Editing Computational fluid dynamics (section)

===Solution algorithms===

Discretization in the space produces a system of [[ordinary differential equations]] for unsteady problems and algebraic equations for steady problems. Implicit or semi-implicit methods are generally used to integrate the ordinary differential equations, producing a system of (usually) nonlinear algebraic equations.  Applying a [[Newton's method#Nonlinear systems of equations|Newton]] or [[Fixed point iteration|Picard]] iteration produces a system of linear equations which is nonsymmetric in the presence of advection and indefinite in the presence of incompressibility. Such systems, particularly in 3D, are frequently too large for direct solvers, so iterative methods are used, either stationary methods such as [[Successive over-relaxation|successive overrelaxation]] or [[Krylov subspace]] methods. Krylov methods such as [[Generalized minimal residual method|GMRES]], typically used with [[Preconditioner|preconditioning]], operate by minimizing the residual over successive subspaces generated by the preconditioned operator.

[[Multigrid method|Multigrid]] has the advantage of asymptotically optimal performance on a number of problems. Traditional{{According to whom|date=November 2010}} solvers and preconditioners are effective at reducing high-frequency components of the residual, but low-frequency components typically require a number of iterations to reduce. By operating on multiple scales, multigrid reduces all components of the residual by similar factors, leading to a mesh-independent number of iterations.{{Citation needed|date=November 2010}}

For indefinite systems, preconditioners such as [[incomplete LU factorization]], [[Additive Schwarz method|additive Schwarz]], and [[Multigrid method|multigrid]] perform poorly or fail entirely, so the problem structure must be used for effective preconditioning.<ref>{{cite journal |last1=Benzi |first1=Michele |last2=Golub |first2=Gene H. |last3=Liesen |first3=Jörg |title=Numerical solution of saddle point problems |journal=Acta Numerica |date=May 2005 |volume=14 |pages=1–137 |doi=10.1017/S0962492904000212 |bibcode=2005AcNum..14....1B |s2cid=122717775 |citeseerx=10.1.1.409.4160 }}</ref> Methods commonly used in CFD are the [[SIMPLE algorithm|SIMPLE]] and [[Uzawa iteration|Uzawa algorithms]] which exhibit mesh-dependent convergence rates, but recent advances based on block LU factorization combined with multigrid for the resulting definite systems have led to preconditioners that deliver mesh-independent convergence rates.<ref>{{cite journal |last1=Elman |first1=Howard |last2=Howle |first2=V.E. |last3=Shadid |first3=John |last4=Shuttleworth |first4=Robert |last5=Tuminaro |first5=Ray |title=A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier–Stokes equations |journal=Journal of Computational Physics |date=January 2008 |volume=227 |issue=3 |pages=1790–1808 |doi=10.1016/j.jcp.2007.09.026 |bibcode=2008JCoPh.227.1790E |osti=920807 |s2cid=16365489 |url=https://digital.library.unt.edu/ark:/67531/metadc902332/ }}</ref>