The integration methods available in Ansys Polyflow are summarized in Table 29.1: Integration Methods in Ansys Polyflow.
Table 29.1: Integration Methods in Ansys Polyflow
| Integration method |
 | precision |
| Explicit Euler | 0 |
 |
| Crank-Nicolson |
 |
 |
| Galerkin |
 |
 |
| Implicit Euler | 1 |
 |
With the explicit Euler method (
), 
can be evaluated directly from
the known value of 
at the previous time 
(and, hence,
the method is explicit). The other methods, with nonzero 
, are known as implicit,
since 
depends not only on 
at
the previous time, but also on the time derivative of 
at 
, which is itself a function of the unknown 
. Since 
appears on both sides of Equation 29–8, using an implicit
method requires solving for 
.
From this point of view, the implicit method requires more operations.
However, all explicit techniques are only conditionally stable: the
time step size 
must not exceed a certain value.
The maximum admissible 
of an explicit scheme is related
to the size of elements and the speed of the trajectories. With an
explicit technique, the time step should not exceed the time required
for a trajectory to pass through an element (for all elements in the
mesh).
Ansys Polyflow offers only implicit time-marching schemes of
the predictor-corrector family. The predictor is an explicit time-marching
scheme that gives an estimate of 
, denoted hereafter by 
. This value is an
initial guess for the corrector. The corrector itself might be nonlinear,
in which case several iterations may be needed for convergence. However,
in the limit of small time steps, the mass matrix terms dominate the
system (Equation 29–1)
and all problems become linear. Predictor-corrector methods enjoy
the stability properties of implicit techniques. Another advantage
is the automatic control of time step. This is based on the difference
between the predicted value 
and the corrected value 
. After the solution at 
, an estimate of the next time step size 
can be calculated using the
formula
 | (29–9) |
where the coefficients 
and 
depend on the order of the implicit method, and 
is the maximum relative difference between the predicted
and corrected value of 
:
 | (29–10) |
where 
is the 
th component
in the vector 
. The only user-defined parameter in Equation 29–9 is 
, which represents
a relative tolerance of the local truncation error in 
by comparison with the exact
solution 
. The choice
of 
has a significant effect on computational
cost and accuracy.