Finite difference discretization of the Benjamin-Bona-Mahony-Burgers equation

Numerical Methods for Partial Differential Equations

Volumn 24, Issue 1, 2008, Pages 239-248

  Omrani, Khaled ^a   Ayadi, Mekki ^a,b  

^a INSTITUT SUPÉRIEUR DES SCIENCES APPLIQUÉES ET DE TECHNOLOGIE DE SOUSSE   (Tunisia)

^b UNIVERSITY OF MONASTIR   (Tunisia)

Author keywords

BBMB equation; Difference scheme; L convergence; Stability

Indexed keywords

EID: 38049113156     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20256     Document Type: Article

References (30)

1. 1 T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos Trans Roy Soc Lond A 272 ( 1972), 47–78.
2. 2 J. L. Bona and R. smith, The initial‐value problem for the Korteweg‐de Vries equation, Philos Trans Roy Soc Lond A 278 ( 1975), 555–601.
3. 3 C. Yong, L. Biao, and Z. Hongqing, Exact solutions of two nonlinear wave equations with simulation terms of any order, Comm Nonlinear Sci Numer Simul 10 ( 2005), 133–138.
4. 4 L. A. Medeiros and M. Milla Miranda, Weak solutions for a nonlinear dispersive equation, J Math Anal Appl 59 ( 1977), 432–441.
5. 5 L. A. Medeiros and G. Perla Menzela, Existence and uniqueness for periodic solutions of the Benjamin‐Bona‐Mahony equation, SIAM J Math Anal 8 ( 1977), 792–799.
6. 6 M. Mei, Large‐time behavior of solution for generalized Benjamin‐Bona‐Mahony‐Burgers Equations, Nonlinear Anal 33 ( 1998), 699–714.
7. 7 D. H. Peregrine, Calculations of the development of an undular bore, J Fluid Mech 25 ( 1996), 321–326.
8. 8 H. Zhang, G. M. Wei, and Y. T. Gao, On the general form of the Benjamin‐Bona‐Mahony equation in fluid mechanics, Czech J Phys 52 ( 2002), 344–373.
9. 9 M. A. Raupp, Galerkin methods applied to the Benjamin‐Bona‐Mahony equation, Bol Soc Brazil Mat 6 ( 1975), 65–77.
10. 10 L. Wahlbin, Error estimates for a Galerkin method for a class of model equations for long waves, Numer Math 23 ( 1975), 289–303.
11. 11 R. E. Ewing, Time‐stepping Galerkin methods for nonlinear Sobolev partial differential equation, SIAM J Numer Anal 15 ( 1978), 1125–1150.
12. 12 D. N. Arnold, J. Douglas Jr., and V. Thomée, Superconvergence of finite element approximation to the solution of a Sobolev equation in a single space variable, Math Comput 27 ( 1981), 737–743.
13. 13 S. A. V. Manickam, A. K. Pani, and S. K. Chang, A second‐order splitting combined with orthogonal cubic spline collocation method for the roseneau equation, Numer Methods Partial Diff Equat 14 ( 1998), 695–716.
14. 14 K. Omrani, The convergence of the fully discrete Galerkin approximations for the Benjamin‐Bona‐Mahony (BBM) equation, Appl Math Comput 180 ( 2006), 614–621.
15. 15 D. Kaya and I. E. Inan, Exact and numerical traveling wave solutions for nonlinear coupled equations using symbolic computation, Appl Math Comput 151 ( 2004), 775–787.
16. 16 D. Kaya, A numerical simulation of solitary‐wave solutions of the generalized regularized long‐wave equation, Appl Math Comput 149 ( 2004), 833–841.
17. 17 R. D. Richtmyer and K. W. Morton, Difference methods for initial‐value problems, 2nd ed., Wiley, New York, 1967.
18. 18 B. Fornberg, On the instabilty of leap‐flog and Crank‐Nicolson approximations of a nonlinear partial differential equation, Math Comput 27 ( 1973), 45–57.
19. 19 G. D. Akrivis, Finite difference discretization of the Kuramoto‐Sivashinsky equation, Numer Math 63 ( 1992), 1–11.
20. 20 R. Kannan and S. K. Chung, Finite differnce approximate solutions for the two‐dimensional Burger's system, Comput Math Appl 44 ( 2002), 193–200.
21. 21 T. F. Chan and L. Shen, Stability analysis of difference schemes for variable coefficient Schrödinger type equations, SIAM J Numer Anal 24 ( 1981), 336–349.
22. 22 G. Ben‐Yu, P. J. Pascual, M. J. Rodriguez, and L. Vázquez, Numerical solution of the Sine‐Gordon equation, Appl Math Comput 18 ( 1986), 1–14.
23. 23 L. Zhang and Q. Chang, A conservative numerical scheme for a class of nonlinear Schrödinger with wave operator, Appl Math Comput 145 ( 2003), 603–612.
24. 24 Y. Zhou, Application of discrete functional analysis to the finite difference methods, International Academic Publishers, Beijing, 1990.
25. 25 N. Khiari, T. Achouri, M. L. Ben Mohamed, and K. Omrani, Finite difference approximate solutions for the Cahn‐Hilliard equation, Numer Methods Partial Differ Equat 23 ( 2007), 437–455.
26. 26 K. Omrani, On the numerical Approach of the enthalpy method for the Stefan problem, Numer Methods Partial Differ Equat 20 ( 2004), 527–551.
27. 27 K. Omrani, A second order splitting method for a finite difference scheme for the Sivashinsky equation, Appl Math Lett 16 ( 2003), 441–445.
28. 28 K. Omrani, Convergence of Galerkin approximations for the Kuramoto‐Tsuzuki equation, Numer Method Partial Differ Equat 21 ( 2005), 961–975.
29. 29 K. Omrani, A second‐order accurate difference scheme on nonuniform meshes for nonlinear heat‐conduction equation, Far East J Appl Math 20 ( 2005), 355–365.
30. 30 K. Omrani, Numerical methods and error analysis for the nonlinear Sivashinsky equation, Appl Math Comput ( 2007), in press.