Projection method (fluid dynamics): Difference between revisions

Revision as of 02:21, 1 March 2009

Revision as of 16:43, 3 March 2009

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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-–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 Decompositiondecomposition. It states that the vector field <math>\mathbf{u}</math> defined on a 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 = 3^rd | publisher = [[Springer Science+Business Media|Springer-Verlag]] | year = 2000 | isbn = 0387979182}}</ref> Thus,

:<math>

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This is the essence of solenoidal projection method for solving incompressible

Navier-&ndash;Stokes equations.

==Chorin's projection method==