Sixth-order compact finite difference method for solving KDV-Burger equation in the application of wave propagations

Document Type : Research Article


1 Ambo University, College of Natural and Computational Sciences, Department of Math-ematics, Ambo, Ethiopia.

2 Jimma University College of Natural Sciences Department of Mathematics, Jimma, Ethiopia.


Sixth-order compact finite difference method is presented for solving the one-dimensional KdV-Burger equation. First, the given solution domain is discretized using a uniform discretization grid point in a spatial direction. Then, using the Taylor series expansion, we obtain a higher-order finite difference discretization of the KdV-Burger equation involving spatial variables and produce a system of nonlinear ordinary differential equa-tions. Then, the obtained system of a differential equation is solved by using the fourth-order Runge–Kutta method. To validate the applicability of proposed techniques, four model examples are considered. The stability and convergent analysis of the present method is worked by using von Neumann stability analysis techniques by supporting the theoretical and mathematical statements in order to verify the accuracy of the present solution. The quality of the attending method has been shown in the sense of root mean square error L2 and point-wise maximum absolute error L∞. This is used to show, how the present method approximates the exact solution very well and how it is quite efficient and practically well suited for solving the KdV-Burger equation. Numerical results of considered examples are presented in terms of L2 and L∞ in tables. The graph of obtained present numerical and its exact solution are also presented in this paper. The present approximate numeric solvent in the table and graph shows that the numerical solutions are in good agreement with the exact solution of the given model problem. Hence the technique is reliable and capable for solving the one-dimensional KdV-Burger equation.


