A Legendre-Petrov-Galerkin and Chebyshev collocation method for third-order differential equations

SIAM Journal on Numerical Analysis

Volumn 38, Issue 5, 2001, Pages 01425-1438

  Ma, Heping ^a   Sun, Weiwei ^b  

^a SHANGHAI UNIVERSITY   (China)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Korteweg de Vries equation; Legendre Petrov Galerkin and Chebyshev collocation; Third order differential equation

Indexed keywords

EID: 0034543430     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (24)

1. B. K. ALPERT AND V. ROKHLIN, A fast algorithm for the evaluation of Legendre expansions, SIAM J. Sci. Stat. Comput., 12 (1991), pp. 158-179.
2. U. M. ASCHER, S. J. RUUTH, AND B. T. R. WETTON, Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), pp. 797-823.
3. C. BERNARDI AND Y. MADAY, Spectral methods, in Handbook of Numerical Analysis, Vol. V, Techniques of Scientific Computing (Part 2), P. Ciarlet and J. L. Lions, eds., North-Holland, Amsterdam, 1997, pp. 209-486.
4. N. BRESSAN AND D. PAVONI, Truncation versus mapping in the spectral approximation to the Korteweg-de Vries equation, Comput. Methods Appl. Mech. Engrg., 80 (1990), pp. 443-450.
5. C. CANUTO, M. Y. HUSSAINI, A. QUARTERONI, AND T. A. ZANG, Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988.
6. G. F. CAREY AND Y. SHEN, Approximations of the KdV equation by least squares finite elements, Comput. Methods Appl. Mech. Engrg., 93 (1991), pp. 1-11.
7. T. F. CHAN AND T. KERKHOVEN, Fourier methods with extended stability intervals for the Korteweg-de Vries equation, SIAM J. Numer. Anal., 22 (1985), pp. 441-454.
8. E. COUTSIAS, T. HAGSTROM, J. HESTHAVEN, AND D. TORRES, Integration preconditioned for differential operators in spectral τ-methods, in Proceedings of the Third International Conference on Spectral and High Order Methods, A. V. Ilin and L. R. Scott, eds., Houston, TX, 1996, pp. 21-38.
9. K. DJIDJELI, W. G. PRICE, P. TEMAREL, AND E. H. TWIZELL, A linearized implicit pseudospectral method for certain non-linear water wave equations, Comm. Numer. Methods Engrg., 14 (1998), pp. 977-993.
10. K. DJIDJELI, W. G. PRICE, E. H. TWIZELL, AND Y. WANG, Numerical methods for the solution of the third- and fifth-order dispersive Korteweg-de Vries equations, J. Comput. Appl. Math., 58 (1995), pp. 307-336.
11. W. S. DON AND D. GOTTLIEB, The Chebyshev-Legendre method: Implementing Legendre methods on Chebyshev points, SIAM J. Numer. Anal., 31 (1994), pp. 1519-1534.
12. B. FORNBERG AND T. DRISCOLL, A fast spectral algorithm for nonlinear wave equations with linear dispersion, J. Comput. Phys., 155 (1999), pp. 456-467.
13. B. FORNBERG AND G. B. WHITHAM, A numerical and theoretical study of certain nonlinear phenomena, Philos. Trans. Roy. Soc. London Ser. A, 289 (1978), pp. 373-404.
14. B.-Y. GUO, Spectral Methods and Their Applications, World Scientific, Singapore, 1998.
15. J. S. HESTHAVEN, Integration preconditioning of pseudospectral operators. I. Basic linear operators, SIAM J. Numer. Anal., 35 (1998), pp. 1571-1593.
16. W. HUANG AND D. M. SLOAN, The pseudospectral method for third-order differential equations, SIAM J. Numer. Anal., 29 (1992), pp. 1626-1647.
17. H. KREISS, Numerical methods for solving time-dependent problems for partial differential equations, Lecture Notes, Univ. de Montreal, 1978.
18. J. LI, H.-P. MA, AND W.-W. SUN, Error analysis for solving the Korteweg-de Vries equation by a Legendre pseudospectral method, Numer. Methods Partial Differential Equations, to appear.
19. H.-P. MA AND B.-Y. GUO, The Fourier pseudospectral method with a restrain operator for the Korteweg-de Vries equation, J. Comput. Phys., 65 (1986), pp. 120-137.
20. Y. MADAY AND A. QUARTERONI, Error analysis for spectral approximation of the Korteweg-de Vries equation, RAIRO Modél. Math. Anal. Numér., 22 (1988), pp. 499-529.
21. W. J. MERRYFIELD AND B. SHIZGAL, Properties of collocation third-derivative operators, J. Comput. Phys., 105 (1993), pp. 182-185.
22. D. PAVONI, Single and multidomain Chebyshev collocation methods for the Korteweg-de Vries equation, Calcolo, 25 (1988), pp. 311-346.
23. J. SHEN, Efficient spectral-Galerkin method I. Direct solvers for second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), pp. 1489-1505.
24. J. SHEN, Efficient spectral-Galerkin method II. Direct solvers of second- and fourth-order equations using Chebyshev polynomials, SIAM J. Sci. Comput., 16 (1995), pp. 74-87.