The solution of parabolic PDEs (3.6-1) requires for time integration a
suitable time step size . The selection of an appropriate
is closely connected to the method applied for time integration, in our
case the backward Euler method, and the according error estimation. Here, we
will discuss three details of the time step selection: (1) The estimation of an
initial time step, (2) the time step adaption and (3) some considerations
for the last time step.
For the estimation of the initial time step we permit a local
change
(3.6-2) of the concentration of a few
percent (3.6-3). Local means, we estimate the change for each
quantity
for each grid point
.
If the local calculation (3.6-3) is not successful, we try a
global estimation (3.6-4) and if this also fails, we take a
heuristic value .
The time step adaption is based on an accuracy estimation. Let us consider the concentration at one specific grid point with respect to time, then we get (3.6-6) from Taylor series expansion (3.6-5).
From the concentrations at time steps ,
and
we
estimate the second derivative
using divided
differences (see Section 3.4.1). Three criteria are evaluated:
(1) The local truncation error
of the time discretization
(3.6-7), (2) the influence of higher order terms on the
concentration (3.6-8) and (3) a comparison of the scaled higher
order terms with the Newton accuracy
(3.6-9). The weakest
of these criteria determines the accuracy
.
The time step has to be rejected if the accuracy is too small (typically
). For accepted time steps, the new time step size
is predicted by (3.6-11). We have introduced this
nonlinear damping function to avoid oscillations of the time step size and
to limit the time step increase to a factor
(typically
).
This estimated time step size is truncated to a given
minimum time step and is adjusted to the end time, if necessary
(3.6-12) (see Figure 3.6-1).