2. Benchmark solution#

2.1. Calculation method used for the reference solution#

Semi-analytical solution involving error functions:

\(T(x,t)=2\left\{\sqrt{\left[1+2\sqrt{(t/\pi )}\mathrm{exp}(\frac{-{x}^{2}}{\mathrm{4t}})+\mathrm{x.erfc}(\frac{x}{2\sqrt{(t)}})\right]}-1\right\}\)

with \(\mathrm{erfc}(x)=\frac{2}{\pi }{\int }_{x}^{\infty }{e}^{-{t}^{2}}\mathrm{dt}\)

Where \(x=\) is the abscissa

\(t=\) time

This formula is only valid for \(\lambda (T)=\rho c(T)=1.+\mathrm{0.5T}\)

Temperature at points \(A\) (\(x=0\)) and \(E\) (\(x=1\)) at the following moments \(t\): \(t=0.1,0.3,0.5,0.7\) and \(\mathrm{1s}\)

Unknown, due to evaluation of error functions.

Segal, N. Praagman, « A fast implementation of explicit time stepping algorithms with the finite element method for a class of nonlinear evolution problems », Int. J. num. Meth. Engng, vol. 23, pp. 155-168, 1986.