118
5. Solution of Linear Equation Systems
other equations, which may slow or prevent convergence. Unfortunately, it is
hard to analyze the convergence of these methods so the selection of underrelaxation factors is largely empirical.
The multigrid method, which was described above as a convergence accelerator for inner iterations (linear problems), may be applied to coupled
problems. It may also be used to accelerate the outer iterations as will be
described in Chap. 11.
5.4.3 Under-Relaxation
We shall present one under-relaxation technique that is widely used. On the
nth outer iteration, the algebraic equation for a generic variable, 4, at a
typical point P may be written:
where Q contains all the terms that do not depend explicitly on qP; the
coefficients Al and the source Q may involve 4n-1. The discretization scheme
is unimportant here. This equation is linear and the system of equations for
the whole solution domain is solved usually iteratively (inner iterations).
In the early outer iterations, allowing q5 to change by as much as Eq. (5.67)
requires could cause instability, so we allow qY to change only a fraction a+
of the would-be difference:
where
is the result of Eq. (5.67) and the under-relaxation factor satisfies
O
matrix and source vector are updated, the new solution can be written over
it. Replacing qYeW in Eq. (5.68) by
which follows from Eq. (5.67), leads to a modified equation at node P:
where A; and Q; are modified main diagonal matrix elements and source
vector components. This modified equation is solved within inner iterations.
When the outer iterations converge, the terms involving a4 cancel out and
we obtain the solution of the original problem.
5. Solution of Linear Equation Systems
other equations, which may slow or prevent convergence. Unfortunately, it is
hard to analyze the convergence of these methods so the selection of underrelaxation factors is largely empirical.
The multigrid method, which was described above as a convergence accelerator for inner iterations (linear problems), may be applied to coupled
problems. It may also be used to accelerate the outer iterations as will be
described in Chap. 11.
5.4.3 Under-Relaxation
We shall present one under-relaxation technique that is widely used. On the
nth outer iteration, the algebraic equation for a generic variable, 4, at a
typical point P may be written:
where Q contains all the terms that do not depend explicitly on qP; the
coefficients Al and the source Q may involve 4n-1. The discretization scheme
is unimportant here. This equation is linear and the system of equations for
the whole solution domain is solved usually iteratively (inner iterations).
In the early outer iterations, allowing q5 to change by as much as Eq. (5.67)
requires could cause instability, so we allow qY to change only a fraction a+
of the would-be difference:
where
is the result of Eq. (5.67) and the under-relaxation factor satisfies
O
matrix and source vector are updated, the new solution can be written over
it. Replacing qYeW in Eq. (5.68) by
which follows from Eq. (5.67), leads to a modified equation at node P:
where A; and Q; are modified main diagonal matrix elements and source
vector components. This modified equation is solved within inner iterations.
When the outer iterations converge, the terms involving a4 cancel out and
we obtain the solution of the original problem.