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

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) is an element of H-M can be decomposed as the large eddy component v is an element of H-m(m < M) and the small eddy component w is an element of H-m(perpendicular to). 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 H. m. 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.