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

• Published in: Journal of scientific computing Vol. 25; no. 1-2; pp. 29 - 49
• Main Author: Jackiewicz, Z.
• Format: Journal Article
• Language: English
• 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
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/BF02728981

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.