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The '''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 and independently by [[Roger Temam]]<ref>
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{{Citation
| surname1 = Temam
| given1 = R.
| title = Une m&eacute;thode d'approximation des solutions des &eacute;quations Navier-Stokes,
| journal = Bull. Soc. Math. France
| volume = 98
| year = 1968
| pages = 115&ndash;152
| url =
}}</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 velocity and the pressure fields are decoupled.


==The algorithm==
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The algorithm of 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&ndash;Hodge decomposition==
The theoretical background of projection type method is the decomposition theorem of [[Olga Aleksandrovna Ladyzhenskaya|Ladyzhenskaya]] sometimes referred to as Helmholtz&ndash;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>.<ref>{{cite book | title = A Mathematical Introduction to Fluid Mechanics | author1 = Chorin, A. J. | author2 = J. E. Marsden | edition = 3rd | publisher = [[Springer Science+Business Media|Springer-Verlag]] | year = 1993 | isbn = 0-387-97918-2}}</ref> Thus,
 
:<math>
  \mathbf{u} = \mathbf{u}_{\text{sol}} + \mathbf{u}_{\text{irrot}} = \mathbf{u}_{\text{sol}} + \nabla \phi
</math>
 
since <math>\nabla \times \nabla \phi = 0</math> for some scalar function, <math>\,\phi</math>. Taking the
divergence of equation yields
 
:<math>
  \nabla\cdot \mathbf{u} = \nabla^2 \phi \qquad ( \text{since,} \; \nabla\cdot \mathbf{u}_{\text{sol}} = 0 )
</math>
 
This is a [[Poisson equation]] for the scalar function <math>\,\phi</math>. If the vector field <math>\mathbf{u}</math> is known, the above equation can be solved for the scalar function <math>\,\phi</math> and the divergence-free part of <math>\mathbf{u}</math> can be extracted using the relation
 
:<math>
  \mathbf{u}_{\text{sol}} = \mathbf{u} - \nabla \phi
</math>
 
This is the essence of solenoidal projection method for solving incompressible
Navier&ndash;Stokes equations.
 
==Chorin's projection method==
The incompressible Navier-Stokes equation (differential form of momentum equation) may be written as
:<math>
  \frac {\partial \mathbf{u}} {\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u} = - \frac {1}{\rho} \nabla p + \nu \nabla^2 \mathbf{u}
</math>
 
In Chorin's original version of the projection method
,<ref>
{{Citation
  | surname1 = Chorin
| given1 = A. J.
| title = Numerical Solution of the Navier-Stokes Equations
| journal = Math. Comp.
| volume = 22
| year = 1968
| pages = 745&ndash;762
| url =
}}</ref> the intermediate velocity, <math>\mathbf{u}^*</math>, is explicitly computed using the momentum equation ignoring the pressure gradient term:
 
:<math>
  \quad (1) \qquad \frac {\mathbf{u}^* - \mathbf{u}^n} {\Delta t} = -(\mathbf{u}^n \cdot\nabla) \mathbf{u}^n + \nu \nabla^2
  \mathbf{u}^n
</math>
 
where <math>\mathbf{u}^n</math> is the velocity at <math>\,n</math><sup>th</sup> time level. In the next (projection) step, we have
 
:<math>
  \frac {\mathbf{u}^{n+1} - \mathbf{u}^*} {\Delta t} = - \frac {1}{\rho} \, \nabla p ^{n+1}
</math>
 
Re-writing the above equation for the velocity at <math>\,(n+1)</math> level, we have
 
:<math>
  \quad (2) \qquad \mathbf{u}^{n+1} = \mathbf{u}^* - \frac {\Delta t}{\rho} \, \nabla p ^{n+1}
</math>
 
Computing the right-hand side of the above equation requires a knowledge of the pressure, <math>\,p</math>, at <math>\,(n+1)</math> level. This is obtained by taking the [[divergence]] and requiring that <math>\nabla\cdot \mathbf{u}^{n+1} = 0</math>, which is the divergence-free(continuity) condition, thereby deriving the following Poisson equation for <math>\,p^{n+1}</math>,
:<math>
  \nabla ^2 p^{n+1} = \frac {\rho} {\Delta t} \, \nabla\cdot \mathbf{u}^*
</math>
It is instructive to note that, the equation written in the following way
:<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> is <math>\nabla p^{n+1}\cdot \mathbf{n} = 0</math>.
 
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 field 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 ==
 
Typically the projection method operates as a two-stage fractional step scheme, a method which uses multiple calculation steps for each numerical time-step. In many projection algorithms, the steps are split as follows:
 
# First the system is progressed in time to a mid-time-step position, solving the above transport equations for mass and momentum using a suitable advection method. This is denoted the ''predictor'' step.
# At this point an initial projection may be implemented such that the mid-time-step velocity field is enforced as divergence free.
# The ''corrector'' part of the algorithm is then progressed. These use the time-centred estimates of the velocity, density, etc. to form final time-step state.
# A final projection is then applied to enforce the divergence restraint on the velocity field. The system has now been fully updated to the new time.
 
== References ==
{{reflist}}
 
[[Category:Computational fluid dynamics]]
[[Category:Mathematical physics]]
[[Category:Computational physics]]

Revision as of 16:12, 19 February 2014

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