8.6 Finite Volume Methods
237
the whole CV surface is smaller than the integral of the principal term. For
this reason, the underlined term is usually treated explicitly. As shown above,
the derivatives are easily calculated a t the cell face using the derivatives a t
the CV center.
In the above approximations it was assumed that the line connecting
nodes P and E passes through the cell face center 'e'. In that case the approximation of the surface integral is second order accurate (midpoint rule).
When the grid is irregular, the line connecting P and E may not pass through
the cell face center, and the approximations for $, and (&b/dE), used above
are second order accurate a t a point 'e", see Fig. 8.9. The approximation
to the surface integral is no longer second order accurate but the additional
error is small if 'e" is not far from 'e'. If 'e" is close to the corners 'se' or 'ne',
the approximation becomes first order accurate.
Fig. 8.9. An alternative way of calculating cell face values and gradients
The second order accuracy of the midpoint rule integral approximation
can be preserved on irregular grids if the values of the variable and its gradient
are calculated at the cell face center with second order approximations. These
can be obtained by using appropriate shape functions. Alternatively, one can
use values at auxiliary nodes PI and El, which lie at the intersection of the
cell face normal n and straight lines connecting nodes P and N or E and
NE, respectively, see Fig. 8.9. Both the cell face value of the variable and its
gradient can then be approximated as on a Cartesian grid from the values
at P' and El. In order to avoid extended computational molecules in implicit
methods, the deferred correction approach can be used: the implicit terms
are based on the values at nodes P and E ignoring grid irregularity (i.e. using
values at 'e"), while the difference between the implicit term and the more
accurate approximation is treated explicitly. For example, the diffusive flux
approximation can be implemented in the following way, using a modified
version of expression (8.30):
237
the whole CV surface is smaller than the integral of the principal term. For
this reason, the underlined term is usually treated explicitly. As shown above,
the derivatives are easily calculated a t the cell face using the derivatives a t
the CV center.
In the above approximations it was assumed that the line connecting
nodes P and E passes through the cell face center 'e'. In that case the approximation of the surface integral is second order accurate (midpoint rule).
When the grid is irregular, the line connecting P and E may not pass through
the cell face center, and the approximations for $, and (&b/dE), used above
are second order accurate a t a point 'e", see Fig. 8.9. The approximation
to the surface integral is no longer second order accurate but the additional
error is small if 'e" is not far from 'e'. If 'e" is close to the corners 'se' or 'ne',
the approximation becomes first order accurate.
Fig. 8.9. An alternative way of calculating cell face values and gradients
The second order accuracy of the midpoint rule integral approximation
can be preserved on irregular grids if the values of the variable and its gradient
are calculated at the cell face center with second order approximations. These
can be obtained by using appropriate shape functions. Alternatively, one can
use values at auxiliary nodes PI and El, which lie at the intersection of the
cell face normal n and straight lines connecting nodes P and N or E and
NE, respectively, see Fig. 8.9. Both the cell face value of the variable and its
gradient can then be approximated as on a Cartesian grid from the values
at P' and El. In order to avoid extended computational molecules in implicit
methods, the deferred correction approach can be used: the implicit terms
are based on the values at nodes P and E ignoring grid irregularity (i.e. using
values at 'e"), while the difference between the implicit term and the more
accurate approximation is treated explicitly. For example, the diffusive flux
approximation can be implemented in the following way, using a modified
version of expression (8.30):