e aim to convert the scheme (
Alternating
directions1
),(
Alternating
directions2
) to the canonical form by eliminating the
.
Add the equations (
Alternating
directions1
),(
Alternating
directions2
):
and substitute
from (
Alternating
boundary
):
Note that
and
are commutative. We transform the last expression as
follows:
Finally, we write the evolution
equation
where
For stability it is sufficient to
have
Such results follow from the spectral considerations of the Crank-Nicolson
(
Crank Nicolson spectrum
) and implicit
(
Implicit spectrum
) schemes and the minimax
theorem (
Minmax theorem
).
One may ask "Wait a moment, how about the boundary conditions?". The answer is
"Check the trick around the (
Boundary trick
)".
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