uppose the operator of the finite-difference
problem
is time
dependent
According to the convergence theorem (
Lax
convergence theorem
) we need to prove approximation on the solution
and stability. The stability is proved with the same techniques as in the time
independent case. We assume that the spacial operator
has a correct approximation in spacial variables. In this section we
concentrate on the problem of
-directional
approximation. We assume that the solution
possesses smoothness in
-variable
and invoke the Taylor
decomposition
where the higher index refers to the point on the uniform time mesh, the low
index is the derivative in
and the
is the time step. We repeatedly use the fact that
is the solution of
:
With the above results we would like to construct a Crank-Nicolson
scheme:
for some operator
that remains to be determined. We expand the above scheme in power of
and collect the
terms:
Hence we need to
have
and the
-term
expression
gives
Hence, the following requirement should be sufficient for approximation in
direction
Some of the possibilities
are