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

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8 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)29-49
Number of pages21
JournalJournal of Scientific Computing
Issue number1
StatePublished - Oct 1 2005


  • General linear methods
  • Implementation aspects
  • Local error estimation
  • Order and stage order
  • Runge-Kutta stability

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • Engineering(all)
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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