he recipes of this
section were not verified because we are using a procedure that makes the
issue of time step selection insignificant. We will discover that selection of
time step
should be made based on the task of the section
(
Rebalancing wavelet basis
).
However, it is not uncommon to encounter a practical situation when every
optimization must be made. Hence, we leave this discussion in place.
According to the proposition
(
Convergence of
discontinuous Galerkin technique
), we need to keep the product
around a constant value for all
.
There are several problems with such task. If we substitute
with
then we run into problem because
th
derivative of
is zero. We may
differentiate
times
and arrive
to
then we substitute
for
and replace
with
However,
is a sum of wavelets
and the wavelets have a
piecewise polynomial representation of second degree for numerical stability
reasons. Thus, taking
norm
of second derivative is not convenient. Finally, we do not know
until we actually make the time step.
In the section
(
Asymptotic
expansion for Black equation
) we calculate asymptotic of the solution. We
might try to apply the operation
to that. However, calculation of asymptotic of sufficient order requires
taking high order derivatives with respect to
.
The following procedure might work. We make a first time step at arbitrary
small length. The we calculate
and use one step finite difference to calculate
using values
for consecutive
.
Thus we get small error from using finite differences and another error from
being late by one time period. We compensate by taking smaller general level
of
.
Another possibility is finite difference approximation for
after
time steps.
