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270 Polynomial Approximation of Differential Equations
Provided h is sufficiently small, this algorithm gives stable and consistent results.
The restriction on h is severe (h ≈ 1/n 4 ) due to the jump in the initial condition, but
can be relaxed (h ≈ 1/n 2 ) when the heat flux at the free boundary starts decreasing in
magnitude.
Numerical results for T = 1, λ = 5, n = 6, h = 1/800, are now presented to
illustrate the algorithm. The polynomials p (k)
n , k = 2, 10, 100, 200, 400, 800, are
plotted in figure 12.1.1. Some terms of the sequence {γk}0≤k≤800 are given in figure
12.1.2.
Figure 12.1.1 - Heat propagation for Figure 12.1.2 - Approximation of the
the Stefan problem. free boundary.
As predicted by asymptotic estimates we have Γ(t) ≈ σ √ t, where σ > 0 is related to
λ. As the reader can see in figure 12.1.2, this behavior is maintained by the approximate
free boundary.
We conclude by noting that a relation similar to (12.1.6) holds for the discretizing
functions. Actually, using (12.1.8), we have