Unconventional Schemes for a Class of Ordinary Differential Equations - With Applications to the Korteweg-de Vries Equation

Journal of Computational Physics

Volumn 134, Issue 2, 1997, Pages 316-331

  Kahan, William ^a   Li, Ren Chang ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF KENTUCKY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL EFFICIENCY; KORTEWEG-DE VRIES EQUATION; LINEAR SYSTEMS; NUMERICAL METHODS;

KORTEWEG-DE VRIES-EQUATION; PROPERTY; SECOND ORDERS; TIME STEP;

ORDINARY DIFFERENTIAL EQUATIONS;

EID: 0031188104     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcph.1997.5710     Document Type: Article

References (33)

1. de Frutos J., Sanz-Serna J. M. An easily implementable fourth-order method for the time integration of wave problems. J. Comput. Phys. 103:1992;160.
2. Demmel J. Numerical Linear Algebra. 1993;Univ. of CaliforniaDepartment of Mathematics, Berkeley.
3. Forest E., Ruth R. D. Fourth-order symplectic integrator. Physica D. 43:1990;105.
4. Fornberg B. On a Fourier method for the integration of hyperbolic equations. SIAM J. Numer. Anal. 12:1975;509.
5. Fornberg B. The pseudospectral method: Comparisons with finite differences for the elastic wave equation. Geophysics. 52:1987;483.
6. Golub G. H., Van Loan C. F. Matrix Computations. 1989;Johns Hopkins Univ. Press, Baltimore.
7. Gottlieb D., Orszag S. A. Numerical Analysis of Spectral Methods: Theory and Applications. 1977;Soc. Indus. Appl. Math. Philadelphia.
8. Gottlieb D., Turkel E. Spectral Methods for Time-Dependent Partial Differential Equations. Report NASA. CR-172241:1983.
9. Hairer E., Nørsett S. P., Wanner G. Solving Ordinary Differential Equations I. 1992;Springer-Verlag, New York.
10. Hairer E., Wanner G. Solving Ordinary Differential Equations II. 1991;Springer-Verlag, New York.
11. W. Kahan, October 1977, Relaxation methods for solving systems of ordinary differential equations, CS Division, Department of EECS, University of California at Berkeley.
12. Kahan W. Unconventional numerical methods for trajectory calculations. Lectures notes. October 1993.
13. W. Kahan, R.-C. Li, Composition constants for raising the orders of unconventional schemes for ordinary differential equations, Math. Comput.
14. Korteweg D. J., de Vries G. On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves. Phil. Mag. 39:1895;422.
15. Kreiss H.-O., Oliger J. Comparison of accurate methods for the integration of hyperbolic equations. Tellus. 24:1972;199.
16. Lambert J. D. Numerical Methods for Ordinary Differential Systems. 1991;Wiley, New York.
17. R.-C. Li, 1995, Raising the Orders of Unconventional Schemes for Ordinary Differential Equations, Department of Mathematics, University of California at Berkeley, CA.
18. McLachlan R. I. On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput. 16:1995;151.
19. Meyer-Spasche R., Düchs D. A General Method for Obtaining Unconventional and Nonstandard Difference Schemes. Technical Report. IPP 6/3434:1995.
20. Mickens R. E. Nonstandard Finite Difference Models of Differential Equations. 1994;World Scientific, Singapore.
21. Nouri F. Z., Sloan D. M. A comparison of fourier pseudospectral methods for the solution of the Korteweg-de Vries equations. J. Comput. Phys. 83:1989;324.
22. Orszag S. A. Comparison of pseudospectral and spectral approximation. Stud. Appl. Math. 51:1972;253.
23. Ruth R. D. A canonical integration technique. IEEE Trans. Nucl. Sci. NS-30:1983.
24. Saad Y., Schultz M. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statis. Comput. 7:1986;856.
25. Sanz-Serna J. M., Abia L. Order conditions for canonical Runge-Kutta schemes. SIAM J. Numer. Anal. 28:1991;1081.
26. Sanz-Serna J. M., Christie I. Petrov-Galerkin methods for nonlinear dispersive waves. J. Comput. Phys. 39:1981;94.
27. Suzuki M. General theory of higher-order decomposition of exponential operators and symplectic integrators. Phys. Lett. A. 165:1992;387.
28. Tadmor E. The exponential accuracy of Fourier and Chebyshev differencing methods. SIAM J. Numer. Anal. 23:1986;1.
29. Taha T. R., Ablowitz M. J. Analytical and numerical aspects of certain nonlinear evolution equations. III. numerical, Korteweg-de Vries equation. J. Comput. Phys. 55:1984;231.
30. Twizell E. H., Wang Y., Price W. G. Chaos-free numerical solutions of reaction-diffusion equations. Proc. R. Soc. Lond. A. 430:1990;541.
31. Whitham G. B. Linear and Nonlinear Waves. 1974;Wiley, New York.
32. Yoshida H. Construction of higher order symplectic integrators. Phys. Lett. A. 150:1990;262.
33. Zlatev Z., Waśniewski J. Running air pollution models on the connection machine. Math. Comput. Modeling. 20:1993;1.