Projection method (fluid dynamics): Difference between revisions

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The{{Short '''projectiondescription|Method method'''for 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>}}

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 = TemamChorin

| given1 = RA. J.

| title = UneThe m&eacute;thodenumerical d'approximationsolution desof solutions des &eacute;quationsthe Navier-Stokes, equations for an incompressible fluid

| journal = Bull. SocAm. Math. FranceSoc.

| volume = 9873

| year = 19681967

| volumeissue = 226

| pages = 745928&ndash;762931

| doi = 10.1090/s0025S0002-57189904-19681967-024239211853-26

| url =http://math.berkeley.edu/~chorin/chorin67.pdf

}}</ref><ref>

{{Citation

| surname1 = Chorin

| given1 = A. J.

| title = Numerical Solution of the Navier-Stokes Equations

| journal = Math. Comp.

| volume = 22

| year = 1968

| pagesissue = 115&ndash;152104

| urlpages = 745&ndash;762

| 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.

| doi=10.1090/s0025-5718-1968-0242392-2

| doi-access = free

, }}</ref>

}}</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&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,

Thus,

:<math>

\mathbf{u} = \mathbf{u}_{\text{sol}} + \mathbf{u}_{\text{irrot}} = \mathbf{u}_{\text{sol}} + \nabla \phi

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</math>

In [[Alexandre Chorin|Chorin]]'s original version of the projection method, one first computes an intermediate velocity, <math>\mathbf{u}^*</math>, explicitly using the momentum equation by ignoring the pressure gradient term:

,<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 =

| doi=10.1090/s0025-5718-1968-0242392-2

}}</ref> one first computes an intermediate velocity, <math>\mathbf{u}^*</math>, explicitly using the momentum equation by ignoring the pressure gradient term:

:<math>

Line 71 ⟶ 79:

:<math>

\frac {\mathbf{u}^{n+1} - \mathbf{u}^*} {\Delta t} = - \frac {1}{\rho} \, \nabla p ^{n+1}

</math>

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[[Category:Computational fluid dynamics]]

[[Category:Mathematical physics]]

[[Category:Computational physics]]