A two-level correction method in space and time based on Crank–Nicolson scheme for Navier–Stokes equations

Abstract

A fully discrete two-level scheme in space and time is presented for solving the two-dimensional time-dependent Navier–Stokes equations. The approximate solution u M ∈H M can be decomposed as the large eddy component v∈H m (m<M) and the small eddy component . We obtain the large eddy component v by applying the classical Crank–Nicolson (CN) scheme in a coarse-level subspace H m , while the small eddy component w is advanced by the usual semi-implicit Euler scheme by solving a linear equation in an orthogonal complement subspace . Analysis and some numerical experiments show that this two-level scheme can reach the same accuracy as the classical CN scheme with M 2 modes by choosing a suitable m. However, the two-level scheme will involve much less work.