Divergence-Free Wavelet Projection Method for Incompressible Viscous Flow on the Square

We present a wavelet numerical scheme for the discretization of two-dimensional Navier--Stokes equations with Dirichlet boundary condition on the square. This work is an extension to nonperiodic boundary conditions of the previous method of Deriaz and Perrier [E. Deriaz and V. Perrier, Multiscale Model. Simul. , 7 (2008), pp. 1101--1129]. Here the temporal discretization is borrowed from the projection method. The projection operator is defined through a discrete Helmholtz--Hodge decomposition using divergence-free wavelet bases; this prevents the use of a Poisson solver as in usual methods, while improving the accuracy of the boundary condition. The stability and precision order of the new method are stated in the linear case of Stokes equations, confirmed by numerical experiments. Finally, the effectiveness, stability, and accuracy of the method are validated by simulations conducted on the benchmark problem of lid-driven cavity flow at Reynolds number $Re=1000$ and $Re=10000$.