Discrete Operator Splitting
From CFD-Wiki
Different from many other pressure-velocity decoupling method, in Discrete Operator Splitting (DOS) method, the fully coupled discrete system is first obtained (with all boundary conditions taken into account). Next, the coupled system is split, sufficiently and necessarily, into two smaller (more importantly, well-natured) subsytems, which are coupled through source terms. Then, these two subsystems are iterated within each time marching step. DOS requires no numerical boundary conditions. The method is described as follows:
Discretization of the momentum and continuity equation eventually leads to the following system
![\left[
\begin{matrix}
A & G\\
D & 0
\end{matrix}
\right]
\left\{
\begin{matrix}
u \\
p \\
\end{matrix}
\right\}
=
\left\{
\begin{matrix}
S_u \\
S_p \\
\end{matrix}
\right\}](/W/images/math/d/e/2/de24c1f3cdcd0a7cfee1fc2a89e2335c.png)
where 
in the momentum equation is the coefficient matrix for
the velocity, 
is the coefficient matrix for the pressure, 
in the continuity equation is the coefficient matrix for the
velocity, and 
and 
are the right-hand-side known
vectors for the momentum and the continuity equations,
respectively. It is well known that such a system is
ill-conditioned, which causes difficulty to directly or
iteratively solve the whole large system simultaneously. We
split matrix into the diagonal part (
) and the
off-diagonal part
![\Leftrightarrow u+A^{-d}Gp = A^{-d}[S_u-(A-A^d)u]](/W/images/math/7/f/7/7f78267e9ddc1f4bbea9702b26491949.png)
![\Rightarrow
Du+DA^{-d}Gp = DA^{-d}[S_u-(A-A^d)u],](/W/images/math/7/b/d/7bd36eb1172da77aed6b3a517e3abd9b.png)
where 
stands for the inverse of . Let the continuity
equation be
incorporated into the above momentum equation and let the original
momentum equation
retained, we obtain the following system
![DA^{-d}Gp = -S_p+DA^{-d}[S_u-(A-A^d)u],](/W/images/math/0/f/2/0f2cf820fe2adc4758f1dc742563c495.png)
For convenience, we define
so that the discrete forms of the momentum equation and the continuity equation become
Furthermore, let's define
finally we have two well-posed subsystems
So far it has been proved that the original indefinite system implies the two definite subsystems. The reverse MUST be proved, so that they are equivalent to each other. The reverse can be found in the external link of this article.
In CCPPE (Consistent Continuous Pressure Poisson Equation), the continuous pressure poisson equation is first derived and assigned with numerical boundary condtions (not physical boundary conditions), and the resulting system is involved with linear iteration only (suppose the nonlinear term is already linearized in time). In contrast, the DOS involves source-term iteration and linear iteration. However, while in actual practice a strict convergence criterion is imposed on the source-term iteration, only few prescibed number of iterations are imposed on the linear iteration inside the source-term iteration. Actual practice shows that when the solution reaches the level that the source-term iteration is nearly complete, the same solution also reaches the level that the criterion for the linear iteration is easily met. Hence, apart from offering simplicity and no requirement of numerical boundary conditions, DOS is efficient.
