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

Calculation method used for the reference solution
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:math:`{T}_{(x,y,z,t)}={T}_{w}+\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }\sum _{l=1}^{\infty }{a}_{\mathrm{nml}}\mathrm{exp}(-{\kappa }_{\mathrm{mnl}}^{2}\alpha \mathrm{.}t){\mathrm{Tcos}}_{(x,y,z,m,n,l)}`

with :math:`{\mathrm{Tcos}}_{(x,y,z,m,n,l)}=\mathrm{cos}(\frac{(2m-1)\pi x}{2{L}_{1}})\mathrm{cos}(\frac{(2n-1)\pi y}{2{L}_{2}})\mathrm{cos}(\frac{(2l-1)\pi z}{2{L}_{3}})`

:math:`{a}_{\mathrm{mnl}}=\frac{64({T}_{0}-{T}_{w})}{{\pi }^{3}(2m-1)(2n-1)(2l-1)}\mathrm{sin}(\frac{(2m-1)\pi }{2})\mathrm{sin}(\frac{(2n-1)\pi }{2})\mathrm{sin}(\frac{(2l-1)\pi }{2})`

:math:`{\kappa }_{\mathrm{mnl}}={(\frac{(2m-1)\pi }{2{L}_{1}})}^{2}+{(\frac{(2n-1)\pi }{2{L}_{2}})}^{2}+{(\frac{(2l-1)\pi }{2{L}_{3}})}^{2}`

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

Reference values are obtained with :math:`m=n=l=100.`


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