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Hassanien, I. A.; Salama, A. A.; Hosham, H. A.

Fourth-order finite difference method for solving Burgers’ equation. (English) Zbl 1084.65078

Appl. Math. Comput. 170, No. 2, 781-800 (2005).
The main scope of the article is the elaboration of the two level three-point scheme by means of the method of unknown coefficients for an approximate solving of the nonlinear Burgers’ problem. The local error of this scheme is of second-order by the time-step \(\Delta t\) and of fourth-order by the variable space step \(h\). The present method can be written in matrix form \(AI^{m+1}_{i,j}= BU_{i,j-1}\), where \(A\) is a tridiagonal matrix.
The constructed algorithm is applied to the numerical solution of some concrete problems with different Reynolds numbers. The obtained numerical results are compared with the same results by other known methods. The comparison is made only by numerical results and the main properties of other methods are neglected.
Reviewer: Ivan Secrieru (Chişinău)

Cited in 1 Review
Cited in 59 Documents

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)

Keywords:

error bounds; von Neumann stability; convergence; numerical results; comparison
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\textit{I. A. Hassanien} et al., Appl. Math. Comput. 170, No. 2, 781--800 (2005; Zbl 1084.65078)
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References:

[1] Bateman, H., Some recent researches on the motion of fluids, Mon. Weather Rev., 43, 163-170 (1915)
[2] Burgers, J. M., Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion, Trans. Roy. Neth. Acad. Sci. Amsterdam, 17, 1-53 (1939) · Zbl 0061.45709
[3] Burgers, J. M., A mathematical model illustrating the theory of turbulence, (Adv. in Appl. Mech., vol. I (1948), Academic Press: Academic Press New York), 171-199
[4] Cole, J. D., On a quasi-linear parabolic equations occuring in aerodynamics, Quart. Appl. Math., 9, 225-236 (1951) · Zbl 0043.09902
[5] Hopf, E., The partial differential equation \(u_t+ uu_x = μu_{ xx } \), Commun. Pure Appl. Math., 3, 201-230 (1950) · Zbl 0039.10403
[6] Benton, E.; Platzman, G. W., A table of solutions of the one-dimensional Burgers equations, Quart. Appl. Math., 30, 195-212 (1972) · Zbl 0255.76059
[7] Fletcher, C. A., Burgers’ equation: a model for all reasons, (Noye, J., Numerical Solutions of Partial Differential Equations (1982), North-Holland: North-Holland Amsterdam), 139-225
[8] Ames, W. F., Nonlinear Partial Differential Equations in Engineering (1965), Academic Press: Academic Press New York · Zbl 0255.35001
[9] Dagˇ, İ.; Irk, D.; Saka, B., A numerical solution of the Burgers’ equation using cubic B-splines, Appl. Math. Comput., 163, 199-211 (2005) · Zbl 1060.65652
[10] Kutluay, S.; Esen, A., A linearized numerical scheme for Burgers-like equations, Appl. Math. Comput., 156, 295-305 (2004) · Zbl 1061.65081
[11] Kutluay, S.; Esen, A.; Dagˇ, İ., Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method, J. Comput. Appl. Math., 167, 21-33 (2004) · Zbl 1052.65094
[12] Mittal, R. C.; Singhal, P., Numerical solution of Burgers’ equation, Commun. Numer. Meth. Eng., 9, 397-406 (1993) · Zbl 0782.65147
[13] Mittal, R. C.; Singhal, P., Numerical solution of periodic Burgers’ equation, Indian J. Pure Appl. Math., 27, 689-700 (1996) · Zbl 0859.76053
[14] Öziş, T.; Özdeş, A., A direct variational methods applied to Burgers’ equation, J. Comput. Appl. Math., 71, 163-175 (1996) · Zbl 0856.65114
[15] Berger, A. E.; Solomon, J. M.; Ciment, M.; Leventhal, S. H.; Weinberg, B. C., Generalized OCI schemes for boundary layer problems, Math. Comput., 35, 695-731 (1980)
[16] Smith, G. D., Numerical Solution of Partial Differential Equations (1978), Oxford University Press: Oxford University Press Oxford · Zbl 0389.65040
[17] Fröberg, C., Introduction to Numerical Analysis (1969), Addison-Wesley: Addison-Wesley New York
[18] Usmani, R. A.; Marsden, M. J., Numerical solution of some ordinary differential equations occurring in plate deflection theory, J. Eng. Math., 9, 1-10 (1975) · Zbl 0311.65055
[19] Aksan, E. N.; Özdes, A., A numerical solution of Burgers’ equation, Appl. Math. Comput., 156, 395-402 (2004) · Zbl 1061.65085
[20] Caldwell, J.; Wanless, P.; Cook, A. E., Solution of Burgers’ equation for large Reynolds number using finite elements with moving nodes, Appl. Math. Model., 11, 211-214 (1987) · Zbl 0622.76063
[21] Hon, Y. C.; Mao, X. Z., An efficient numerical scheme for Burgers’ equation, Appl. Math. Comput., 95, 37-50 (1998) · Zbl 0943.65101
[22] Iskander, L.; Mohsen, A., Some numerical experiments on the splitting of Burgers’ equation, Numer. Meth. Partial Differen. Equat., 8, 267-276 (1992) · Zbl 0753.76117
[23] Kutluay, S.; Bahadir, A. R.; Özdeş, A., Numerical solution of one-dimensional Burgers equation: explicit and exact-explicit finite difference methods, J. Comput. Appl. Math., 103, 251-261 (1999) · Zbl 0942.65094
[24] Özis, T.; Aksan, E. N.; Özdes, A., A finite element approach for solution of Burgers’ equation, Appl. Math. Comput., 139, 417-428 (2003) · Zbl 1028.65106
[25] Ali, A. H.A.; Gardner, G. A.; Gardner, L. R.T., A collocation solution for Burgers’ equation using cubic B-spline finite elements, Comput. Meth. Appl. Mech. Eng., 100, 325-337 (1992) · Zbl 0762.65072
[26] Dogan, A., A Galerkin finite element approach to Burgers’ equation, Appl. Math. Comput., 157, 331-346 (2004) · Zbl 1054.65103
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.