Long-time $L^2$&$H^1$-stability of the Family of DLN Methods for the Two-dimensional Incompressible Navier-Stokes Equations

Abstract

In this report, we study the long-time stability of the family of one-leg DLN methods for the two-dimensional incompressible Navier-Stokes equations. The family of DLN methods (with one parameter $\theta$), non-linear energy stable ($G$-stable) and second-order accurate under arbitrary time grids, has been widely applied to the simulations of various fluid models with success. We derive a new version of the $G$-stability identity for the family of DLN methods under uniform time grids and mild time constraints. Then we utilize this crucial auxiliary tool and the discrete uniform Gr\"onwall inequality lemma to prove the uniform-in-time stability of the numerical solutions. Essentially, the bounds are independent of the time interval and the initial conditions, consistent with the theories of the continuous case.