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5. Solution of Linear Equation Systems
5.6 Deferred-Correction Approaches
If all terms containing the nodal values of the unknown variable are kept on
the left-hand side of Eq. (3.42), the computational molecule may become very
large. Since the size of the computational molecule affects both the storage
requirements and the effort needed to solve the linear equation system, we
would like to keep it as small as possible; usually, only the nearest neighbors
of node P are kept on the left hand sides of the equations. However, approximations which produce such simple computational molecules are usually not
accurate enough, so we are forced to use approximations that refer to more
nodes than just the nearest neighbors.
One way around this problem would be to leave only the terms containing
the nearest neighbors on the left-hand side of Eq. (3.42) and bring all other
terms to the right-hand side. This requires that these terms be evaluated
using values from the previous iteration. However, this is not a good practice
and may lead to the divergence of the iterations because the terms treated
explicitly may be substantial. To prevent divergence, strong under-relaxation
of the changes from one iteration to the next would be required (see Sect.
5.4.3), resulting in slow convergence.
A better approach is to compute the terms that are approximated with
a higher-order approximation explicitly and put them on the right-hand side
of the equation. Then one takes a simpler approximation to these terms (one
that gives a small computational molecule) and puts it both on the left-hand
side of the equation (with unknown variable values) and on the right-hand
side (computing it explicitly using existing values). The right-hand side is now
the difference between two approximations of the same term, and is likely to
be small. Thus it should cause no problems in the iterative solution procedure.
Once the iterations converge, the low order approximation terms drop out
and the obtained solution corresponds to the higher-order approximation.
Since iterative methods are usually necessary due to the non-linearity of
the equations to be solved, adding a small term to the part treated explicitly
increases the computing effort by only a small amount. On the other hand,
both the memory and computing time required are greatly reduced when
the size of the computational molecule in the part of the equation treated
implicitly is small.
We shall refer to this technique very often. It is used when treating
higher-order approximations, grid non-orthogonality, and corrections needed
to avoid undesired effects like oscillations in the solution. Because the righthand side of the equation can be regarded as a "correction" this method is
called deferred correction. Here its use will be demonstrated in conjunction
with Pad6 schemes in FD (see Sect. 3.3.3) and with higher-order interpolations in FV-methods (see' Sect. 4.4.4).
If Pad6 schemes are to be used in implicit FD-methods, deferred correction
must be employed since approximation of the derivative at one node involves
derivatives at neighboring nodes. One approach is to use the "old values"
5. Solution of Linear Equation Systems
5.6 Deferred-Correction Approaches
If all terms containing the nodal values of the unknown variable are kept on
the left-hand side of Eq. (3.42), the computational molecule may become very
large. Since the size of the computational molecule affects both the storage
requirements and the effort needed to solve the linear equation system, we
would like to keep it as small as possible; usually, only the nearest neighbors
of node P are kept on the left hand sides of the equations. However, approximations which produce such simple computational molecules are usually not
accurate enough, so we are forced to use approximations that refer to more
nodes than just the nearest neighbors.
One way around this problem would be to leave only the terms containing
the nearest neighbors on the left-hand side of Eq. (3.42) and bring all other
terms to the right-hand side. This requires that these terms be evaluated
using values from the previous iteration. However, this is not a good practice
and may lead to the divergence of the iterations because the terms treated
explicitly may be substantial. To prevent divergence, strong under-relaxation
of the changes from one iteration to the next would be required (see Sect.
5.4.3), resulting in slow convergence.
A better approach is to compute the terms that are approximated with
a higher-order approximation explicitly and put them on the right-hand side
of the equation. Then one takes a simpler approximation to these terms (one
that gives a small computational molecule) and puts it both on the left-hand
side of the equation (with unknown variable values) and on the right-hand
side (computing it explicitly using existing values). The right-hand side is now
the difference between two approximations of the same term, and is likely to
be small. Thus it should cause no problems in the iterative solution procedure.
Once the iterations converge, the low order approximation terms drop out
and the obtained solution corresponds to the higher-order approximation.
Since iterative methods are usually necessary due to the non-linearity of
the equations to be solved, adding a small term to the part treated explicitly
increases the computing effort by only a small amount. On the other hand,
both the memory and computing time required are greatly reduced when
the size of the computational molecule in the part of the equation treated
implicitly is small.
We shall refer to this technique very often. It is used when treating
higher-order approximations, grid non-orthogonality, and corrections needed
to avoid undesired effects like oscillations in the solution. Because the righthand side of the equation can be regarded as a "correction" this method is
called deferred correction. Here its use will be demonstrated in conjunction
with Pad6 schemes in FD (see Sect. 3.3.3) and with higher-order interpolations in FV-methods (see' Sect. 4.4.4).
If Pad6 schemes are to be used in implicit FD-methods, deferred correction
must be employed since approximation of the derivative at one node involves
derivatives at neighboring nodes. One approach is to use the "old values"