n this section we present a general way to construct a finite difference
approximation for a solution of the heat equation via a procedure with linear
dependency of the amount of computation on the size of the lattice.
We are considering the following boundary problem for the heat
equation:
We introduce the uniform
lattices
the lattice function
and notation
.
The introduced above notations
and
refer to the
variable.
Consider the
scheme


(Alternating directions1)



(Alternating directions2)

in all internal points of the lattice
.
The
refers to the operator
acting in the
index. The initial conditions
are
and the boundary conditions
are


(Alternating boundary1)



(Alternating boundary2)

The key observation about the scheme
(
Alternating
directions1
)(
Alternating boundary1
)
is that the (
Alternating directions1
)
is a one dimensional implicit scheme in the
direction
while the (
Alternating directions2
) is
the implicit scheme in the
direction.
Hence, starting from
we use (
Alternating directions1
) to
find the
through the factorization procedure of the previous section. Afterwards, we
similarly use (
Alternating
directions2
) to find
.
To understand the boundary condition
(
Alternating boundary1
) subtract
(
Alternating directions1
) from
(
Alternating directions2
) and obtain


(Alternating boundary)

