Source term linearization

=Introduction=
In seeking the solution of the general transport equation for a scalar <math>\phi</math>, the main objective is to correctly handle the non-linearities by transforming them into linear form and then iteratively account for the non-linearity. The source term plays a central role in this respect when it is non-linear. For example, in radiation heat transfer, the source term in energy equation is expressed as fourth powers in the temperature. <br>
When the source is constant and independent of the conserved scalar, the finite volume method assumes that the value of S prevails of the control volume and thus can be easily integrated. For a given control volume P, we obtain <br>
<math>\int_{\Omega} S d\Omega = S\Omega \,</math> <br>

==Picard's Method==
Picard's method is the most popular method used in conjunction with the finite volume method. For a given control volume P, we start by writing the source term as <br>
<math> S = S_C + S_PT_P \,</math> <br>
where <math>S_C</math> denotes the '''constant''' part of S and <math>S_P</math> denotes the coefficient of <math>\phi_P</math> (not the value of S at P). This allows us to place <math>S_P</math> in the coefficients for <math>\phi_P</math>. <br>

Let <math>\phi_P^*</math> denote the value of <math>\phi_P</math>at the current iteration. We now write a Taylor series expansion of S about <math>\phi_P^*</math> as <br>
<math> S = S^* + \left (  \frac {\partial S}{\partial \phi} \right ) ^* \left ( \phi_P - \phi_P^* \right ) </math> <br>
therefore<br>
<math> S_C = S^* - \left (  \frac {\partial S}{\partial \phi} \right ) ^* \phi_P^*</math> <br>
<math> S_P = \left(  \frac {\partial S}{\partial \phi} \right ) ^* </math> <br>
where <math>\left  (  \frac {\partial S}{\partial \phi} \right ) ^* </math> is the gradient of S evaluated at <math>\phi_P^*</math>. <br>

==Example==
As an illustrative example, consider <math> S = -T^3 + 10 \,</math>. Following Picard's method, we have <br>
<math>\left( \frac {\partial S}{\partial \phi} \right ) = -3T^2 </math> <br>
<math> S_C = -T_P^{*3} +10  + 3T_P^{*2}T_P^* = 2T_P^{*3} +10 </math> <br>
<math> S_P = -3T_P^{*2} </math> <br>

==References==
#{{reference-book|author=Patankar, S.V.|year=1980|title=Numerical Heat Transfer and Fluid Flow|rest=ISBN 0070487405, Hemisphere Publishing Corporation, USA.}}
#{{reference-paper|author=Murthy, Jayathi Y.|year=1998|title=Numerical Methods in Heat, Mass, and Momentum Transfer|rest=Draft Notes, Purdue University ([http://widget.ecn.purdue.edu/%7Ejmurthy/me608/main.pdf/ download])}} 
#{{reference-paper|author=[http://webfea-lb.fea.aub.edu.lb/fea/me/CFD/ Darwish, Marwan]|year=2003|title=CFD Course Notes|rest=Notes, American University of Beirut}}