Methods for the numerical solution of the Benjamin-Bona-Mahony-Burgers equation

Numerical Methods for Partial Differential Equations

Volumn 24, Issue 6, 2008, Pages 1501-1516

  Kadri, Tlili ^a   Khiari, Noomen ^c   Abidi, Faycal ^b   Omrani, Khaled ^c  

^a FACULTÉ DES SCIENCES DE TUNIS   (Tunisia)

^b UNIVERSITY OF CARTHAGE   (Tunisia)

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

Author keywords

Backward Euler; BBMB equation; Error estimates; Existence; Finite element method; Fully discrete schemes; Semidiscrete schemes; Two step backward method; Uniqueness; Weak formulation

Indexed keywords

EID: 56249134570     PISSN: 0749159X     EISSN: 10982426     Source Type: Journal    
DOI: 10.1002/num.20330     Document Type: Article

References (29)

1. 1 A. Dura and M. A. Lopez‐Marcos, Numerical behavior of stable and unstable solitary waves, App Numer Math 42 (2002), 95–116.
2. 2 M. Mei, Large‐time behavior of solution for generalized Benjamin‐Bona‐Mahony‐Burgers Equations, Nonlinear Anal 33 (1998), 699–714.
3. 3 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.
4. 4 D. H. Peregrine, Calculations of the development of an undular bore, J Fluid Mech 25 (1966), 321–326.
5. 5 T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos Trans R Soc Lond A 272 (1972), 47–78.
6. 6 J. L. Bona and R. Smith, The initial‐value problem for the Korteweg‐de Vries equation, Philos Trans R Soc Lond A 278 (1975), 555–601.
7. 7 L. A. Medeiros and M. M. Miranda, Weak solutions for a nonlinear dispersive equation, J Math Anal Appl 59 (1977), 432–441.
8. 8 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.
9. 9 Y. Chen, B. Li, and H. Zhang, Exact solutions of two nonlinear wave equations with simulation terms of any order, Comm Nonlinear Sci Numer Simulation 10 (2005), 133–138.
10. 10 R. K. Mohanty, An O (k 2 + h 2) finite difference method for one‐space Burgers equation in polar coordinates, Numer Meth Partial Diff Eq 12 (1996), 579–583.
11. 11 R. E. Mickens, A finite difference scheme for travelling wave solutions to Burgers equation, Numer Meth Partial Diff Eq 14 (1998), 815–820.
12. 12 R. Kannan and S. K. Chung, Finite difference approximate solutions for the two‐dimensional Burgers' system, Comput Math Appl 44 (2002), 193–200.
13. 13 T. Achouri, N. Khiari, and K. Omrani, On the convergence of difference schemes for the Benjamin‐Bona‐Mahony (BBM) equation, Appl Math Comput 182 (2006), 999–1005.
14. 14 K. Omrani and M. Ayadi, Finite difference discretization of the Benjamin‐Bona‐Mahony‐Burgers (BBMB) equation, Numer Meth Partial Diff Eq 24 (2008), 239–248.
15. 15 L. Wahlbin, Error estimates for a Galerkin method for a class of model equations for long waves, Numer Math 23 (1975), 289–303.
16. 16 R. E. Ewing, Time‐stepping Galerkin methods for nonlinear Sobolev partial differential equation, IAM J Numer Anal 15 (1978), 1125–1150.
17. 17 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.
18. 18 K. Omrani, The convergence of the fully discrete Galerkin approximations for the Benjamin‐Bona‐Mahony (BBM) equation, Appl Math Comput 180 (2006), 614–621.
19. 19 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.
20. 20 D. Kaya, A numerical simulation of solitary‐wave solutions of the generalized regularized long‐wave equation, Appl Math Comput 149 (2004), 833–841.
21. 21 K. Al‐Khaled, S. Momani, and A. Alawneh, Approximate wave solutions for generalized Benjamin‐Bona‐Mahony‐Burgers equations, Appl Math Comp 171 (2005), 281–292.
22. 22 V. Thomée, Galerkin finite element methods for parabolic problems, Lecture notes in mathematics 1054, Springer‐verlag, Berlin, 1984.
23. 23 S. K. Chang and S. N. Ha, Finite element Galerkin solutions for the Roseneau equation, Appl Anal 54 (1994), 39–56.
24. 24 A. K. Pani and H. Saranga, Finite element Galerkin method for the “Good” Boussinesq equation, Nonlinear Anal 29 (1997), 973–956.
25. 25 P. G. Ciarlet, The finite element method for elliptic problems, North‐Holland, Amsterdam, 1978.
26. 26 S. Kesevan, Topics in functional analysis and applications, Wiley Eastern Limited, New Delhi, 1989.
27. 27 K. Omrani, Numerical methods and error analysis for the nonlinear Sivashinsky equation, Appl Math Comput 189 (2007), 949–962.
28. 28 W. McLean and V. Thomée, Numerical solution of an evolution equation with a positive type memory term, J Aust Math Soc Ser B 35 (1993), 23–70.
29. 29 K. Omrani, Convergence of Galerkin approximations for the Kuramoto‐Tsuzuki equation, Numer Meth Partial Diff Eq 21 (2005), 961–975.