Numerical solutions of the equal width wave equation using an adaptive method of lines

Adaptive Method of Lines

Volumn , Issue , 2001, Pages 65-116

  Hamdi, S ^a   Gottlieb, J J ^a   Hansen, J S ^a  

^a UNIVERSITY OF TORONTO   (Canada)

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[No Author keywords available]

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EID: 84981201622     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter

References (32)

1. Kh.O. Abdulloev, H. Bogolubsky, and V.G. Markhankov, One more example of inelastic soliton interaction, Phys. Lett. A, 56, 427-428, 1976.
2. U.M. Ascher, and L.R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia, 1988.
3. T.B. Benjamin, J.L. Bona, and J.L. Mahoney, Model equations for long waves in nonlinear dispersive media, Phil. Trans. Roy. Soc. Lond. A, 272, 47-78, 1995.
4. Yu. Berezin and V.I. Karpman, Nonlinear evolution of disturbances in plasmas and other dispersive media, Soviet Physics JETP, 24, 1049-1056, 1967.
5. J.L. Bona, W.G. Pritchard, and L.R. Scott, Numerical schemes for a model for nonlinear dispersive waves, J. Comput. Phys., 60, 167-186, 1985.
6. K.E. Brenan, S.L. Campbell, and L.R. Petzold, The Numerical Solution of Initial Value Problems in Differential-Algebraic Equations, Elsevier Science Publishing Co., 1989.
7. R.P. Brent, Algorithms for Minimization without Derivatives, Prentice-Hall, Englewood Cliffs, NJ, 1973.
8. B. Fornberg, Calculation of weights in finite difference formulas, SIAM Rev., 40(3), 685-691, 1998.
9. G.E. Forsythe, M.A. Malcolm, and C.B. Moler, Computer Methods for Mathematical Computations, Prentice-Hall, Englewood Cliffs, NJ, 1977.
10. L.R.T. Gardner, G.A. Gardner, A. Ayoub, and N.K. Amein, Simulations of the EW undular bore, Commun. Numer. Meth. Engng., 13, 583-592, 1997.
11. L.R.T. Gardner, G.A. Gardner, and A. Dogan, A least-squares finite element scheme for the RLW equation, Commun. Numer. Meth. Engng., 12, 795-804, 1996.
12. L.R.T. Gardner and G.A. Gardner, Solitary waves of the regularized long-wave equation, J. Comput. Phys., 91, 441-459, 1990.
13. C.W. Gear, The simultaneous numerical solution of differential-algebraic equations, IEEE Trans. Circuit Theory, CT-18: 89-95, 1971.
14. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, 1991.
15. S. Hamdi, J.J. Gottlieb, and J.S. Hansen, Computation of blow-up solutions of the generalized Korteweg-de Vries equation using adaptive method of lines, in J. Militzer, ed., 7th Annual Conference of the CFD Society of Canada, 921-926, June 1999.
16. S. Hamdi, J.J. Gottlieb, and J.S. Hansen, Numerical solution of the equal width wave equation using the method of lines, in 8th Annual Conference of the CFD Society of Canada, 129-134. Centre de Recherche en Calcul Appliqué, 2000.
17. A.C. Hindmarsh, ODEPACK: Asystematized collection ofODEsolvers, in R.S. Steplemman, ed., Scientific Computing, 55-64. North-Holland, Amsterdam, 1983.
18. P.C. Jain, R. Shankar, and T.V. Singh, Numerical solutions of regularized long wave equation, Commun. Numer. Meth. Engng., 9, 579-586, 1993.
19. D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and a new type of long stationary wave, Phil. Mag., 39, 422-443, 1895.
20. J.C. Lewis and J.A. Tjon, Resonant production of solitons in the RLWequation, Phys. Lett. A, 73, 275-279, 1979.
21. P.J. Morrison, J.D. Meiss, and J.R. Carey, Scattering of RLW solitary waves, Physica D, 11, 324-336, 1984.
22. P.J. Olver, Euler operators and conservation laws of the BBM equations, Math. Proc. Camb. Phil. Soc., 85, 143-159, 1979.
23. D.H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25, 321-330, 1966.
24. L.R. Petzold, A Description of DASSL: A differential/algebraic system solver, in Scientific Computing, R.S. Stepleman et al., eds., North Holland, Amsterdam, 1983.
25. M.A. Revilla, Simple time and space adaptation in one-dimensional evolutionary partial differential equation, Int. J. Numer. Methods Eng., 23, 2263-2275, 1986.
26. A.R. Santarelli, Numerical analysis of the regularized long-wave equation: Anelastic collision of solitary waves, Nuovo Cimento B, 46, 179-188, 1978.
27. J.M. Sanz-Serna and I. Christie, Asimple adaptive technique for nonlinearwave problems, J. Comput. Phys., 67, 348-360, 1986.
28. P. Saucez, A.V.Wouwer, andW.E. Schiesser, Some observations on a static spatial remeshing method based on equidistribution principles, J. Comput. Phys., 128, 274-288, 1996.
29. P. Saucez, A.V. Wouwer, and W.E. Schiesser, An adaptive method of lines solution of the Korteweg-de Vries equation, Computers Math. Applic., 35, 13-25, 1998.
30. W.E. Schiesser, Method of lines solution of the Korteweg-de Vries equation, Computers Math. Applic., 28, 147-154, 1994.
31. A.B. White, On the numerical solution of initial/boundary-value problems in one space dimension, SIAM J. Numer. Anal., 19, 683-697, 1982.
32. G.B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974.