Order conditions for general linear methods

Angelamaria Cardone, Zdzislaw Jackiewicz, James H. Verner, Bruno Welfert

Research output: Contribution to journalArticlepeer-review

16 Scopus citations


Abstract We describe the derivation of order conditions, without restrictions on stage order, for general linear methods for ordinary differential equations. This derivation is based on the extension of the Albrecht approach proposed in the context of Runge-Kutta and composite and linear cyclic methods. This approach was generalized by Jackiewicz and Tracogna to two-step Runge-Kutta methods, by Jackiewicz and Vermiglio to general linear methods with external stages of different orders, and by Garrappa to some classes of Runge-Kutta methods for Volterra integral equations with weakly singular kernels. This leads to general order conditions for many special cases of general linear methods such as diagonally implicit multistage integration methods, Nordsieck methods, and general linear methods with inherent Runge-Kutta stability. Exact coefficients for several low order methods with some desirable stability properties are presented for illustration.

Original languageEnglish (US)
Article number10156
Pages (from-to)44-64
Number of pages21
JournalJournal of Computational and Applied Mathematics
StatePublished - Dec 15 2015


  • General linear methods
  • Nordsieck methods
  • Order conditions
  • Two-step Runge-Kutta formulas

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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