×
Bartoszewski, Zbigniew; Jackiewicz, Zdzisław

Explicit Nordsieck methods with extended stability regions. (English) Zbl 1246.65119

Appl. Math. Comput. 218, No. 10, 6056-6066 (2012).
Summary: We describe the construction of explicit Nordsieck methods of order \(p\) and stage order \(q = p\) with large regions of absolute stability. We also discuss error propagation and estimation of local discretization errors. The error estimators are derived for examples of general linear methods constructed in this paper. Some numerical experiments are presented which illustrate the effectiveness of proposed methods.

Cited in 6 Documents

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations

Keywords:

explicit Nordsieck methods; general linear methods; order conditions; region of absolute stability; error bounds; numerical experiments

Software:

RODAS
Cite Review PDF
Full Text: DOI

References:

[1] Abdulle, A., Fourth order Chebyshev methods with recurrence relation, SIAM J. Sci. Comput., 23, 6, 2041-2054 (2002) · Zbl 1009.65048
[2] Abdulle, A.; Medovikov, A. A., Second order Chebyshev methods based on orthogonal polynomials, Numer. Math., 90, 1-18 (2001) · Zbl 0997.65094
[3] Braś, M.; Jackiewicz, Z., Nordsieck methods with computationally verified algebraic stability, Appl. Math. Comput., 217, 8598-8610 (2011) · Zbl 1217.65145
[4] Butcher, J. C., Numerical Methods for Ordinary Differential Equations (2003), John Wiley & Sons: John Wiley & Sons Chichester · Zbl 1032.65512
[5] Butcher, J. C.; Jackiewicz, Z., A new approach to error estimation for general linear methods, Numer. Math., 95, 487-502 (2003) · Zbl 1032.65088
[6] Butcher, J. C.; Jackiewicz, Z.; Wright, W. M., Error propagation for general linear methods for ordinary differential equations, J. Complexity, 23, 560-580 (2007) · Zbl 1131.65068
[7] R. D’Ambrosio, G. Izzo, Z. Jackiewicz, Search for highly stable two-step Runge-Kutta methods, Appl. Numer. Math., in press.; R. D’Ambrosio, G. Izzo, Z. Jackiewicz, Search for highly stable two-step Runge-Kutta methods, Appl. Numer. Math., in press. · Zbl 1252.65119
[8] D’Ambrosio, R.; Jackiewicz, Z., Continuous two-step Runge-Kutta methods for ordinary differential equations, Numer. Algorithms, 54, 169-193 (2010) · Zbl 1191.65092
[9] Hairer, E.; Norsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I. Solving Ordinary Differential Equations I, Nonstiff Problems (1993), Springer Verlag: Springer Verlag Berlin, Heidelberg, New York · Zbl 0789.65048
[10] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II. Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems (1996), Springer Verlag: Springer Verlag Berlin, Heidelberg, New York · Zbl 0859.65067
[11] Jackiewicz, Z., General Linear Methods for Ordinary Differential Equations (2009), John Wiley & Sons: John Wiley & Sons Hoboken, New Jersey · Zbl 1211.65095
[12] Jeltsch, R.; Nevanlinna, O., Largest disk of stability of explicit Runge-Kutta methods, BIT, 18, 500-502 (1978) · Zbl 0399.65051
[13] Jeltsch, R.; Nevanlinna, O., Stability of explicit time discretizations for solving initial value problems, Numer. Math., 37, 61-91 (1981) · Zbl 0457.65054
[14] Medovikov, A. A., High order explicit methods for parabolic equations, BIT, 38, 372-390 (1998) · Zbl 0909.65060
[15] Owren, B.; Zennaro, M., Order barriers for continuous explicit Runge-Kutta methods, Math. Comput., 56, 645-661 (1991) · Zbl 0718.65051
[16] Owren, B.; Zennaro, M., Derivation of efficient, continuous, explicit Runge-Kutta methods, SIAM J. Sci. Statist. Comput., 13, 1488-1501 (1992) · Zbl 0760.65073
[17] B. Owren and M. Zennaro, Continuous explicit Runge-Kutta methods. Computational ordinary differential equations (London, 1989), 97-105, Inst. Math. Appl. Conf. Ser. New Ser., 39, Oxford University Press, New York 1992.; B. Owren and M. Zennaro, Continuous explicit Runge-Kutta methods. Computational ordinary differential equations (London, 1989), 97-105, Inst. Math. Appl. Conf. Ser. New Ser., 39, Oxford University Press, New York 1992. · Zbl 0774.65050
[18] Prothero, A.; Robinson, A., On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations, Math. Comput., 28, 125, 145-162 (1974) · Zbl 0309.65034
[19] Shampine, L. F., Numerical Solution of Ordinary Differential Equations (1994), Chapman & Hall: Chapman & Hall New York, London · Zbl 0826.65082
[20] Shampine, L. F.; Gordon, M. K., Computer Solution of Ordinary Differential Equations. Computer Solution of Ordinary Differential Equations, The Initial Value Problem (1975), W.H. Freeman and Company: W.H. Freeman and Company San Francisco · Zbl 0347.65001
[21] W. Wright, General linear methods with inherent Runge-Kutta stability, Ph.D. thesis, University of Auckland, New Zealand, 2002.; W. Wright, General linear methods with inherent Runge-Kutta stability, Ph.D. thesis, University of Auckland, New Zealand, 2002. · Zbl 1016.65049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.