space-direction finite difference operator,
is independent from
We seek for efficient ways to convert to a finite difference scheme in
We integrate over
and approximate the
is the step
If the operator
is positive definite the the norm of
is less then
and the scheme is stable.
The predictor-corrector way to improve scheme's efficiency is the following.
We find a separation
for the implicit part of the scheme. Observe that the Crank-Nicolson may be
written in two steps as
The last equation is equivalent
We aim to replace the last equation with the
To see that we preserve the second order of approximation we
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
Make the change of
and the stability follows from the stabilization scheme considerations.