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'''False diffusion''' is a type of error observed when the  [[upwind scheme]] is used to approximate the [[convection]] term in [[convection–diffusion equation]]s. The more accurate [[Finite difference#central difference|central difference scheme]] can be used for the [[convection]] term, but  for grids with cell [[Peclet number]] more than 2, the central difference scheme is unstable and the simpler upwind scheme is often used. The resulting error from the upwind differencing scheme has a diffusion-like appearance in two or three dimensional co-ordinate systems and is referred as "false diffusion". False-diffusion errors in numerical solutions of convection-diffusion problems, in two- and three-dimensions, arise from the numerical approximations of the convection term in the conservation equations. Over the past 20 years many [[numerical analysis|numerical]] techniques have been developed to solve convection-diffusion equations and none are problem free, but false diffusion is one of the most serious problems and a major topic of controversy and confusion among [[numerical analysis|numerical analysts]].

==Definition==
False diffusion is defined as an error having a diffusion-like appearance, obtained when the ''upwind scheme'' is used in multidimensional cases to solve for the distribution of transported properties flowing non-orthogonally to one or more of the system's major axes. The error is absent when the flow is orthogonal or parallel to each major axis.

==Example==
[[File:Upwind differencing scheme.jpg|thumb|Fig 1:Flow domain illustrating false diffusion]]
In figure 1, ''u''&nbsp;=&nbsp;2 and ''v''&nbsp;=&nbsp;2&nbsp;m/s everywhere so the [[velocity field]] is uniform and [[parallel (geometry)|parallel]] to the [[diagonal]] (XX). The boundary conditions for  [[temperature]] on north and west wall is 100 ̊C and for east and south wall is 0 ̊C. This region is meshed into 10×10 equal grids. Take two cases, (i) with [[diffusion coefficient]] ≠ 0 and, case (ii) with diffusion coefficient = 0.

===Case (i)===
[[File:Convection.jpg|thumb|Fig 2: West face is at 100°C while south wall is at 0°C. Heat is diffused across the diagonal XX]]
In this case, heat from west and south wall is carried by [[convection]] flow towards north and east wall. Heat is also diffused across the diagonal XX from upper to lower triangle. Figure 2 shows the approximate temperature distribution.

===Case (ii)===
In this case heat from west and south wall is convected by flow towards north and east. There will be no diffusion across the diagonal XX but, when the ''upwind scheme'' is applied the results are similar to case (i) where actual diffusion is occurring. This error is known as false diffusion.

==Background==
In early approaches, [[derivatives]] in the ''[[differential form]]'' of the governing ''[[transport equation]]'' were replaced by finite difference approximations, usually central differencing approximations with second order accuracy. However, for large Peclet numbers (generally >&nbsp;2) this approximation gave inaccurate results. It was recognized independently by several investigators [1,2]<ref>{{cite journal |author= R. Courant,E.Isaacson and M.Rees|title= On the solution of non-linear hyperbolic differential equations by finite difference, Comm. Pure Appl. Math. 5(1952) 243–255}}</ref><ref>{{cite journal |author= K.E.Torrance,  |title= Comparison of finite difference computations of natural convection J.Res N.B.S 72B(1968)281–301.}}</ref> that the less expensive but only first order accurate ''upwind scheme'' can be employed but that this scheme produces results with false diffusion for  multidimensional cases. Many new schemes have been developed to counter false diffusion but a reliable, accurate and economical discretisation scheme is still unavailable.

== Reducing errors ==
[[File:Mesh size 8X8.jpg|thumb|Fig 3(a): Mesh size of 8×8]]
[[File:Result with grid size 8X8.jpg|thumb|Fig 3(b): Result of upwind scheme with mesh size 8X8]]
[[File:Mesh size 64X64.jpg|thumb|Fig 4(a): Mesh size of 64×64]]
[[File:Result with grid size 64X64.jpg|thumb|Fig 4(b): Result of upwind scheme with mesh size 64×64]]

=== Finer mesh===
False diffusion with the ''upwind scheme'' is reduced by increasing the mesh density. In the results of figure 3 and 4 the false diffusion error is lowest in figure 4(b) with finer mesh size.

=== Other schemes ===
False diffusion error also can be reduced by using schemes such as the ''[[power law scheme]]'', ''[[QUICK scheme]]'', ''exponential scheme'', and ''SUCCA'', and others.<ref>{{cite book|last=Versteeg|first=H.K.|title=An introduction to computational fluid dynamics : the finite volume method|year=2007|publisher=Prentice Hall|location=Harlow|isbn=9780131274983|edition=2nd |coauthors=Malalasekera, W.}}</ref><ref>{{cite book|last=Patankar|first=Suhas V.|title=Numerical heat transfer and fluid flow|year=1980|publisher=Taylor & Francis|location=Bristol, PA|isbn=9780891165224|edition=14. printing.}}</ref>

===Improving the upwind scheme===
False diffusion with the simple ''upwind scheme'' occurs because the scheme does not take into account grid/flow direction inclination. An approximate expression for the false-diffusion term in two dimensions has been given by Vahl Davis and Mallinson(1972)<ref>{{cite book|last=Patankar|first=Suhas V.|title=Numerical heat transfer and fluid flow page no:108|year=1980|publisher=Taylor & Francis|location=Bristol, PA|isbn=9780891165224|edition=14. printing.}}</ref>

{{NumBlk|:|<math>{\tau _{fc}^\star= \frac {\rho u \,\Delta x \,\Delta y \sin 2\theta}{4(\Delta y \sin ^3 \theta+\Delta x \cos ^3 \theta)}}</math>|{{EquationRef|1}}}} 

where ''U'' is the resultant velocity and ''θ'' is the angle made by the velocity vector with the ''x'' direction. False diffusion is absent when the resultant flow is aligned with either of the sets of grid lines and is greatest when the flow direction is 45˚ to the grid lines.

