SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Contributed Talk

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On the construction of sequential second derivative general linear methods

G. Hojjati and A. Abdi

Abstract
Second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of SGLMs have been introduced in [1] in four types, together with their intended applications (for nonstiff or stiff ODEs) and architectures (sequential or parallel). Order barriers for parallel SDIMSIMs, of type four and generalized type four and also for sequential SDIMSIMs of type 2 with Runge-Kutta stability (RKS) property have been discussed in [1, 2]. The coefficients of SGLMs are given by six matrices, instead of four matrices for GLMs, which are obtained by solving linear system of order conditions and usually nonlinear system of RKS conditions. In this survey, we introduce a new formula which causes the order conditions and the stability matrix of SGLMs take simpler form and decreases the complexity of finding coefficients matrices. The constructed $A$-stable methods are also $L$-stable.

Bibliography
[1] A. Abdi and G. Hojjati, An extension of general linear methods, Numer. Algor. 57 (2011), pp. 149-167.
[2] A. Abdi and G. Hojjati, Maximal order of second derivative general linear methods with Runge-kutta stability, Appl. Numer. Math. 61 (2011), pp. 1046-1058.

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