Application of the adaptive method of lines to nonlinear wave propagation problems

Adaptive Method of Lines

Volumn , Issue , 2001, Pages 39-64

  Vande Wouwer, Alain ^a   Saucez, Philippe ^a   Schiesser, William ^b  

^a FACULTÉ POLYTECHNIQUE DE MONS   (Belgium)

^b LEHIGH UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85059723793     PISSN: None     EISSN: None     Source Type: Book    
DOI: None     Document Type: Chapter

References (29)

1. M. Berzins and R.M. Furzeland, A User’s Manual for SPRINT - A Versatile Software Package for Solving Systems of Algebraic Ordinary and Partial Differential Equations, Thornton Research Centre, Shell Maatschappij (1985, 86 and 89).
2. M. Delfour, M. Fortin, and G. Payne, Finite difference solution of a nonlinear Schrödinger equation, J. Comp. Phys., 44, (1981), 277-288.
3. B. Fornberg, Generation of finite difference formulas on arbitrarily spaced grids, Math. Comp., 51, (1988), 699-706.
4. C.S. Gardner, J. Green, M. Kruskal, and R. Muira, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19, (1967), 1095-1097.
5. B. Garcia-Archilla and J.M. Sanz-Serna, A finite difference formula for the discretization of d3/dx3 on nonuniform grids, Math. Comp., 57, (1991), 239-257.
6. D.F. Griffiths, A.R. Mitchell, and J.Ll. Morris, A numerical study of the nonlinear Schrödinger equation, Comput. Meth. Appl. Mech. Eng., 45, (1984), 177-215.
7. E. Hairer and G. Wanner, Solving ordinary differential equations II. Stiff and differential-algebraic problems, Springer-Verlag, Berlin, 1991.
8. B.M. Herbst, J.Ll. Morris, and A.R. Mitchell, Numerical experience with the nonlinear Schrödinger equation, J. Comp. Phys., 60, (1985), 282-305.
9. A. Jeffrey and S. Xu, Exact solutions to the Korteweg-de Vries-Burgers Equation, Wave Motion, 11, (1989), 559-564.
10. R.S. Johnson, A nonlinear equation incorporating damping and dispersion, J. Fluid Mech., 42, (1970), 49-60.
11. D.J. Kaup and A.C. Newell, An exact solution for the derivative nonlinear Schrödinger equation, J. Math. Phys., 19, (1978), 798-801.
12. J. Kautsky and N. K. Nichols, Equidistributing meshes with constraints, SIAM J. Sci. Stat. Comput., 1, (1980), 499-511.
13. S. Kichenassamy, Nonlinear Wave Equations, Marcel Dekker, 1996.
14. D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Philos. Mag., 39, (1895), 422-443.
15. J.W. Miles, An envelope soliton problem, SIAM J. Appl. Math., 41, (1981), 227-230.
16. E. Mjolhus, On the modulational instability of hydromagnetic waves parallel to the magnetic field, J. Plasma Phys., 16, (1976), 321-334.
17. F.Z. Nouri and D.M. Sloan, A comparison of Fourier pseudospectral methods for the solution of the Korteweg-de Vries equation, J. Comp. Phys., 83, (1989), 324-344.
18. P. Rosenau and J.M. Hyman, The compacton: a soliton with compact support, Phys. Rev. Lett., 70, (1993), 564.
19. J.M. Sanz-Serna, Methods for the numerical solution of the nonlinear Schrödinger equation, Math. Comp., 43, (1984), 21-27.
20. J.M. Sanz-Serna and J.G. Verwer, Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation, IMA J. Numer. Anal., 6, (1986), 25-42.
21. J.M. Sanz-Serna and I. Christie, Asimple adaptive technique for nonlinearwave problems, J. Comp. Phys., 67, (1986), 348-360.
22. P. Saucez, A. VandeWouwer andW.E. Schiesser, Some observations on a static spatial remeshing method based on equidistribution principles, J. Comp. Phys., 128, (1996), 274-288.
23. P. Saucez, A. VandeWouwer, andW.E. Schiesser, An adaptive method of lines solution of the Korteweg-de Vries equation, Comp. Math. Applic., 35, (1998), 13-25.
24. G. Steinebach, Die Linienmethode und ROW-Verfahren zur Abfluss- und Prozess-simulation in Fliessgewässern am Beispiel von Rhein und Mosel, Ph.D. Thesis, Darmstadt Technical University, Germany, 1995.
25. C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation - Self-Focusing and Wave Collapse, Springer-Verlag, Berlin, 1999.
26. T.R. Taha and M.J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equation. II. Numerical, nonlinear Schrödinger equation, J. Comp. Phys., 55, (1984), 203-230.
27. T.R. Taha and M.J. Ablowitz, Analytical and numerical aspects of certain nonlinear evolution equation. III. Numerical, nonlinear Korteweg-de Vries equation, J. Comp. Phys., 55, (1984), 231-253.
28. G.B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974.
29. V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation ofwaves in nonlinear media, Soviet Phys. JEPT, 34, (1972), 62-69.