Analysis of a Chebyshev-based backward differentiation formulae and relation with Runge-Kutta collocation methods

International Journal of Computer Mathematics

Volumn 88, Issue 3, 2011, Pages 555-577

  Ramos, Higinio ^a   Garcia Rubio R ^b  

^a Campus Viriato   (Spain)

^b UNIVERSITY OF ALICANTE   (Spain)

Author keywords

BDF type methods; Chebyshev nodes; initial value problems; Runge Kutta collocation methods

Indexed keywords

EID: 79251501017     PISSN: 00207160     EISSN: 10290265     Source Type: Journal    
DOI: 10.1080/00207161003631877     Document Type: Article

References (43)

1. L. Aceto and C. Magherini, On the relations between B2VMs and Runge-Kutta collocation methods, J. Comput. Appl. Math. 231 (1) (2009), pp. 11-23.
2. P. Amodio and F. Mazzia, Boundary value methods based on Adams-type methods, Appl. Numer. Math. 18 (1995), pp. 23-35.
3. P. Amodio and L. Brugnano, A note on the efficient implementation of implicit methods for ODEs, J. Comput. Appl. Math. 87 (1997), pp. 1-9. (Pubitemid 127411602)
4. J. Alvarez and J. Rojo, An improved class of generalized Runge-Kutta methods for stiff problems. Part II: The separated system case, Appl. Math. Comp. 159 (2004), pp. 717-758.
5. J.P. Berrut and L.N. Trefethen, Barycentric Lagrange interpolation, SIAM Rev. 46 (2004), pp. 501-517.
6. P.N. Brown, G.D. Byrne, and A.C. Hindmarsh, VODE: A variable-coefficient ODE solver, SIAM J. Sci. Statist. Comput. 10 (1989), pp. 1038-1051.
7. L. Brugnano and D. Trigiante, On the potentiality of sequential and parallel codes based on extended trapezoidal rules (ETRs), Appl. Numer. Math. 25 (1997), pp. 169-184.
8. L. Brugnano and D. Trigiante, Solving Differential Problems by Multistep Initial and Boundary Value Methods, Gordon & Breach, Amsterdam, 1998.
9. J.C. Butcher, General linear methods, Comput. Math. Appl. 31 (1996), pp. 105-112.
10. J.C. Butcher, Numerical Methods for Ordinary Differential Equations,Wiley, Chichester, UK, 2003.
11. M. Calvo, S. González-Pinto, and J.I. Montijano, On the iterative solution of the algebraic equations in fully implicit Runge-Kutta methods, Numer. Algor. 23 (2000), pp. 97-113.
12. J.R. Cash, Stable Recursions: With Applications to the Numerical Solution of Stiff Systems, Academic Press, New York, 1979.
13. J.R. Cash, On the integration of stiff systems of O.D.E.s using extended backward differentiation formulae, Numer. Math. 34 (1980), pp. 235-246.
14. J.R. Cash, Efficient numerical methods for the solution of stiff initial-value problems and differential algebraic equations, Proc. R. Soc. Lond. A 459 (2003), pp. 797-815.
15. G.J. Cooper and J.C. Butcher, An iteration scheme for implicit Runge-Kutta methods, IMA J. Numer. Anal. 3 (1983), pp. 127-140.
16. C.F. Curtiss and J.O. Hirschfelder, Integration of stiff equations, Proc. Natl Acad. Sci. 38 (1952), pp. 235-243.
17. L. Fox and I.B. Parker, Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London, 1968.
18. C. Fredebeul, A-BDF: A generalization of the backward differentiation formulae, SIAM J. Numer. Anal. 35 (1998), pp. 1917-1938.
19. C.W. Gear, Algorithm 407-DIFSUB for solution of ordinary differential equations, Comm.ACM14 (1971), pp. 185- 190.
20. N. Guzel and M. Bayram, On the numerical solution of stiff systems, Appl. Math. Comput. 170 (2005), pp. 230-236.
21. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer, Berlin, 1996.
22. E. Hairer and G. Wanner, Stiff differential equations solved by Radau methods, J. Comp. Appl. Math. 111 (1999), pp. 93-111.
23. E. Hairer, S.P. Norsett, and G.Wanner, Solving Ordinary Differential Equations I, Springer, Berlin, 1993.
24. P. Henrici, Discrete variable Methods in Ordinary Differential Equations, JohnWiley, New York, 1962.
25. D.J. Higham, Analysis of the Enright-Kamel partitioning method for stiff ODEs, IMA J. Numer. Anal. 9 (1989), pp. 1-14.
26. D.J. Higham, Stiffness of ODEs, BIT 33 (1993), pp. 285-303.
27. A.C. Hindmarsh, LSODE and LSODI, two initial value ordinary differential equations solvers, ACM SIGNUM Newsl. 15 (1980), pp. 10-11.
28. F. Iavernaro and F. Mazzia, Solving ordinary differential equations by generalized Adams methods: Properties and implementation techniques, Appl. Numer. Math. 28 (1998), pp. 107-126.
29. L.W. Jackson and S.K. Kenue, A fourth order exponentially fitted method, SIAM J. Numer. Anal. 11 (1974), pp. 965-978.
30. J.D. Lambert, Numerical Methods for Ordinary Differential Systems, JohnWiley, UK, 1991.
31. J.D. Lambert and S.T. Sigurdsson, Multistep methods with variable matrix coefficients, SIAM J. Numer. Anal. 9 (1972), pp. 715-733.
32. W. Liniger and R.A.Willoughby, Efficient Numerical Integration Of Ordinary Differential Equations, Res. Rep. RC, IBM,Yorktown Heights, New York, 1970.
33. J. Martin-Vaquero and J. Vigo-Aguiar, Exponential fitting BDF algorithms: Explicit and implicit 0-stable methods, J. Comp. Appl. Math. 196 (2006), pp. 100-113.
34. J.C. Mason and D.C. Handscomb, Chebyshev Polynomials, CRC Press, New York, 2003.
35. L.R. Petzold,A Description ofDASSL:A Differential Algebraic System Solver. IMACS.,Trans. Scientific Computing, North-Holland, Amsterdam, 1993.
36. H. Ramos andVigo-Aguiar, J. A fourth-order Runge-Kutta method based on BDF-type Chebyshev approximations, J. Comp. Appl. Math. 204 (2007), pp. 124-136.
37. H. Ramos and J.Vigo-Aguiar, An almost L-stable BDF-type method for the numerical solution of stiff ODEs arising from the method of lines, Numer. Meth. Part. Diff. Eq. 23 (2007), pp. 1110-1121.
38. D.L. Richardson, D. Schmitd, and J. Mitchell, Improved Chebysehv methods for the numerical integration of firstorder differential equations. AAS/AIAA Space Flight Mechanics Meeting, paper no. 98-208, Monterey, California, 1998.
39. J. Vigo-Aguiar and H. Ramos, Dissipative Chebyshev exponential-fitted methods for numerical solution of secondorder differential equations, J. Comput. Appl. Math. 158 (2003), pp. 187-211.
40. J. Vigo-Aguiar and D.L. Richardson, Modification of the Richardson-Panowsky methods for precise integration of satellite orbits, Comput. Math. Appl. 45 (2003), pp. 25-36.
41. J.Vigo-Aguiar and H. Ramos, A family of A-stable Runge-Kutta collocation methods of higher order for initial-value problems, IMA J. Numer. Anal. 27 (2007), pp. 798-817.
42. J.Vigo-Aguiar, J.Martin-Vaquero, and H. Ramos, Exponential fitting BDF-Runge-Kutta algorithms, Comput. Phys. Commun. 178 (2008), pp. 15-34.
43. X.-Y.Wu and J.-L. Xia, Two low accuracy methods for stiff systems, Appl. Math. Comput. 123 (2001), pp. 141-153.