onsider the
following boundary
problem:
where the
is the unknown function, the functions
are given and regular, and the variable
and
lie in the domain
We set up the lattice
and approximate
with the
.
We arrive to the following ODE
problem
for
,
where
,
,
.
Let us consider the boundary conditions. The matrix of the Laplacian
has the
form
Note what happens to the finite difference approximation of the second
derivative on the edges of the matrix. Obviously, we do not approximate it if
are some non zero values. We perform the following trick. We
set
Then the
equation
will describe exactly the same
if we choose
according
to
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|
(Boundary trick)
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Set up the lattice
covering the
and integrate the
-th
equation over the interval
.
We
have
where
The integral may be approximated by one of the quadrature
formulas
The resulting schemes are called implicit, explicit and Crank-Nicolson schemes
respectively. The expressions for the schemes
are
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|
(Implicit scheme)
|
|
|
(Explicit scheme)
|
|
|
(Krank Nicolson)
|
with the boundary
conditions
in every case.
|
