5.5 Diagonal dominance

Let us return to the transport equation in Sec. 5.1

@T- @t + r (uT ) r ( rT ) = 0 \relax \special {t4ht=

If the Gauss-Seidel method is applied to solve the equation, a convergent solution is only guaranteed if the diagonal dominance condition of Sec. 5.3 is satisfied.

The contributions to the diagonal and off-diagonal coefficients are summarised below for each term in the equation.

PICT\relax \special {t4ht=

The Laplacian term eqn (note the sign), is discretised by Eq. (3.2 ) and Eq. (3.5 ), providing a positive contribution to the diagonal coefficient (relating to eqn) equal to the sum of the magnitude of negative off-diagonal coefficients (for eqn).

The contribution from the advection term eqn, described by Eq. (3.8 ), is demonstrated above using the linear interpolation scheme of Eq. (3.4 ). The sign of eqn depends on whether the flux in incoming or outgoing from the cell. Some contributions to off-diagonal coefficients are thus negative and some positive, with the diagonal contribution tending to zero as the mesh becomes more regular and flow becomes more uniform.

PICT\relax \special {t4ht=

Advection with linear interpolation maintains diagonal equality while its positive off-diagonal contributions offset the negative contributions from the Laplacian term.

But once the advection contributions exceed the magnitude of the Laplacian contributions, diagonal equality is not achieved. This occurs when eqn (= 2 for a regular mesh), where the “numerical” Péclet number eqn is specified at cell faces from its definition in Sec. 2.21 with eqn.

Diagonal equality is achieved with upwind interpolation and any scheme that provides an explicit correction on upwind, e.g. linear upwind. This is conditional on mass conservation being satisfied, as discussed in Sec. 3.22 . If mass is not conserved, the bounded advection scheme, described in that section, is required to ensure diagonal equality.

The time derivative eqn increases the diagonal coefficient only, so promotes diagonal dominance. Notably, the diagonal contribution of eqn from eqn becomes larger than eqn from advection with upwind when eqn.

In summary, diagonal dominance is not guaranteed due to the contribution of the advection term.

Notes on CFD: General Principles - 5.5 Diagonal dominance