# Discrete Operator Splitting

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. The method is described as follows.

Discretization of the momentum and continuity equation eventually leads to the following system

$\left[ \begin{matrix} A & G\\ D & 0 \end{matrix} \right] \left\{ \begin{matrix} u \\ p \\ \end{matrix} \right\} = \left\{ \begin{matrix} S_u \\ S_p \\ \end{matrix} \right\}$

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. And 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.