The time-efficient DLN algorithms for the Navier-Stokes equations

Dahlquist, Liniger, and Nevanlinna proposed a two-step time-stepping scheme for systems of ordinary differential equations (ODEs) in 1983. The little-explored variable time-stepping scheme has advantages in numerical simulations for its fine properties such as unconditional G-stability and second-order accuracy. We simplify its implementation through time filters (pre-filter and post-filter) on a certain first-order implicit method. The adaptivity algorithm for this variable time-stepping scheme, highly reducing computation cost as well as keeping time accuracy, has been applied to systems of ODEs and flow models. We have applied fully-implicit and semi-implicit DLN algorithms to Navier-Stokes equations (NSE) for stability and error analysis. Moreover the DLN-ensembele algorithm has been proposed to solve multiple NSEs at one time. Some time adaptivity mechanisms for the variable steps DLN are designed to improve the time efficiency in the numerical simulations.