e start
from the problem
(
Backward
discontinuous Galerkin timediscretization
): We seek
such that for every
and any
,
where
is input data. By inclusion
,
the function
has the
form
The functions
are sought as linear combinations of a wavelet
basis:
for some finite index set
.
A system of linear equations is obtained by
selecting
for all
and all
.
We substitute definitions into
and
calculate
We treat the pairs
,
as a single indexes
.
The above equation has the
form
where
is a column
,
is a
matrix


(Matrix of discontinuous Galerkin)

and the column
comes from previous step or input
data:
We will discover in the following sections that we can frequently do
successful calculations using
.
We adapt the formulas to such
case:
We
get
The column
is
defined
For
we
have
For
we use result of the previous
step
In either
case
