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 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".

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 perpendicular to the diagonal (XX).

Heat is also diffused across the diagonal XX from upper to lower triangle.

In this case heat from west and south walls is convected by flow towards north and east.

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 > 2) this approximation gave inaccurate results.

False diffusion with the upwind scheme is reduced by increasing the mesh density.

Neglecting the higher order in equation (2a), the error of convected flux due to this approximation is

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.

Considering the SW corner inflow for cell P, the SUCCA equations for the convective transport of the conserved species

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.

Fig 1:Flow domain illustrating false diffusion
Fig 2: West face is at 100°C while south wall is at 0°C. Heat is diffused across the diagonal XX
Fig 3(a): Mesh size of 8×8
Fig 3(b): Result of upwind scheme with mesh size 8X8
Fig 4(a): Mesh size of 64×64
Fig 4(b): Result of upwind scheme with mesh size 64×64
Fig 5: SUCCA grid cell cluster
Fig 6: Comparison of different schemes