### Chorin-Temem Projection method

#### by Yi Zhang

This post is a expansion of the topic raised by two old posts, in which the projection method and Sobolev space are introduced. During the analysis of Navier-Stokes equation of incompressible flow, the function space is referred to as

to which a Hilbert space is endowed with norm:

This induced vector fields like:

and

is the closure of with respect to norm ;

is the closure of with respect to norm of .

So is a closed subspace in , and the orthogonal decomposition can be performed on it:

It was proved by Ladyzhenskaya that, following Helmholtz decomposition, there is:

To apply this conclusion to the velocity field of a incompressible flow, assign

as the velocity field, then we have

;

.

And the mapping is called projection due its obvious geometric implication.

In Chorin-Temam projection method, is first calculated as without divergence-free constraint:

where and can be selected on the choices of using forward or backward etc. plan.

then is obtained from as its projection using pressure as in the decomposition:

or just

One can immediately see that this projection method is first order in time, and velocity only satisfies the Dirichlet boundary condition in normal direction, those are some drawbacks of this method.