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We consider the solution on the time interval , partitioned
into time steps as
. Considering the thermal conductivities
to be isotropic and constant, we proceed as described in section 4
and write (24) at time step as:
and we consider the general Trapezoidal rule:
Substituting this equation into the above yields, and after proceeding to the variational
formulation, we have the following expression:
where we have assumed that
.
We now focus on the
terms in the above which arise on the freezing front
. These
terms consist of a pair of heat fluxes:
which are inherently associated
with the motion of the interface through (26).
We note that we need to apply a time-step algorithm to the equation
describing the motion of the interface. To this purpose, we rewrite
equation (26) at time step as:
And now let's ignore conventional
wisdom, and use a different algorithm , the Forward Euler algorithm ( which
is the case of the general Trapezoidal rule algorithm when
), yields:
which can be rewritten as:
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Renaud Merle
2000-08-26