Main Subjects

1.Ahmad, I., Ahmad, H., Inc, M., Rezazadeh, H., Akbar, M.A., Khater, M.M., Akinyemi, L. and Jhangeer, A. Solution of fractional-order Korteweg-de Vries and Burgers’ equations utilizing local meshless method, J. Ocean Eng. Sci. (2021).
2. Ahmad, H., Khan, T.A., Stanimirovic, P.S. and Ahmad, I. Modified vari-ational iteration technique for the numerical solution of fifth order KdV-3. Aliyi, K. and Muleta, H. Numerical method of the line for solving onedimensional initial-boundary singularly perturbed Burger equation, Indian J. Adv. Math. 1(2) (2021), 4–14.
4. Aliyi, K., Shiferaw, A. and Muleta, H. Radial basis functions based differ-ential quadrature method for one dimensional heat equation, American Journal of Mathematical and Computer Modelling, 6 (2021), 35–42.
5. Benney, D.J. Long waves on liquid films, J. Math. Phys. 45 (1966), 150–155.
6. Chen, B. and Xie, Y.C. Exact solutions for wick-type stochastic coupled Kadomtsev- Petviashili equations, J. Phys. A, 38 (2005), 815–822.
7. Chen, B. and Xie, Y.C. Exact solutions for generalized stochastic Wick-type KdV-mKdV equations , Chaos Solitons Fractals, 23 (2005), 281–287.
8. Chen, B. and Xie, Y.C. Periodic-like solutions of variable coefficient and Wick type stochastic NLS equations, J. Comput. Appl. Math. 203 (2007), 249–263.
9. Feng, Z. and Qing-guo, M. Burgers-Korteweg-de Vries equation and its travelling solitary waves , Sci. China Ser. A, 50 (3) (2007), 412-422.
10. Gao, G. A theory of interaction between dissipation and dispersion of turbulence , Sci. Sinica Ser. A, 28 (1985), 616–627.
11. Ghany, H.A., and Fathallah, A. Exact solutions for KDV-Burger equa-tions with an application of white-noise analysis , Int. J. Pure Appl. Math. 78(1) (2012), 17–27.
12. Gowrisankar, S. and Natesan, S. Uniformly convergent numerical method for singularly perturbed parabolic initial-boundary-value problems with equidistributed grids, Int. J. Comput. Math. 91 (2014), 553–577.
13. Grad, H. and Hu, P.N. Unified shock profile in a plasma, Phys Fluids, 10 (1967), 2596–2602.
14. Gurlu, Y.U. and Kaya, D. Analytic method for solitary solutions of some partial differential equations, Phys. Lett. A, 370 (2007), 251–259.
15. Hixon, R. and Turkel, E. Compact implicit MacCormack-type schemes with high accuracy, J. Comput. Phys. 158 (2000), 51–70.
16. Hu, P.N. Collisional theory of shock and nonlinear waves in a plasma. ,Phys. Fluids. 15 (1972), 854–864.
17. Johnson, R.S. A nonlinear equation incorporating damping and dispersion , J. Fluid Mech. 42 (1970), 49–60.
18. Johnson, R.S. Application of He’s homotopy perturbation method to non-linear integro-differential equations, Appl. Math. Comput. 188(1) (2007), 538–548.
19. Johnson, R.S. A modern introduction to the mathematical theory of water waves, Cambridge, Cambridge University Press, 1997.
20. Kashchenko, S.A. Normal form for the KdV-Burgers equation, Dokl. Math. 93 (2016), 331–333.
21. Kaya, D. An application of the decomposition method for the two-dimensional KdV-Burgers equation , Comput. Math. Appl. 48 (2004), 1659–1665.
22. Koroche, A. K. Numerical solution for one dimensional linear types of parabolic partial differential equation and application to heat equation, Mathematics and Computer Science, 5 (2020), 76–85.
23. Korteweg, D.J. and de Vries, G. On the change of form of long waves advancing in a Rectangular canal and on a new type of long stationary waves, Lond. Edinb. Dublin philos. mag. j. sci. 39 (1895), 22–43.
24. Kudryashov, N.A. On ”new travelling wave solutions” of the KdV and the KdV-Burgers equations, Commun. Nonlinear Sci. Numer. Simul. 14(5) (2009), 1891–1900.
25. Kutluay, S., Esen, A. and Dag, I. Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method, J. Comput. Appl. Math. 167 (2004), 21–33.
26. Li, J. and Visbal, M.R. High-order compact schemes for nonlinear dis-persive waves, J. Sci. Comput., 26 ( 2006), 1–23.
27. Navon, I.M. and Riphagen, H.A. An implicit compact fourth order algo-rithm for solving the shallow-water equations in conservation-law form , Mon. Weather Rev. 107 (1979), 1107–1127.
28. Navon, I.M. and Riphagen, H.A. SHALL4 An implicit compact fourth-order Fortran program for solving the shallow-water equations in conservation-law form , Comput. Geosci. 12 (1986), 129–150.
29. Shang, J.S. High-order compact-difference schemes for time-dependent Maxwell equations, J. Comp. Phys. 153 (1999), 312–333.
30. Shi, Y.F. , Xu, B. and Guo, Y. Numerical solution of Korteweg-de Vries-Burgers equation by the compact-type CIP method, Adv. Differ. Equ. (2015), 1–9.
31. Spotz, W.F. and Carey, G.F. Extension of high order compact schemes to time dependent problems , Numer. Methods Partial Differential Equations 17 (2001), 657–672.
32. Visbal, M.R. and Gaitonde, D.V. Very high-order spatially implicit schemes for computational acoustics on curvilinear meshes, J. Comput. Acoust. 9 (2001), 1259–1286.
33. Wadati, M. Deformation of solitons in random media, J. Phys. Soc. Japan 59 59 (1990), 4201–4203.
34. Wadati, M. and Akutsu, Y. Stochastic Korteweg de Vries equation, J. Phys. Soc. Japan, 53 (1984), 3342–3350.
35. Wazzan, L. A modified tanh-coth method for solving the KdV and the KdV–Burgers equations , Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 443–450.
36. van Wijngaarden, L. On the motion of gas bubbles in a perfect fluid , Arch. Mech. (Arch. Mech. Stos.) 34(3) (1982), 343–349 (1983).
37. Xiang, T. A summary of the Korteweg-de Vries Equation, (2015).
38. Xie, Y.C. Exact solutions for stochastic KdV equations, Phys. Lett. A, 310 (2003), 161–167.