A numerical method based on Crank-Nicolson scheme for Burgers’ equation. (English) Zbl 1112.65081

The authors present a solution based on Crank-Nicolson finite difference method for one-dimensional Burgers’ equation. The Burgers equation arises frequently in mathematical modeling of problems in fluid dynamics. The Hopf-Cole transformation is used to linearize the Burgers’ equation, the resulting heat equation is discretized by using the Crank-Nicolson finite difference scheme. This method is shown to be unconditionally stable and second order accurate in space and time. Numerical results obtained by the present method are compared with exact solution for different values of the Reynolds number.


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76M20 Finite difference methods applied to problems in fluid mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI


[1] Ames, W.F., Nonlinear partial differential equations in engineering, (1965), Academic press New York · Zbl 0255.35001
[2] Bateman, H., Some recent researches in motion of fluids, Mon. weather rev., 43, 163-170, (1915)
[3] Burgers, J.M., A mathematical model illustrating the theory of turbulence, Advances in applied mechanics, 1, (1948), Academic Press New York, pp. 171-199
[4] Benton, E.R.; Platzman, G.W., A table of solutions of the one-dimensional Burgers equation, Quart. appl. math., 30, 195-212, (1972) · Zbl 0255.76059
[5] Cole, J.D., On a quasilinear parabolic equation occurring in aerodynamics, Quart. appl. math., 9, 225-236, (1951) · Zbl 0043.09902
[6] Evans, D.J.; Abdullah, A.R., The group explicit method for the solution of Burgers equation, Computing, 32, 239-253, (1984) · Zbl 0523.65071
[7] Hopf, E., The partial differential equation ut+uux=νuxx, Commun. pure appl. math., 3, 201-230, (1950)
[8] Lagerstrom, P.A.; Cole, J.D.; Trilling, L., Problems in the theory of viscous compressible fluids, Calif. inst. technol., (1949)
[9] Mittal, R.C.; Singhal, Poonam, Numerical solution of Burgers equation, Commun. numer. methods eng., 9, 397-406, (1993) · Zbl 0782.65147
[10] Mittal, R.C.; Singhal, Poonam, Numerical solution of periodic Burgers equation, Indian J. pure appl. math., 27, 689-700, (1996) · Zbl 0859.76053
[11] Kutluay, S.; Bahadir, A.R.; Ozdes, A., Numerical solution of one-dimensional Burgers equation: explicit and exact explicit methods, J. comp. appl. math., 103, 251-261, (1998) · Zbl 0942.65094
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.