### Chorin-Temem Projection method

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

$H(\text{div};\Omega)\triangleq\{v\in L^2(\Omega)|\nabla\cdot v\in L^2(\Omega)\}$

to which a Hilbert space is endowed with norm:

$\|v\|_{\text{div};\Omega}^2=\int_{\Omega}{v\cdot v}d\Omega+\int_{\Omega}{(\nabla\cdot v)^2}d\Omega=(v,v)+(\nabla\cdot v,\nabla\cdot v)$

This induced vector fields like:

$J(\Omega)\triangleq\{v\in D(\Omega)|\nabla\cdot v=0\}$

$J_0^1(\Omega)\triangleq\{v\in H_0^1(\Omega)|\nabla\cdot v=0\}$

$J_0(\Omega)\triangleq\{v\in L^2(\Omega)|\nabla\cdot v=0 \text{ in }\Omega, n\cdot v=0 \text{ on } \Gamma\}$

and

$H(\text{div};\Omega)$ is the closure of $D(\bar{\Omega})$ with respect to norm $\|\cdot\|_{\text{div};\Omega}$;

$J_0(\Omega)$ is the closure of $J(\Omega)$ with respect to norm of $L^2(\Omega)$.

So $J_0(\Omega)$ is a closed subspace in $L^2(\Omega)$, and the orthogonal decomposition can be performed on it:

$L^2(\Omega)=J_0(\Omega)\bigoplus J_0^{\perp}(\Omega)$

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

$J_0^{\perp}(\Omega)=\{w\in L^2(\Omega)|w=\nabla p,p\in H^1(\Omega)\}$

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

$L^2(\Omega)$ as the velocity field, then we have

$u=u^d+\nabla \phi,\forall u\in L^2(\Omega)$;

$\nabla\cdot u^d=0,n\cdot u^d|_{\Gamma}=0$.

And the mapping $\mathcal{P}: u^d=\mathcal{P}(u)$ is called projection due its obvious geometric implication.

In Chorin-Temam projection method, $u^{n+1}$ is first calculated as $u^{n+1/2}$ without divergence-free constraint:

$(u^{n+1/2}-u^n)/\Delta t+(u^{\ast}\cdot \nabla)u^{\ast\ast}-\nu \nabla^2 u^{\ast\ast}=f^{n+1},\text{ in }\Omega$

$u^{n+1/2}=u_D^{n+1},\text{ in }\Gamma$

where $u^{\ast}$ and $u^{\ast\ast}$ can be selected on the choices of using forward or backward etc. plan.

then $u^{n+1}$ is obtained from $u^{n+1/2}$ as its $\mathcal{P}$ projection using pressure $p$ as $\phi$ in the decomposition:

$(u^{n+1}-u^{n+1/2})/\Delta t+\nabla p^{n+1}=0\Leftrightarrow u^{n+1}=u^{n+1/2}-\nabla (\Delta t p^{n+1}),\text{ in }\Omega$

$\nabla\cdot u^{n+1}=0,\text{ in }\Omega$

$n\cdot u^{n+1}=n\cdot u_D^{n+1},\text{ in }\Gamma$

or just

$u^{n+1}=\mathcal{P}u^{n+1/2}$

$\Delta t\nabla p^{n+1}=(\mathcal{I}-\mathcal{P})u^{n+1/2}$

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.