Editing Computational fluid dynamics (section)

===Discretization methods===
{{further|Discretization of Navier–Stokes equations}}

The stability of the selected discretisation is generally established numerically rather than analytically as with simple linear problems.  Special care must also be taken to ensure that the discretisation handles discontinuous solutions gracefully.  The [[Euler equations (fluid dynamics)|Euler equations]] and [[Navier–Stokes equations]] both admit shocks and contact surfaces.

Some of the discretization methods being used are:

==== Finite volume method ====

{{Main|Finite volume method}}

The finite volume method (FVM) is a common approach used in CFD codes, as it has an advantage in [[Random-access memory|memory]] usage and solution speed, especially for large problems, high [[Reynolds number]] turbulent flows, and source term dominated flows (like combustion).<ref>{{cite book|last=Patankar|first=Suhas V.|author-link=Suhas Patankar|title=Numerical Heat Transfer and Fluid FLow|year=1980|publisher=Hemisphere Publishing Corporation
|isbn=978-0891165224}}</ref>

In the finite volume method, the governing partial differential equations (typically the Navier-Stokes equations, the mass and energy conservation equations, and the turbulence equations) are recast in a conservative form, and then solved over discrete control volumes. This [[discretization]] guarantees the conservation of fluxes through a particular control volume. The finite volume equation yields governing equations in the form,
:<math>\frac{\partial}{\partial t}\iiint Q\, dV + \iint F\, d\mathbf{A} = 0,</math>
where <math>Q</math> is the vector of conserved variables, <math>F</math> is the vector of fluxes (see [[Euler equations (fluid dynamics)|Euler equations]] or [[Navier–Stokes equations]]), <math>V</math> is the volume of the control volume element, and <math>\mathbf{A}</math> is the surface area of the control volume element.

==== Finite element method ====

{{Main|Finite element method}}

The finite element method (FEM) is used in structural analysis of solids, but is also applicable to fluids.  However, the FEM formulation requires special care to ensure a conservative solution. The FEM formulation has been adapted for use with fluid dynamics governing equations.<ref>{{Cite web |title=Detailed Explanation of the Finite Element Method (FEM) |url=https://www.comsol.com/multiphysics/finite-element-method |access-date=2022-07-15 |website=www.comsol.com}}</ref><ref name=":0">{{Cite book |last=Anderson |first=John David |url=https://books.google.com/books?id=phG_QgAACAAJ |title=Computational Fluid Dynamics: The Basics with Applications |date=1995 |publisher=McGraw-Hill |isbn=978-0-07-113210-7 |language=en}}</ref> Although FEM must be carefully formulated to be conservative, it is much more stable than the finite volume approach.<ref>{{cite journal| title=k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutions| last1=Surana| first1=K.A.| last2=Allu| first2=S.| last3=Tenpas| first3=P.W.| last4=Reddy| first4=J.N.| journal=International Journal for Numerical Methods in Engineering| volume=69| issue=6| pages=1109–1157|date=February 2007| doi=10.1002/nme.1801|bibcode = 2007IJNME..69.1109S | s2cid=122551159}}</ref> FEM also provides more accurate solutions for smooth problems comparing to FVM. <ref>{{cite journal |last1=Surana |first1=KS |last2=Allu |first2=S |last3=Tenpas |first3=PW |last4=Reddy |first4=JN |title=k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutions |journal=International Journal for Numerical Methods in Engineering |volume=69 |issue=6 |pages=1109–1157 |year=2007 |publisher=Wiley Online Library|doi=10.1002/nme.1801 |bibcode=2007IJNME..69.1109S }}</ref> Another advantage of FEM is that it can handle complex geometries and boundary conditions. However, FEM can require more memory and has slower solution times than the FVM.<ref>{{cite journal |last1=Surana |first1=KS |last2=Allu |first2=S |last3=Tenpas |first3=PW |last4=Reddy |first4=JN |title=k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutions |journal=International Journal for Numerical Methods in Engineering |volume=69 |issue=6 |pages=1109–1157 |year=2007 |publisher=Wiley Online Library|doi=10.1002/nme.1801 |bibcode=2007IJNME..69.1109S }}</ref>

In this method, a weighted residual equation is formed:

:<math>R_i = \iiint W_i Q \, dV^e</math>

where <math>R_i</math> is the equation residual at an element vertex <math>i</math>, <math>Q</math> is the conservation equation expressed on an element basis, <math>W_i</math> is the weight factor, and <math>V^{e}</math> is the volume of the element.

