×
Butcher, John

General linear methods for ordinary differential equations. (English) Zbl 1159.65333

Math. Comput. Simul. 79, No. 6, 1834-1845 (2009).
Summary: General linear methods were introduced as the natural generalizations of the classical Runge-Kutta (RK) and linear multistep methods. They have potential applications, especially for stiff problems. This paper discusses stiffness and emphasises the need for efficient implicit methods for the solution of stiff problems. In this context, a survey of general linear methods is presented, including recent results on methods with the inherent RK stability property.

Cited in 4 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

Keywords:

general linear methods; stiff differential equations; inherent Runge-Kutta stability; linear multistep methods; implicit methods
PDF BibTeX XML Cite
\textit{J. Butcher}, Math. Comput. Simul. 79, No. 6, 1834--1845 (2009; Zbl 1159.65333)
Full Text: DOI
  • logo of hbz
  • logo of WorldCat

References:

[1] Butcher, J.C., On the convergence of numerical solutions of ordinary differential equations, Math. comp., 20, 1-10, (1966) · Zbl 0141.13504
[2] Butcher, J.C., An algebraic theory of integration methods, Math. comp., 26, 79-106, (1972) · Zbl 0258.65070
[3] Butcher, J.C., The order of numerical methods for ordinary differential equations, Math. comp., 27, 793-806, (1973) · Zbl 0278.65073
[4] Butcher, J.C., Order conditions for a general class of numerical methods for ordinary differential equations, (), 35-40 · Zbl 0278.65074
[5] Butcher, J.C., The order of differential equation methods, Lect. notes math., 362, 72-75, (1974) · Zbl 0309.65029
[6] Butcher, J.C., Order conditions for general linear methods for ordinary differential equations, Isnm, 19, 77-81, (1974) · Zbl 0279.65055
[7] Butcher, J.C., An application of the runge – kutta space, Bit, 24, 425-440, (1984) · Zbl 0555.65045
[8] Butcher, J.C., General linear methods: a survey, Appl. numer. math., 1, 273-284, (1985) · Zbl 0619.65057
[9] Butcher, J.C., The numerical analysis of ordinary differential equations: runge – kutta and general linear methods, (1987), John Wiley & Sons Chichester · Zbl 0616.65072
[10] Butcher, J.C., General linear methods, Comput. math. appl., 31, 105-112, (1996) · Zbl 0874.65057
[11] Butcher, J.C., Runge – kutta methods as mathematical objects, (), 39-56
[12] Butcher, J.C., General linear methods for stiff differential equations, Bit, 41, 240-264, (2001) · Zbl 0983.65085
[13] Butcher, J.C., Numerical methods for ordinary differential equations, (2003), J. Wiley Ltd. Chichester · Zbl 1032.65512
[14] Butcher, J.C., High order A-stable numerical methods for stiff problems, J. sci. comput., 25, 51-66, (2005) · Zbl 1203.65106
[15] Butcher, J.C., General linear methods, Acta numer., 15, 157-256, (2006) · Zbl 1113.65072
[16] Butcher, J.C.; Chartier, P., A generalization of singly-implicit runge – kutta methods, Appl. numer. math., 24, 343-350, (1997) · Zbl 0906.65076
[17] Butcher, J.C.; Jackiewicz, Z., Implementation of diagonally implicit multistage integration methods for ordinary differential equations, SIAM J. numer. anal., 34, 2119-2141, (1997) · Zbl 0892.65044
[18] Butcher, J.C.; Jackiewicz, Z., A reliable error estimation for diagonally implicit multistage integration methods, Bit, 41, 656-665, (2001) · Zbl 0998.65081
[19] Butcher, J.C.; Jackiewicz, Z., Error estimation for Nordsieck methods, Numer. algorithms, 31, 75-85, (2002) · Zbl 1012.65083
[20] Butcher, J.C.; Jackiewicz, Z., A new approach to error estimation for general linear methods, Numer. math., 95, 487-502, (2003) · Zbl 1032.65088
[21] Butcher, J.C.; Jackiewicz, Z., Construction of general linear methods with runge – kutta stability properties, Numer. algorithms, 36, 53-72, (2004) · Zbl 1055.65083
[22] Butcher, J.C.; Podhaisky, H., On error estimation in general linear methods for stiff odes, Appl. numer. math., 56, 345-357, (2006) · Zbl 1089.65080
[23] Butcher, J.C.; Wright, W.M., A transformation relating explicit and diagonally-implicit general linear methods, Appl. numer. math., 44, 313-327, (2003) · Zbl 1016.65048
[24] Butcher, J.C.; Wright, W.M., The construction of practical general linear methods, Bit, 43, 695-721, (2003) · Zbl 1046.65054
[25] Gear, C.W., The numerical integration of ordinary differential equations, Math. comp., 21, 146-156, (1967) · Zbl 0155.20601
[26] Hairer, E.; Wanner, G., On the Butcher group and general multi-value methods, Computing, 1-15, (1974) · Zbl 0293.65050
[27] Nordsieck, A., On numerical integration of ordinary differential equations, Math. comp., 16, 22-49, (1962) · Zbl 0105.31902
[28] Stoffer, D., General linear methods: connection to one step methods and invariant curves, Numer. math., 64, 395-408, (1993) · Zbl 0810.65076
[29] W.M. Wright, General linear methods for ordinary differential equations: theoretical and practical investigation, MSc thesis, The University of Auckland, New Zealand, 1999.
[30] Wright, W.M., Explicit general linear methods with inherent runge – kutta stability, Numer. algorithms, 31, 381-399, (2002) · Zbl 1016.65049
[31] W.M. Wright, General linear methods with inherent Runge-Kutta stability, PhD thesis, Department of Mathematics, The University of Auckland, Auckland, 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.