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Discrete Operator Splitting

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</math>
</math>
 +
where <math>A</math> in the momentum equation is the coefficient matrix for
 +
the velocity, <math>G</math> is the coefficient matrix for the pressure, <math>D</math>
 +
in the continuity equation is the coefficient matrix for the
 +
velocity, and <math>S_u</math> and <math>S_p</math> 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 <math>A</math> into the diagonal part (<math>A^d</math>) and the
 +
off-diagonal part
 +
:<math>
 +
(A-A^d+A^d)u+Gp = S_u
 +
</math>
 +
:<math>
 +
\Leftrightarrow A^d u+Gp = S_u-(A-A^d)u
 +
</math>
 +
:<math>
 +
\Leftrightarrow u+A^{-d}Gp = A^{-d}[S_u-(A-A^d)u]
 +
</math>
 +
:<math>
 +
\Rightarrow
 +
Du+DA^{-d}Gp = DA^{-d}[S_u-(A-A^d)u],
 +
</math>
 +
where <math>A^{-d}</math> stands for the inverse of <math>A^d</math>. Let the continuity
 +
equation be
 +
incorporated into the above momentum equation and let the original
 +
momentum equation
 +
retained, we obtain the following system
 +
:<math>
 +
DA^{-d}Gp = -S_p+DA^{-d}[S_u-(A-A^d)u],
 +
</math>
 +
:<math>
 +
Au = S_u-Gp
 +
</math>
 +
For convenience, we define
 +
:<math>
 +
D^{*} \equiv D A^{-d},
 +
</math>
 +
so that the discrete forms of the momentum equation and the
 +
continuity equation become
 +
:<math>
 +
D^{*}Gp = -(S_p-D^*S_u)-D^*(Au-A^d u)
 +
</math>
 +
:<math>
 +
Au = S_u-Gp
 +
</math>
 +
Furthermore, let's define
 +
:<math>
 +
L \equiv D^*G
 +
</math>
 +
:<math>
 +
S^*_p \equiv S_p-D^*S_u,
 +
</math>
 +
finally we have two well-posed subsystems
 +
:<math>
 +
Lp = -S^*_p-D^*(Au-A^d u)\equiv b_p(u)
 +
</math>
 +
:<math>
 +
Au = S_u-Gp\equiv b_u(p)
 +
</math>
 +
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. 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.
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.

Revision as of 16:53, 26 January 2008

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\}

where A in the momentum equation is the coefficient matrix for the velocity, G is the coefficient matrix for the pressure, D in the continuity equation is the coefficient matrix for the velocity, and S_u and S_p 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 A into the diagonal part (A^d) and the off-diagonal part

(A-A^d+A^d)u+Gp = S_u
\Leftrightarrow A^d u+Gp = S_u-(A-A^d)u
\Leftrightarrow u+A^{-d}Gp = A^{-d}[S_u-(A-A^d)u]
\Rightarrow Du+DA^{-d}Gp = DA^{-d}[S_u-(A-A^d)u],

where A^{-d} stands for the inverse of A^d. 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],
Au = S_u-Gp

For convenience, we define

D^{*} \equiv D A^{-d},

so that the discrete forms of the momentum equation and the continuity equation become

D^{*}Gp = -(S_p-D^*S_u)-D^*(Au-A^d u)
Au = S_u-Gp

Furthermore, let's define

L \equiv D^*G
S^*_p \equiv S_p-D^*S_u,

finally we have two well-posed subsystems

Lp = -S^*_p-D^*(Au-A^d u)\equiv b_p(u)
Au = S_u-Gp\equiv b_u(p)

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. 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.


External link

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