et
be a
spacedirection finite difference operator,
Assume that
is independent from
.
We seek for efficient ways to convert to a finite difference scheme in
direction.
We integrate over
:
and approximate the
integral
where the
is the step
.
Hence,
If the operator
is positive definite the the norm of
is less then
and the scheme is stable.
The predictorcorrector way to improve scheme's efficiency is the following.
We find a separation
for the implicit part of the scheme. Observe that the CrankNicolson may be
written in two steps as
We transform
further
The last equation is equivalent
to
We aim to replace the last equation with the
equation
To see that we preserve the second order of approximation we
compute
The resulting scheme is
These should be more efficient because we split the inversion into two,
presumably, simpler components. The last step is explicit.
To explore stability we put all steps
together
Hence
Make the change of
function
then
and the stability follows from the stabilization scheme considerations.
