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during the solution phase of the system of equations and is equivalent
to an Euler integration in time (Conte and de Boor, 1980). The
disadvantage of the explicit solution is that the numerical error may
propagate through time and grow. The maximum time step is functionally
related to the boundary conditions and thermal properties of the soil.
The system of equations can also be changed such that all
derivatives are evaluated at the next time step in which none of the
state variables are known. This implicit representation requires a
relatively complicated iterative or matrix solution technique for the
system of equations. The advantage of implicit solution methods is
that the value of the state variables do not depend upon previous
values implying that the only source of error would be due to round-off
or truncation errors and would not propagate or grow with time and
would yield an unconditionally stable solution (Conte and de Boor,
1980).
Another technique which has some of the desirable characteristics
of both the explicit and implicit methods is the alternating direction
(ADI) method. This is accomplished by incrementally marching through
space in one direction (z=0 to zo) evaluating the derivatives
containing the previous node (j-1) at time, t + 1/2 dt, and those
containing the following node (j+1) at time, t. Then returning in the
opposite direction (z=zo to 0), evaluate the derivatives containing the
node j+1, at t=t+dt, and the derivatives containing j-1 at t=t+1/2dt.
This technique uses a relatively simple algorithm similar to that for
the explicit methods because all of the values of the state variables
used in estimating the state variable at the next time step are known
quantities. However, the number of equations to be evaluated is twice