onsider the
equation (
Evolution with penalty
term
) involving explicit time discretization of first
order:
The explicit formulation is selected for calculational convenience. At the
initial time step
,
the column
satisfies the condition
.
After evolution for one step, it falls into
.
Under explicit formulation, it takes at least another time step for the
penalty term to take affect and the solution
already deviated from
.
Clearly, we would have to make very small time steps to keep the deviation
small. Furthermore, we will pick up discrepancy on every step and accumulate
it.
Therefore, we arrive to implicit
formulation:
The
and
superscripts over
term
mark discrepancy of grids. The
is constructed with respect to an adaptive basis selection
:
and
,
are connected similarly via
.
The
transformation
adapts
to the grid
.
Hence, before performing the operation
,
we apply such transformation and remove griddependent details from
consideration:


(Same grid reduction)

where
is calculated by projecting final payoff on
:
To see that such transformation is correct, put
.
