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The time integration of 1D thermal conduction equation of grand
surface (A.55) is performed by
the Crank-Nicolson scheme.
The space differencing in (A.55) is evaluated by the second
order centered scheme.
The grand temperature and depth are evaluated on the grid point and
the heat flux is evaluated on the half grid point.
The number of vertical grid point is  and the suffix
of the lowest grid point is j = 1.
The  is assumed to the surface temperature
The finite difference 1D thermal conduction equation is represented as follows.
or,
where  ,  .
This equation can be represented in matrix form as follows.
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(B.61) |
where  .
The elements of  are represented as follows.
Considering the boundary condition
(A.56),
(B.61) is modified as follows.
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(B.62) |
Therefore, the grand temperature is given by the solution of the
following matrix equation.
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(B.63) |
where the elements of are modified as follows.
 is a column vector whose dimension is
are represented as follows.
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