Blended General Linear Methods based on Boundary Value Methods in the GBDF family

TL;DR

This paper develops a new family of high-order, L-stable General Linear Methods derived from Boundary Value Methods, specifically the GBDF family, and introduces blended versions for improved efficiency in solving stiff ODE initial value problems.

Contribution

It introduces a novel class of high-order, L-stable GLMs based on GBDF boundary value methods and their blended variants for efficient stiff ODE solving.

Findings

The new GLMs are of arbitrarily high order and L-stable.

Blended methods enable efficient nonlinear problem splitting.

Numerical tests demonstrate the methods' excellent potential.

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. In particular, 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.