Blended General Linear Methods based on Boundary Value Methods in the Generalized BDF family

Keywords: Numerical methods for ordinary differential equations, General
Linear Methods, Boundary Value Methods (BVMs), Generalized Backward
Differentiation Formulae (GBDF), Blended Implicit Methods, blended iteration.

Abstract: Among the methods for solving ODE-IVPs, the class of General Linear Methods
(GLMs) is able to encompass most of them, ranging from Linear Multistep Formulae
(LMF) to RK formulae. Moreover, it is possible to obtain methods able to overcome
typical drawbacks of the previous classes of methods. For example, order barriers for
stable LMF and the problem of order reduction for RK methods. Nevertheless, these goals
are usually achieved at the price of a higher computational cost. Consequently, many
efforts have been made in order to derive GLMs with particular features, to be exploited
for their efficient implementation.
In recent years, the derivation of GLMs from particular Boundary Value Methods (BVMs),
namely the family of Generalized BDF (GBDF), has been proposed for the numerical
solution of stiff ODE-IVPs [11]. In particular, in [8], this approach has been recently
developed, resulting in a new family of L-stable GLMs of arbitrarily high order, whose
theory is here completed and fully worked-out. Moreover, for each one of such methods, it
is possible to define a corresponding Blended GLM which is equivalent to it from the point
of view of the stability and order properties. These blended methods, in turn, allow the
definition of efficient nonlinear splittings for solving the generated discrete problems.
A few numerical tests, confirming the excellent potential of such blended methods, are also
reported.

 

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