The quasilinear heat conduction equation 523
engineering and technology. In tackling nonlinear problen1s some prelim-
inary tests may be of help in verifying the quality of numerical methods.
The traditional way of covering this is to compare numerical solutions of
a simple specimen problem with known analytical solutions of the same
problem.
It is worth noting here that Newton's method is quite applicable for
solving proble1n ( 35) in addition to the well-established method of itera-
tions.- Calculations of the ternperature waves. Of special interest is the case
where the coefficient k(u) is a function of temperature such that
( 41)As we have mentioned above, the process of heat conducting emerged in
that case with a finite velocity and the derivative (hl/ ox tends to= behind
the front for O' > 1.
The temperature waves can be found through the use of the scheme b)
relating to "continuous through execution" ones. No fixation of the front
applies here. Under the guidelines of the preceding section with regard
to problem (15)-(17) having the exact solution given by formula (18), the
calculations permit us to discover that almost everywhere except several
near-front nodes the approximate solution deviates very slightly fron1 the
exact one, not exceeding 0.002 for x 0 = 0.5, O' = 2, D = 5 and the total
nmnber of nodal points N = 50. In so doing the number of the necessary
iterations is no greater than 3 and t < 0.2. When the temperature wave is
n1oving frmn the left to the right along the zero te1nperature background,
more and more grid intervals are captured in a step-by-step fashion in the
process of numerical solutions in a number of different ways in connection
with possible cmnpu tations of the coefficient a; ( y).
Apparently, the n1ain idea behind this approach needs certain clari-
fication. For example, formula (38) necessitates i1nposing a nonzero back-
ground temperature prior to the front. In spite of this fact, there are some
delays in introducing new intervals, thus causing large deviations of a solu-
tion in a vicinity of the front. Formula (37) is useless for very large values of
the index O' (O' > 20). Formula (36) has the best accuracy and reproduces
rather accurately without concern of the background temperature.- The Stephan problem (problem of the phase transition). Subsequent
considerations include two phases with the coefficients of heat conductiv-
ity k 1 (1l), k 2 (1l) and of heat capacity c 1 (1l), c 2 (u), in either of which it is