Let and
be the computed
solution at
. The TR substep goes from
to
via
The BDF2 substep goes from to
via
The constant multiplying the truncation error is minimized if
. This value of gamma also produces identical
jacobians for both the TR and BDF2 steps.
The local truncation error (LTE) can be used to estimate a reasonable next time step. One approach is to use the divided difference formula
is used to estimate the LTE. Note that C is the constant multiplying the highest order truncation term in the method
The LTE is compared to an allowable error tolerance to produce a RMS measure of the normalized pointwise error. Let
and
Then, the RMS measure of the normalized pointwise error is given by
A candidate next time step is chosen as
If r is reasonably small (say less than 2), the error in the
timestep is assumed to be acceptable and the next time step is
chosen to be . The 2 times cap on
increases is to avoid stepsize oscillations. If r is too large,
the current timestep is repeated with
.
A second timestep estimator is based on a LTE estimate computed
using a Milne's device. A second TR step is taken from to
. The solutions of the BDF2 step and the
second TR step are compared and used to estimate the truncation
error. See [4] for more details.