Benchmark solution
=====================

Calculation method used for the reference solution
--------------------------------------------------------

:math:`T(x,y,z,t)={T}_{0}+{\mathrm{2q}}_{w}\frac{\sqrt{\alpha t}}{\lambda }(A+B+C)` with:

:math:`A=\sum _{m=0}^{\infty }\left[\mathrm{i.erfc}\left[\frac{(\mathrm{2m}-1){L}_{1}+x}{2\sqrt{\alpha t}}\right]+\mathrm{i.erfc}\left[\frac{(\mathrm{2m}-1){L}_{1}-x}{2\sqrt{\alpha t}}\right]\right]`

:math:`B=\sum _{m=0}^{\infty }\left[\mathrm{i.erfc}\left[\frac{(\mathrm{2m}-1){L}_{2}+y}{2\sqrt{\alpha t}}\right]+\mathrm{i.erfc}\left[\frac{(\mathrm{2m}-1){L}_{2}-y}{2\sqrt{\alpha t}}\right]\right]`

:math:`C=\sum _{m=0}^{\infty }\left[\mathrm{i.erfc}\left[\frac{(\mathrm{2m}-1){L}_{3}+z}{2\sqrt{\alpha t}}\right]+\mathrm{i.erfc}\left[\frac{(\mathrm{2m}-1){L}_{3}-z}{2\sqrt{\alpha t}}\right]\right]`

:math:`\alpha =\frac{\lambda }{\rho {C}_{p}}`

Reference values are obtained with :math:`m=1000.`


Benchmark results
----------------------

Temperature at points: :math:`O(\mathrm{0,0}\mathrm{,0})`, :math:`H(0.5\mathrm{,0}.8\mathrm{,1}\mathrm{.})` and :math:`C(1.\mathrm{,1}.6\mathrm{,2}\mathrm{.})`


Uncertainty about the solution
---------------------------

Analytical solution.


Bibliographical references
---------------------------


* M.J. Chang, L.C. Chow, W.S. Chang, "Improved alternating direction implicit for solving transient three dimensional heat diffusion problems", Numerical Heat Transfer, vol. 19, vol. 19, pp 69-84, 1991.