Reference problem
=====================


Geometry
---------


.. image:: images/10000A9A00001BB3000014ACE499FE74620A6962.svg
    :width: 430
    :height: 259


.. _RefImage_10000A9A00001BB3000014ACE499FE74620A6962.svg:

:math:`\stackrel{ˉ}{\mathrm{AB}}=\stackrel{ˉ}{\mathrm{BC}}=L=0.1m`

:math:`x(\mathrm{M1})=0.02m`

:math:`x(\mathrm{M2})=0.08m`


Material properties
-----------------------

:math:`\lambda =1W/m°C`

:math:`\rho {C}_{P}=1000J/{m}^{3}°C`


Boundary conditions and loads
-------------------------------------

exchange :math:`\lambda {\frac{\partial T}{\partial n}\mid }_{x=\pm L}=h({T}_{\mathrm{ext}}-T(x,t))`

with: :math:`h=100.W/{m}^{2}°C`

     :math:`\{\begin{array}{}{T}_{\mathrm{ext}}{(C)}_{t=0}=100.\\ {T}_{\mathrm{ext}}{(C)}_{t={10}^{-2}}=0.\end{array}`


Initial conditions
--------------------

:math:`T(x,0)=100°C` for everything :math:`x`

Discretization in time (t):


.. csv-table::

    "10 steps for", ":math:`[0.,\mathrm{1.D}–2]` ", "either", ":math:`\Delta t={10}^{–3}s`"
    "9 steps for", ":math:`[1D–2,\mathrm{1.D}–1]` ", "either", ":math:`\Delta t={10}^{–2}s`"
    "9 steps for", ":math:`[\mathrm{1.D}–1,1.]` ", "either", ":math:`\Delta t={10}^{–1}s`"
    "5 steps for", ":math:`[1.,2.]` ", "either", ":math:`\Delta t=2.{10}^{–1}s`"
    "8 steps for", ":math:`[2.,10.]` ", "either", ":math:`\Delta t=1s`"