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In [[computational fluid dynamics]] (CFD), '''SIMPLE algorithm''' is a widely used [[numerical algorithm|numerical procedure]] to solve the [[Navier-Stokes equation]]s. ''SIMPLE'' is an acronym for Semi-Implicit Method for Pressure Linked Equations.

The SIMPLE algorithm was developed by Prof. [[Brian Spalding]] and his student [[Suhas Patankar]] at [[Imperial College London|Imperial College]], London in the early 1970s. Since then it has been extensively used by many researchers to solve different kinds of fluid flow and heat transfer problems.<ref>{{cite web|url=https://engineering.purdue.edu/ME608/webpage/project-reports/SIMPLE-DrivenCavity.pdf |title=SIMPLE solver for driven cavity flow problem |format=PDF |date= |accessdate=2011-08-21}}</ref><ref>{{cite web|url=http://www.opensourcecfd.com/conference2008/2007/media/proceedings/OFIC-07_ManganiLuca.pdf?phpMyAdmin=f7f6d5b5be922e9db00b8b978b532798 |title=Heat transfer applications in turbomachinery |date= |accessdate=2011-08-21}}</ref>

Many popular books on computational fluid dynamics discuss the SIMPLE algorithm in detail.<ref>{{cite book |last=Patankar |first=S. V. | authorlink = Suhas Patankar |title=Numerical Heat Transfer and Fluid Flow |publisher=[[Taylor & Francis]] |year=1980 |isbn=978-0-89116-522-4}}</ref><ref>{{cite book |last=Ferziger |first=J. H. | authorlink = J. H. Ferziger |coauthors = Peric, M. |title=Computational Methods for Fluid Dynamics|publisher=[[ Springer-Verlag]] |year=2001 |isbn= 978-3-540-42074-3}}</ref> This algorithm forms the basis of Commercial CFD packages.<ref>{{cite web|url=http://www.salihnet.freeservers.com/engineering/cfd/cfd_history.html |title=History of CFD |publisher=Salihnet.freeservers.com |date= |accessdate=2011-08-21}}</ref>

A modified variant is the ''SIMPLER'' algorithm (SIMPLE Revised), that was introduced by Patankar in 1979.<ref>{{cite book |last=Tannehill|first=J. C.| authorlink = |coauthors = [[Dale A. Anderson|Anderson, D. A.]]; Pletcher, R. H. |title=Computational Fluid Mechanics and Heat Transfer |publisher=[[Taylor & Francis]] |year=1997 |isbn=}}</ref>

== Algorithm ==
The algorithm is [[iterative]]. The basic steps in the solution update are as follows:

#Set the boundary conditions.
#Compute the gradients of velocity and pressure.
#Solve the discretized momentum equation to compute the intermediate velocity field.
#Compute the uncorrected mass fluxes at faces.
#Solve the pressure correction equation to produce cell values of the pressure correction.
#Update the pressure field: <math> p^{k + 1} = p^k + \text{urf} \cdot p^{'} </math> where urf is the under-relaxation factor for pressure.
#Update the boundary pressure corrections <math> p_b^{'} </math>.
#Correct the face mass fluxes: <math>\dot m_f^{k + 1} = \dot m_f^{*} + \dot m_f^{'} </math>
#Correct the cell velocities: <math> \vec v^{k + 1} = \vec v^{*} - \frac{{\text{Vol} \ \nabla p^{'} }}{{\vec a_P^v }} </math> ; where <math> {\nabla p^{'} } </math> is the gradient of the pressure corrections, <math> {\vec a_P^v } </math> is the vector of central coefficients for the discretized linear system representing the velocity equation and Vol is the cell volume.
#Update density due to pressure changes.

==References==
{{reflist}}

[[Category:Computational fluid dynamics]]

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