e apply recipes of the section
(
Reduction
to system of linear algebraic equations
) to the problem
(
Example strong Black problem
2
).
We introduce the time
mesh
The location of the points
is to be chosen according to the statement
(
Convergence of
discontinuous Galerkin technique
) and will be addressed in the section
(
Adaptive time step for
Black PDE
).
First, we apply results of the section
(
Decomposition
of payoff function in one dimension section
) to the function
of the problem (
Example strong
Black problem 2
). Thus, we have a wavelet basis
and we start from
decomposition
for some index set
According to the results around the formula
(
Matrix of discontinuous
Galerkin
), we perform a single time step transformation by solving the
equation
where we introduce double indexes
,
,
is a column
,
is a
matrix
For the present problem, the operator
does not have
dependency.
Hence
and
where the functions
are second degree piecewise polynomials constructed to be in
.
The column
comes from previous step or input
data:
Thus, for
,
we put
and for
the
is taken from the previous step.
We introduce the
notations
Then
We arrange indexes
by changing the spacial indexes
before changing the time indexes
.
Then the matrix
has block structure. For
we
get
For any matrixes
and
,
we introduce the notation
"
":
Note
that
We
get
where the last line is valid for all
.
For
we
calculate
Let
then
We
obtain
Note that
is a strongly diagonal matrix and
is a well conditioned matrix. We act on the last equation by
and
obtain
The summary above shows that if the operator
is
independent
then we can solve the problem with
using the same manipulations as in the case
without noticeable increase in memory requirements. The space and time parts
of the problem may be processed separately. This observation extends to a
situation when there is multiplicative separation of space and time dependent
expressions in the PDE, for
example