==== Finite difference method ====

{{Main|Finite difference method}}

The finite difference method (FDM) has historical importance<ref name=":0" /> and is simple to program.  It is currently only used in few specialized codes, which handle complex geometry with high accuracy and efficiency by using embedded boundaries or overlapping grids (with the solution interpolated across each grid).{{Citation needed|date=November 2010}} 
:<math> 
\frac{\partial Q}{\partial t}+
\frac{\partial F}{\partial x}+
\frac{\partial G}{\partial y}+
\frac{\partial H}{\partial z}=0
</math>
where <math>Q</math> is the vector of conserved variables, and <math>F</math>, <math>G</math>, and <math>H</math> are the fluxes in the <math>x</math>, <math>y</math>, and <math>z</math> directions respectively.

==== Spectral element method ====

{{Main|Spectral element method}}

Spectral element method is a finite element type method. It requires the mathematical problem (the partial differential equation) to be cast in a weak formulation. This is typically done by multiplying the differential equation by an arbitrary test function and integrating over the whole domain. Purely mathematically, the test functions are completely arbitrary - they belong to an infinite-dimensional function space. Clearly an infinite-dimensional function space cannot be represented on a discrete spectral element mesh; this is where the spectral element discretization begins. The most crucial thing is the choice of interpolating and testing functions. In a standard, low order FEM in 2D, for quadrilateral elements the most typical choice is the bilinear test or interpolating function of the form <math>v(x,y) = ax+by+cxy+d</math>. In a spectral element method however, the interpolating and test functions are chosen to be polynomials of a very high order (typically e.g. of the 10th order in CFD applications). This guarantees the rapid convergence of the method. Furthermore, very efficient integration procedures must be used, since the number of integrations to be performed in numerical codes is big. Thus, high order Gauss integration quadratures are employed, since they achieve the highest accuracy with the smallest number of computations to be carried out.
At the time there are some academic CFD codes based on the spectral element method and some more are currently under development, since the new time-stepping schemes arise in the scientific world.

==== Lattice Boltzmann method ====

{{Main|Lattice Boltzmann methods}}

The lattice Boltzmann method (LBM) with its simplified kinetic picture on a lattice provides a computationally efficient description of hydrodynamics.
Unlike the traditional CFD methods, which solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy) numerically, LBM models the fluid consisting of fictive particles, and such particles perform consecutive propagation and collision processes over a discrete lattice mesh. In this method, one works with the discrete in space and time version of the kinetic evolution equation in the Boltzmann [[Bhatnagar–Gross–Krook operator|Bhatnagar-Gross-Krook (BGK)]] form.

==== Vortex method ====

The vortex method, also Lagrangian Vortex Particle Method, is a [[Meshfree methods|meshfree]] technique for the simulation of incompressible turbulent flows. In it, [[vorticity]] is discretized onto [[Lagrangian and Eulerian specification of the flow field|Lagrangian]] particles, these computational elements being called vortices, vortons, or vortex particles.<ref>{{cite book |last1=Cottet |first1=Georges-Henri |last2=Koumoutsakos |first2=Petros D. |date=2000 |title=Vortex Methods: Theory and Practice |location=Cambridge, UK |publisher=Cambridge Univ. Press |isbn=0-521-62186-0
}}</ref> Vortex methods were developed as a grid-free methodology that would not be limited by the fundamental smoothing effects associated with grid-based methods. To be practical, however, vortex methods require means for rapidly computing velocities from the vortex elements – in other words they require the solution to a particular form of the [[N-body problem]] (in which the motion of N objects is tied to their mutual influences). This breakthrough came in the 1980s with the development of the [[Barnes–Hut simulation|Barnes-Hut]] and [[fast multipole method]] (FMM) algorithms. These paved the way to practical computation of the velocities from the vortex elements.

Software based on the vortex method offer a new means for solving tough fluid dynamics problems with minimal user intervention.{{Citation needed|date=November 2010}}  All that is required is specification of problem geometry and setting of boundary and initial conditions. Among the significant advantages of this modern technology;
* It is practically grid-free, thus eliminating numerous iterations associated with RANS and LES.
* All problems are treated identically. No modeling or calibration inputs are required.
* Time-series simulations, which are crucial for correct analysis of acoustics, are possible.
* The small scale and large scale are accurately simulated at the same time.

==== Boundary element method ====

{{Main|Boundary element method}}

In the boundary element method, the boundary occupied by the fluid is divided into a surface mesh.

==== High-resolution discretization schemes ====

{{Main|High-resolution scheme}}

High-resolution schemes are used where shocks or discontinuities are present. Capturing sharp changes in the solution requires the use of second or higher-order numerical schemes that do not introduce spurious oscillations. This usually necessitates the application of [[flux limiters]] to ensure that the solution is [[total variation diminishing]].{{Citation needed|date=November 2010}}