Robust Linearly-Implicit Backward Difference Formulas for Navier–Stokes Equations: Computations of Steady and Unsteady Flows

Abstract

We propose a linearly-implicit method (called LBDFT) to solve the incompressible Navier–Stokes equations. Linearly-implicit methods have an algorithmic complexity that lies between fully-implicit and semi-implicit time-stepping schemes. In LBDFT, the nonlinear advection in the Navier–Stokes equations is split into three linear terms using a Taylor series expansion. One term is taken explicitly, and the other two are updated with the linear diffusion term at each time step. An additional variant of linearly-implicit methods, referred to as LBDFE, was also included in this study. It is based on an extrapolation formula similar to those used in semi-implicit methods. For the sake of comparison, we recall a set of fully-implicit and semi-implicit time-stepping methods, for the most part based on backward differentiation formulae (BDF). These methods are compared in terms of accuracy, stability, computing time, and ability to compute various flows. We first use a two-dimensional manufactured problem to assess the order convergence (in time) of the methods. The second test case is a two-dimensional unsteady flow around a cylinder at $Re=100$. We observed that linearly-implicit methods are more CPU efficient compared to fully-implicit BDF, both at second- and third-order accuracy. Our third test case explores the ability of the methods to compute steady flows at high Reynolds numbers, in our case a steady two-dimensional lid-driven cavity for $Re=\mathrm{1,000}$ to $65,000$. Our LBDFT method was the most efficient for capturing these flows, with solutions almost identical to published results, while LBDFE methods do not work at all. LBDFT was able to compute steady cavity flows for $Re=100,000$ to $500,000$. Our last test case explores unsteady flows at large Reynolds numbers. Both LBDFT and LBDFE methods do not require any stabilization strategies for an unsteady two-dimensional lid-driven cavity at $Re=100,000$. We generated so-called Schlieren plots, which are comparable with the vorticity plots found in the literature. It was observed that the linearly-implicit methods allow significantly larger critical time steps (approximately 40–50 times higher) compared to the semi-implicit methods, the latter needing a grad-div stabilization term to maintain their stability.