Construction and implementation of general linear methods for ordinary differential equations: A review

It it the purpose of this paper to review the results on the construction and implementation of diagonally implicit multistage integration methods for ordinary differential equations. The systematic approach to the construction of these methods with Runge-Kutta stability is described. The estimation...

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Bibliographic Details
Published inJournal of scientific computing Vol. 25; no. 1-2; pp. 29 - 49
Main Author Jackiewicz, Z.
Format Journal Article
LanguageEnglish
Published New York Springer Nature B.V 01.11.2005
Subjects
Differential equations
Implicit methods
Mathematical analysis
Nonlinear equations
Nonlinear systems
Runge-Kutta method
Stability
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Summary:It it the purpose of this paper to review the results on the construction and implementation of diagonally implicit multistage integration methods for ordinary differential equations. The systematic approach to the construction of these methods with Runge-Kutta stability is described. The estimation of local discretization error for both explicit and implicit methods is discussed. The other implementations issues such as the construction of continuous extensions, stepsize and order changing strategy, and solving the systems of nonlinear equations which arise in implicit schemes are also addressed. The performance of experimental codes based on these methods is briefly discussed and compared with codes from Matlab ordinary differential equation (ODE) suite. The recent work on general linear methods with inherent Runge-Kutta stability is also briefly discussed.
ISSN:0885-7474
1573-7691
DOI:10.1007/BF02728981