The construction of practical general linear methods. (English) Zbl 1046.65054

The purpose of the material is to give a framework for the derivation of diagonally implicit general linear methods possesing the property of “inherent Runge-Kutta stability” (IRKS).
These methods are chosen in such a way that the stage order \(q\) and the order \(p\) are equal and the information passed from step to step takes the form of a Nordsieck vector, this choice making possible to obtain asymptotically correct error estimates, high order interpolants and a convenient means of changing stepsize.
The first section contains a brief introduction to the wide class of the methods characterized by the IRKS property.
A review of general linear methods is given in the second section. One presents also a reformulation of the IRKS property which is a more general form than that used by W. M Wright [Numer. Algorithms 31, 381–399 (2001; Zbl 1016.65049)]. This section contains a survey of order conditions, stability and other related properties.
The third section focusses on the transformations between method arrays in order to find methods with the IRKS property. Details of the relationship between stability matrices and stability functions of the original and transformed methods and also a result which aids in the process of deriving methods, are exposed.
Canonical forms of methods are developed in section four, where the main theorem of the work is stated, which can generate algorithms used to derive a method.
Section five is devoted to show that the methods developed by J. C. Butcher [BIT 41, 240–264 (2001; Zbl 0983.65085)] and by J. C. Butcher and W. M. Wright [Appl. Numer. Math. 44, 313–327 (2003; Zbl 1016.65048)] are special cases of the canonical class of methods with IRKS property which are generated in this paper.
In the sixth section a discussion is given of how to derive some good methods and about the main allenge to obtain optimal ones.
The appendix contains tools and notations which are used. The properties of doubly companion matrices and other classes of matrices which play a role in the present paper, and the alternative conditions for a matrix to have zero spectral radius are also reviewed.


65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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