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In computational fluid dynamics the use of the central difference scheme for convection term in the general transport equation makes the numerical scheme to be unstable for the grid having cell peclet no. More than 2. The upwind schemes give stability but it introduces artificial diffusion. The resulting error from the upwind differencing scheme has diffusion –like appearance in two dimension or three dimension co-ordinate system is referred as FALSE DIFFUSION(artificial diffusion).

DEFINITION
It is defined as an error having diffusion like appearance, obtained when Upwind Differencing Scheme is used for solving distribution of transported properties, in cases when the flow is not aligned with the grid lines. It happens only in multidimensional cases. Let us understand false diffusion with an example. In 'Fig. 1' we have domain where u=2 and v=2 m/s everywhere so the velocity field is uniform and 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 8x8 equal grids. Take two cases, (i) with diffusion coefficient ≠ 0 and, case (ii) with diffusion coefficient =0.

Case (i)
Physically in this case, heat from west and south wall is convected by flow towards north and east wall. Heat is also diffused across the diagonal XX from upper to lower triangle. 'Fig 2' shows the approximate temperature distribution for this case.

Case(ii)
Physically, 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 we apply Upwind scheme to this case we get results which shows similar behaviour to case (i) where actual diffusion was happening.

Hence this error is known as False diffusion.

BACKGROUND
In early methods it was usual to replace derivatives in the differential form of governing transport equations by finite differencing approximations. Central differencing approximation leading to equations that were second order accurate was normally used. However it was found that for large Peclet number (generally >2) this approximation gave inaccurate results. It was recognized independently by several investigators [1,2] that less expensive and only first order accurate upwind scheme can be employed but in this scheme False diffusion occurs for multidimensional cases. Many new schemes have been developed till date to counter false diffusion but reliable, accurate and economical discretisation scheme is still awaited.

HOW TO CONTROL FALSE DIFFUSION
(1) Refining the mesh: By increasing the mesh density, the false diffusion error is reduced as clearly depicted in the fig 4 and fig 5 (2) Using other schemes like power law scheme, QUICK scheme, exponential scheme, etc.

(3) Improvising upwind scheme:  Simple upwind scheme does not take into account grid/flow direction inclination due to which false diffusion error is coming. An approximate expression for the false-diffusion term in two dimensions has been given by Vahl Davis and Mallinson(1972). Where U is the resultant velocity and θ is the angle made by the velocity vector with the x direction. No false diffusion is present when the resultant flow is along one of the sets of the grid lines ; on the other hand the false diffusion is most serious when the flow direction makes an angle of 45˚ with the grid lines.

(4) Determining the accuracy of approximation for the convection term using Taylor expansion series for $${\phi_{W}}$$ and $${\phi_{P}}$$ at the time t+kt are.

According to Upwind Approximation for Convection(UAC),$${\phi_{wk} = \phi_{Wk}}$$. Neglecting the higher order in equation (2a). the error of convected flux due to this approximation is$${-\rho_{w} u_{w}\Delta y_{i}\left(\frac{\partial x_{i}}{2}\right)\left(\frac{\partial \phi}{\partial x}\right)_{wk}}$$.It has the form of flux of $${\phi}$$ by false diffusion with a diffusion co-efficient

The subscript fc is a reminder that this is a False Diffusion arising from the estimate of the Convected flux at the instant $${t+k\Delta t}$$ using UAC.

(5) SUCCA(skew upwind corner convection algorthim) is used to reduce the false diffusion errors. This scheme takes into account of local flow direction by introducing the influence of upwind corner cells into the discretized conservation equation in general governing transport equation. Here SUCCA(skew upwind corner convection algorthim) is applied within nine cell grid cluster in fig .Considering the SW corner inflow for cell P the SUCCA algorthim is written for Convective transport of the conserved species $${\phi}$$ as.

i.e.,

i.e.,

This formulation satisfies all the criteria of convergence and stability.

CONCLUSION
The several Scheme which has been described above among these SUCCA Scheme is the best scheme to reduce the false diffusion error.