e intend to
apply Newton technique to the equation
(
Implicit equation with
penalty term
):
The equation has the
form
with
We introduce the convenience
notations
and drop the upperscript
:
We calculate the matrix of first
derivatives:
where
is a Gram
matrix
is a diagonal matrix with small
values
and the matrix
has spectral radius close to 1 (from above).
The Newton procedure is the
iteration
where we
use
We start Newton iterations
from
For small time step, the initial position
is sufficiently close to the root of
for convergence of Newton iterations.
Due to smallness of the function
and
boundedness of other components, the matrix
is safely invertible for wide range of parameters.
Implementation of such approach is implemented in the script soNewton1.py,
located in the directory OTSProjects/python/wavelets2. Experimentation shows
that in order to keep the discrepancy
to acceptable size, one has to take
in the area of 1.e3. If we do not decrease the time step with
then the Newton procedure diverges. Such extreme stiffness is not a feature of
the problem under consideration but a limitation of the Newton technique. We
propose a more robust procedure in the following section.
