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Abstract: Among the methods for solving ODE-IVPs, the class of General Linear MethodsGLMs is able to encompass most of them, ranging from Linear MultistepFormulae LMF to RK formulae. Moreover, it is possible to obtain methods ableto overcome typical drawbacks of the previous classes of methods. For example,order barriers for stable LMF and the problem of order reduction for RKmethods. Nevertheless, these goals are usually achieved at the price of ahigher computational cost. Consequently, many efforts have been made in orderto derive GLMs with particular features, to be exploited for their efficientimplementation. In recent years, the derivation of GLMs from particularBoundary Value Methods BVMs, namely the family of Generalized BDF GBDF, hasbeen proposed for the numerical solution of stiff ODE-IVPs. In particular, thisapproach has been recently developed, resulting in a new family of L-stableGLMs of arbitrarily high order, whose theory is here completed and fullyworked-out. Moreover, for each one of such methods, it is possible to define acorresponding Blended GLM which is equivalent to it from the point of view ofthe stability and order properties. These blended methods, in turn, allow thedefinition of efficient nonlinear splittings for solving the generated discreteproblems. A few numerical tests, confirming the excellent potential of suchblended methods, are also reported.



Autor: Luigi Brugnano, Cecilia Magherini

Fuente: https://arxiv.org/



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