A time-accurate, adaptive discretization for fluid flow problems

TL;DR

This paper introduces a simple, stable, and adaptive time discretization method for Navier-Stokes equations that improves accuracy without increasing computational complexity, suitable for fluid flow simulations.

Contribution

A novel two-step linear time filter applied to backward Euler enhances time accuracy and stability with minimal implementation changes.

Findings

Method achieves predicted convergence rates

Improves accuracy of flow quantity predictions

Maintains unconditional energy stability

Abstract

This report presents a low computational and cognitive complexity, stable, time accurate and adaptive method for the Navier-Stokes equations. The improved method requires a minimally intrusive modification to an existing program based on the fully implicit / backward Euler time discretization, does not add to the computational complexity, and is conceptually simple. The backward Euler approximation is simply post-processed with a two-step, linear time filter. The time filter additionally removes the overdamping of Backward Euler while remaining unconditionally energy stable, proven herein. Even for constant stepsizes, the method does not reduce to a standard / named time stepping method but is related to a known 2-parameter family of A-stable, two step, second order methods. Numerical tests confirm the predicted convergence rates and the improved predictions of flow quantities such as drag and lift.