 817–832.
[18] Rashidinia J., Jamalzadeh S., Esfahani F., Numerical Solution of One-dimensional Telegraph Equation using Cubic B-spline Collocation Method, Journal of Interpolation and Approximation in Scientific Computing, 2014, 2014, 1-8.
[19] Gho J., Majid A.A., Ismail A.I., Cubic B-Spline Collocation Method for One-Dimensional Heat and Advection-Diffusion Equations, Journal of Applied Mathematics, 2011, 2011, 1-8.
[20] Shallal A.A., Ali K.K., Raslan K.R., Taqi A.H., Septic B-spline collocation method for numerical solution of the coupled Burgers, Arab Journal of Basic and Applied Sciences, 26(1), 2019, 331-341.
[21] Rohila R., Mittal R.C., Numerical study of reaction diffusion Fisher’s equation by fourth order cubic B-spline collocation method, Mathematical Sciences, 12(2), 2018, 79-89.
[22] Kadalbajoo M.K., Arora P., Yadaw A., Fitted Collocation Method for a Singularly Perturbed Time-Dependent Convection-Diffusion Problem, Proceedings of Dynamic Systems and Applications, 6, 2012, 197–204.
[23] Kadalbajoo M.K., Arora P., Gupta V., Collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers, Computers and Mathematics with Applications, 61(6), 2011, 1595–1607.
[24] Roos H.-G., Uzelac Z., The SDFEM for a convection-diffusion problem with two small parameters, Computers and Mathematics with Applications, 3(3), 2003, 443–458.
[25] Das P., Mehrmann V., Numerical solution of singularly perturbed convection-diffusion-reaction problems with two small parameters, BIT Numerical Mathematics, 56(1), 2016, 51–76,.
[26] Prenter P.M., Splines and variational methods, John Wiley & Sons, New York, 1989.
Journal of Applied and Computational Mechanics
Volume 7, Issue 2
April 2021
Pages 535-545