False diffusion

False diffusion is a type of error observed when the upwind scheme is used to approximate the convection term in convection–diffusion equations. The more accurate 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 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 analysts.

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.

In figure 1, 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 10×10 equal grids. Take two cases, (i) with diffusion coefficient ≠ 0 and, case (ii) with diffusion coefficient = 0.

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.

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