Order reduction phenomenon for general linear methods

Michał Braś, Angelamaria Cardone, Zdzislaw Jackiewicz, Bruno Welfert

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


The order reduction phenomenon for general linear methods (GLMs) for stiff differential equations is investigated. It turns out that, similarly as for standard Runge–Kutta methods, the effective order of convergence for a large class of GLMs applied to stiff differential systems, is equal to the stage order of the method. In particular, it is demonstrated that the global error ‖e[n]‖ of GLMs of order p and stage order q applied to the Prothero–Robinson test problem y(t)=λ(y(t)−φ(t))+φ(t), t∈[t0,T], y(t0)=φ(t0), is O(hq)+O(hp) as h→0 and hλ→−∞. Moreover, for GLMs with Runge–Kutta stability which are A(0)-stable and for which the stability function R(z) of the underlying Runge–Kutta methods, (i.e., the corresponding RK methods which have the same absolute stability properties as the GLMs), is such that R(∞)≠1, the global error satisfies ‖e[n]‖=O(hq+1)+O(hp) as h→0 and hλ→−∞. These results are confirmed by numerical experiments.

Original languageEnglish (US)
Pages (from-to)94-114
Number of pages21
JournalApplied Numerical Mathematics
StatePublished - Sep 1 2017


  • General linear methods
  • Linear stability analysis
  • Order conditions
  • Order reduction phenomenon
  • Prothero–Robinson problem
  • Stiff differential systems

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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