234
8. Complex Geometries
this approach is that an oscillatory solution may be generated in the course
of the iteration procedure and the oscillations will not be detected. How this
can be avoided is described below.
For explicit methods, this approach is very simple and effective. It is, however, not suitable for implementation in an implicit method since it produces
large computational molecules. The deferred correction approach described
in Sect. 5.6 offers a way around this problem and helps to eliminate the oscillations. It consists of using a simple approximation to the diffusive flux
implicitly and creating a right hand side which is the difference between the
correct and approximate fluxes. With good choices of approximations the
convergence of the implicit method is not impaired by the deferred correction.
A good approximation for the implicit part of the method is easily found.
If we use the local (n, t , s) orthogonal coordinate system attached to the cell
face center, then only the derivative in the n-direction contributes t o the
diffusive flux:
On a Cartesian grid, n = x at the "en face and we can use the central
difference approximation:
where L P , ~
is the distance between nodes E and P, IrE - rpI ( L P , ~
= Ax on
a uniform Cartesian grid). The interpolated cell center gradient gives (on a
uniform Cartesian grid):
An oscillatory distribution of 4 in x-direction shown in Fig. 8.8 will not
contribute t o this gradient, since both 4E - dw and dEE - 4 p are zero and
so are the gradients a t each cell center. However, the gradients are large a t
cell faces. Oscillations do indeed develop during the iteration process. The
obvious deferred correction approach:
in which 'impl' and 'expl' denote the implicit (using Eq. (8.26)) and the
explicit (using Eq. (8.27)) flux approximation and 'old' means value from
previous iteration, allows oscillatory solutions to develop. A similar problem
appears in the derivation of the pressure-correction equation for colocated
arrangements and was discussed in Chap. 7.
8. Complex Geometries
this approach is that an oscillatory solution may be generated in the course
of the iteration procedure and the oscillations will not be detected. How this
can be avoided is described below.
For explicit methods, this approach is very simple and effective. It is, however, not suitable for implementation in an implicit method since it produces
large computational molecules. The deferred correction approach described
in Sect. 5.6 offers a way around this problem and helps to eliminate the oscillations. It consists of using a simple approximation to the diffusive flux
implicitly and creating a right hand side which is the difference between the
correct and approximate fluxes. With good choices of approximations the
convergence of the implicit method is not impaired by the deferred correction.
A good approximation for the implicit part of the method is easily found.
If we use the local (n, t , s) orthogonal coordinate system attached to the cell
face center, then only the derivative in the n-direction contributes t o the
diffusive flux:
On a Cartesian grid, n = x at the "en face and we can use the central
difference approximation:
where L P , ~
is the distance between nodes E and P, IrE - rpI ( L P , ~
= Ax on
a uniform Cartesian grid). The interpolated cell center gradient gives (on a
uniform Cartesian grid):
An oscillatory distribution of 4 in x-direction shown in Fig. 8.8 will not
contribute t o this gradient, since both 4E - dw and dEE - 4 p are zero and
so are the gradients a t each cell center. However, the gradients are large a t
cell faces. Oscillations do indeed develop during the iteration process. The
obvious deferred correction approach:
in which 'impl' and 'expl' denote the implicit (using Eq. (8.26)) and the
explicit (using Eq. (8.27)) flux approximation and 'old' means value from
previous iteration, allows oscillatory solutions to develop. A similar problem
appears in the derivation of the pressure-correction equation for colocated
arrangements and was discussed in Chap. 7.