Refactorization of a variable step, unconditionally stable method of Dahlquist, Liniger and Nevanlinna

TL;DR

This paper demonstrates that a complex, unconditionally stable variable step method can be simplified to backward Euler with additional steps, making it easier to implement and more accurate for stiff simulations.

Contribution

It proves the equivalence of the Dahlquist, Liniger and Nevanlinna method to backward Euler with added steps, simplifying implementation and improving accuracy.

Findings

Method is equivalent to backward Euler with added steps

Simplifies implementation in complex and legacy codes

Increases accuracy over first-order and constant step second-order methods

Abstract

The one-leg, two-step time-stepping scheme proposed by Dahlquist, Liniger and Nevanlinna has clear advantages in complex, stiff numerical simulations: unconditional $G$-stability for variable time-steps and second-order accuracy. Yet it has been underutilized due, partially, to its complexity of direct implementation. We prove herein that this method is equivalent to the backward Euler method with pre- and post arithmetic steps added. This refactorization eases implementation in complex, possibly legacy codes. The realization we develop reduces complexity, including cognitive complexity and increases accuracy over both first order methods and constant time steps second order methods.