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Algebraically stable general linear methods and the G-matrix

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Abstract

The standard algebraic stability condition for general linear methods (GLMs) is considered in a modified form, and connected to a branch of Control Theory concerned with the discrete algebraic Riccati equation (DARE). The DARE theory shows that, for an algebraically stable method, there is a minimal G-matrix, G *, satisfying an equation, rather than an inequality. This result, and another alternative reformulation of algebraic stability, are applied to construct new GLMs with 2 steps and 2 stages, one of which has order p=4 and stage order q=3. The construction process is simplified by method-equivalence, and Butcher’s simplified order conditions for the case p≤q+1.

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Authors and Affiliations

  1. Department of Mathematics, University of Bath, Claverton Down, Bath, BA2 7AY, UK

    L. L. Hewitt & A. T. Hill

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  1. L. L. Hewitt
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  2. A. T. Hill
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Correspondence to L. L. Hewitt.

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Communicated by S. Nørsett.

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Hewitt, L.L., Hill, A.T. Algebraically stable general linear methods and the G-matrix. Bit Numer Math 49, 93–111 (2009). https://doi.org/10.1007/s10543-008-0207-7

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  • DOI: https://doi.org/10.1007/s10543-008-0207-7

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