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

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.