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<ref>▼
In [[computational fluid dynamics]], the '''projection method''', also called '''Chorin's projection method''', is an effective means of [[Numerical analysis|numerically]] solving time-dependent [[incompressible flow|incompressible fluid-flow]] problems. It was originally introduced by [[Alexandre Chorin]] in 1967<ref>
{{Citation
| surname1 = Chorin
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| volume = 73
| year = 1967
| issue = 6
| pages = 928–931
| doi = 10.1090/S0002-9904-1967-11853-6
| url =http://math.berkeley.edu/~chorin/chorin67.pdf
}}</ref>
{{Citation
| surname1 = Chorin
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| volume = 22
| year = 1968
| issue = 104
| pages = 745–762
| url =
| doi=10.1090/s0025-5718-1968-0242392-2
| doi-access = free
▲ }}</ref>
as an efficient means of solving the incompressible [[Navier-Stokes equation]]s. The key advantage of the projection method is that the computations of the [[Flow velocity|velocity]] and the pressure fields are decoupled.
==The algorithm==
The algorithm of the projection method is based on the [[Helmholtz decomposition]] (sometimes called Helmholtz-Hodge decomposition) of any vector field into a [[solenoidal field|solenoidal]] part and an [[irrotational field|irrotational]] part. Typically, the algorithm consists of two stages. In the first stage, an intermediate velocity that does not satisfy the incompressibility constraint is computed at each time step. In the second, the pressure is used to project the intermediate velocity onto a space of divergence-free velocity field to get the next update of velocity and pressure.
==Helmholtz–Hodge decomposition==
The theoretical background of projection type method is the decomposition theorem of [[Olga Aleksandrovna Ladyzhenskaya|Ladyzhenskaya]] sometimes referred to as Helmholtz–Hodge Decomposition or simply as Hodge decomposition. It states that the vector field <math>\mathbf{u}</math> defined on a [[simply connected space|simply connected]] ___domain can be uniquely decomposed into a divergence-free ([[Solenoidal vector field|solenoidal]]) part <math>\mathbf{u}_{\text{sol}}</math> and an [[Conservative vector field#Irrotational vector fields|irrotational]] part <math>\mathbf{u}_{\text{irrot}}</math>.
Thus,
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:<math>
\frac {\mathbf{u}^{n+1} - \mathbf{u}^*} {\Delta t} = - \frac {1}{\rho} \, \nabla p ^{n+1}
</math>
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\nabla ^2 p^{n+1} = \frac {\rho} {\Delta t} \, \nabla\cdot \mathbf{u}^*
</math>
It is instructive to note that the equation written as
:<math>
\mathbf{u}^* = \mathbf{u}^{n+1} + \frac {\Delta t}{\rho} \, \nabla p ^{n+1}
</math>
is the standard Hodge decomposition if boundary condition for <math>\,p</math> on the ___domain boundary, <math>\partial \Omega</math> are <math>\nabla p^{n+1}\cdot \mathbf{n} = 0</math>. In practice, this condition is responsible for the errors this method shows close to the boundary of the ___domain since the real pressure (i.e., the pressure in the exact solution of the Navier-Stokes equations) does not satisfy such boundary conditions.
For the explicit method, the boundary condition for <math>\mathbf{u}^*</math> in equation (1) is natural. If <math>\mathbf{u}\cdot \mathbf{n} = 0</math> on <math>\partial \Omega</math>, is prescribed, then the space of divergence-free vector fields will be orthogonal to the space of irrotational vector fields, and from equation (2) one has
:<math>
\frac {\partial p^{n+1}} {\partial n} = 0 \qquad \text{on} \quad \partial \Omega
</math>
The explicit treatment of the boundary condition may be circumvented by using a [[staggered grid]] and requiring that <math>\nabla\cdot \mathbf{u}^{n+1}</math> vanish at the pressure nodes that are adjacent to the boundaries.
A distinguishing feature of Chorin's projection method is that the velocity field is forced to satisfy a discrete continuity constraint at the end of each time step.
== General method ==
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