===Determining the accuracy of approximation for the convection term===
Using ''[[Taylor series]]'' for <math> \phi_{W} </math> and <math> \phi_P </math> at the time ''t''&nbsp;+&nbsp;''kt'' are

{{NumBlk|:|<math> \phi_{Wk} = \phi_{wk} - \left(\frac{\delta x_i}{2}\right)\left(\frac{\partial \phi}{\partial x}\right)_{wk}+\frac{1}{2!}\left(\frac{\delta x_i}{2} \right)^2 \left(\frac{\partial^2\phi}{\partial x^2}\right)_{wk} + \cdots. </math>|{{EquationRef|2a}}}}

{{NumBlk|:|<math>\phi_{Pk}=\phi_{wk} - \left(\frac{\delta x_i}{2}\right) \left(\frac{\partial \phi}{\partial x}\right)_{wk} + \frac{1}{2!}\left(\frac{\delta x_i}{2} \right)^2 \left(\frac{\partial^2\phi}{\partial x^2}\right)_{wk}+\cdots.</math>|{{EquationRef|2b}}}}

according to the upwind approximation for convection (UAC),<math>{\phi_{wk} = \phi_{Wk}}</math>. Neglecting the higher order in equation (2a), the error of convected flux due to this approximation is <math>{-\rho_w u_w \delta y_i\left(\frac{\delta x_i}{2} \right) \left( \frac{\partial \phi}{\partial x}\right)_{wk}}</math>. It has the form of  flux of <math>{\phi}</math> by false diffusion with a diffusion co-efficient<ref>{{cite journal |author=G.D Raithby |title= A critical evaluations of upstream differencing applied to problems involving fluid flow , COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING , 9(1976) 75–103}}</ref>
{{NumBlk|:|<math>\tau_{fc,UAC}^\star ={\rho_w u_w \left(\frac{\Delta x_i}{2}\right)}</math>|{{EquationRef|3}}}}
The subscript ''fc'' is a reminder that this is a false diffusion arising from the estimate of the convected flux at the instant <math>t+k\,\Delta t</math> using UAC.

===''Skew upwind corner convection algorithm'' (''SUCCA'')===
[[File:SUCCA scheme.jpg|thumb|Fig 5:''SUCCA'' grid cell cluster]]
''SUCCA'' takes the local flow direction into account by introducing the influence of upwind corner cells into the discretized conservation equation in the general governing transport equation. In Fig 5, ''SUCCA'' is applied within nine cell grid cluster. Considering the SW corner inflow for cell P, the ''SUCCA'' equations for the convective transport of the conserved species <math>{\phi}</math> are

{{NumBlk|:|<math>C_P \phi _P= \left(\dot m_w - \frac {(\dot m_s)^2} {\dot m_w} \right) \phi_W + \left(\dot m_s + \frac {(\dot m_s) ^2} {\dot m_w} \right)\phi_{SW} + 0.\phi_S\text{ for } 0<\theta\le 45</math>|{{EquationRef|4}}}}

i.e.,

{{NumBlk|:|<math>C_P \phi_P = C_w \phi_W + C_sw \phi_{SW}</math>|{{EquationRef|5}}}}

{{NumBlk|:|<math>C_P \phi _P= \left(\dot m_s - \frac {(\dot m_w)^2} {\dot m_s} \right)\phi_W +{\left(\dot m_w+  \frac {(\dot m_w) ^2} {\dot m_s} \right) \phi _{SW}} + 0.\phi_W \text{ for }45<\theta< 90 </math>|{{EquationRef|6}}}}

i.e.,

{{NumBlk|:|<math>C_P \phi_P = C_s \phi_S + C_sw \phi_{SW}</math>|{{EquationRef|7}}}}

This formulation satisfies all the criteria of [[Convergence (mathematics)|convergence]] and stability.<ref>{{cite journal |author=C.Carey, T.J.Scanlon and S.M.Fraser |title= SUCCA- An alternative scheme to reduce the effects of multidimensional false diffusion, Appl. Math Modelling,1993 ,Vol.17, May 263–270}}</ref>

[[File:Comparison of different schemes.jpg|thumb|Fig 6: Comparison of different schemes]]
In Fig. 6, as mesh is refined, the ''upwind scheme'' gives more accurate results but ''SUCCA'' offers a nearly exact solution and is more useful in avoiding multidimensional false diffusion errors.

== See also ==

*[[Computational fluid dynamics]]
*[[Navier–Stokes equations]]
*[[Finite volume method]]
*[[Taylor series]]

==References==
{{reflist}}

==Further reading==
*{{Citation
| publisher = Taylor & Francis Group
| isbn = 9780891165224
| last = Patankar
| first = Suhas V.
| title = Numerical Heat Transfer and Fluid Flow
| year = 1980
}}
*{{Citation
| publisher = Springer
| isbn = 978-3-540-77853-3
| last = Wesseling
| first = Pieter
| title = Principles of Computational Fluid Dynamics
| year = 2001
}}
*{{Citation
| publisher = Cambridge University Press
| isbn = 9780521853262
| last = Date
| first = Anil W.
| title = Introduction to Computational Fluid Dynamics
| year = 2005
}}

[[Category:Computational fluid dynamics| ]]
[[Category:Numerical differential equations]]