Projection method (fluid dynamics)

The projection method is an effective means of numerically solving time-dependent incompressible fluid-flow problems. It was originally introduced by Alexandre Chorin in 1967 as an efficient means of solving the incompressible Navier-Stokes equations. The key advantage of the projection method is that the computations of the velocity and the pressure fields are decoupled.

The algorithm of projection method is based on the Helmholtz decomposition (sometimes called Helmholtz-Hodge decomposition) of any vector field into a solenoidal part and an 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.

The theoretical background of projection type method is the decomposition theorem of Ladyzhenskaya sometimes referred to as Helmholtz–Hodge Decomposition or simply as Hodge decomposition. It states that the vector field ${\displaystyle \mathbf {u} }$ defined on a simply connected domain can be uniquely decomposed into a divergence-free (solenoidal) part ${\displaystyle \mathbf {u} _{\text{sol}}}$ and an irrotational part ${\displaystyle \mathbf {u} _{\text{irrot}}}$. .

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