n this section we
consider time discretization for the
expression
The
depends on a function
that comes from the equation where
participates. Theory of numerical methods for ODE have a similar problem and
an effective solution. We review the theory.
Consider the
ODE
The time evolution happens backwards from
to
.
Taylor decomposition applies to
and
in all arguments. The
is an input value.
We introduce a time mesh
and a mesh function
defined
recursively
Thus
We proceed to estimate the magnitude of the difference
for all
.
We
calculate
We subtract and use
:
For a general time step we
calculate
thus
Then
assuming that all
are of generally the same magnitude.
Following recipes of RungeKutta technique, we introduce a better
approximation
as
follows:
We seek parameters
that deliver the smallest difference
.
We calculate the evolution equation for
:
Therefore


(Evolution of y)

We use Taylor expansion for
:
and substitute the time derivatives using the defining equation for
:
and put all
together:


(Evolution of u)

By comparing the formulas (
Evolution of y
) and
(
Evolution of u
), we
require
We
set
then
At initial time step
we get
and then the higher order propagates.
The crucial requirement is smoothness of
.
In our case, see the expression for
,
the function
would jump. Hence, the expression
is not small for those values of
and
on opposite sides of the jump. Given the nature of the problem, it would be
typical.
One might try to replace the operation
within the function
with some smooth function with similar
properties:
The function
must be zero where
is zero. But then, to be smooth, it must increase gently on the other side of
the
area. One could argue that it would hurt the purpose of the penalty term,
dampening convergence of the procedure and requiring higher
.
But then, one could also argue that this is what we want: we are not certain
about desired strength of the penalty term, thus, we would like to employ a
graduate procedure of high order.
In the following chapter
(
Smooth version of
penalty term
) we will point out a reason why, in fact, we must do such
modification.
