International Journal of Nonlinear Analysis and Applications

A second order fitted operator finite difference scheme for a modified Burgers equation

Document Type : Research Paper

Authors

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

2 Pure and Applied Analytics Focus Area, School of Mathematical and Statistical Sciences, North West University, Mafikeng Campus, Private Bag X2046, Mmabatho, 2735, South Africa

Abstract

In this paper,  a  one-dimensional modified Burgers'  equation is considered for different  Reynolds numbers. For very high Reynolds numbers,  the solution possesses a multiscale character in some part of the independent domain and thus can be classified as a  singularly perturbed problem. A numerical scheme that uses a fitted operator finite difference scheme to solve the spatial derivatives and the implicit Euler scheme for the time derivative is proposed to solve the modified  Burgers'  equation via Rothe's method. It is important to note that the proposed fitted operator finite difference scheme is based on the midpoint upwind scheme. The stability of the scheme is established and the error associated with each discretisation is estimated. Numerical simulations are carried out to validate the theoretical findings.

Keywords

[1] S. Bendaasa, N. Alaab, Periodic wave shock solutions of Burgers equations, A new approach, Int. J. Nonlinear
Anal. Appl. 10(1) (2019) 119–129.
[2] A. G. Bratsos, A fourth-order numerical scheme for solving the modified Burgers equation, Comput. Math. Appl.
60(5) (2010) 1393–1400.
[3] J.D. Cole, On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9(3) (1951)
225–236.
[4] F. Erdogan and M.G. Sakar, A quasilinearization technique for the solution of singularly perturbed delay differential equation, Math. Natural Sci. 2(1) (2018) 1–7.
[5] S.L. Harris, Sonic shocks governed by the modified Burgers’ equation, Euro. J. Appl. Math. 7(2) (1996) 201–222.
[6] M.K. Kadalbajoo and A. Awasthi, The partial differential equation ut + uux = uxx, Commun. Pure Appl. Math.
3(3) (1950) 201–230.
[7] M.K. Kadalbajoo and A. Awasthi, Uniformly convergent numerical method for solving modified Burgers’ equations
on a non-uniform mesh, J. Numerical Math. 16(3) (2008) 217–235.
[8] V. Gupta and M. Kadalbajoo, Numerical approximation of modified Burger’s equation via hybrid finite difference
scheme on layer-adaptive mesh, Neural Paral. Sci. Comput. 18 (2010) 167–194.
[9] 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.
[10] R.E. Mickens, Non-Standard Finite Difference Models of Differential Equations, World Scientific, Singapore,
1994.
[11] G.A. Nariboli and W.C. Lin, A new type of Burgers’ equation, Z. Angew. Math. Mech. 53(8) (1973) 505–510.
[12] L. Liu, G. Long and Z. Cen, A robust adaptive grid method for a nonlinear singularly perturbed differential
equation with integral boundary condition, Numerical Algor. 83(2) (2020) 719–739.
[13] K.C. Patidar, High order fitted operator numerical method for self-adjoint singular perturbation problems, Appl.
Math. Comput. 171(1) (2005) 547–566.
[14] A.S.V. Ravi Kanth and P.M.M. Kumar, Numerical technique for solving nonlinear singularly perturbed delay
differential equations, Math. Model. Anal. 23(1) (2018) 64–78.
[15] 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.
[16] Y. Ucar, N. M. Yagmurlu and O. Tasbozan, Numerical solutions of the modified Burgers’ equation by finite
difference methods, J. Appl. Math. Stat. Inf. 13(1) (2017) 19–30.
[17] H. Zeidabadi, R. Pourgholi and S. H. Tabasi, Solving a nonlinear inverse system of Burgers equations, Int. J.
Nonlinear Anal. Appl. 10(1) (2019) 35–54.
International Journal of Nonlinear Analysis and Applications
Volume 12, Special Issue
December 2021
Pages 689-698
  • Receive Date: 09 September 2020
  • Revise Date: 24 November 2020
  • Accept Date: 11 December 2020