https://www.cfd-online.com/W/index.php?title=Special:Contributions/Continillo&feed=atom&limit=50&target=Continillo&year=&month=CFD-Wiki - User contributions [en]2017-03-29T06:02:57ZFrom CFD-WikiMediaWiki 1.16.5https://www.cfd-online.com/Wiki/Source_term_linearizationSource term linearization2007-02-22T10:25:42Z<p>Continillo: /* Picard's Method */</p>
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<div>=Introduction=<br />
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><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><br />
<math>\int_{\Omega} S d\Omega = S\Omega \,</math> <br><br />
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==Picard's Method==<br />
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><br />
<math> S = S_C + S_PT_P \,</math> <br><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><br />
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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><br />
<math> S = S^* + \left ( \frac {\partial S}{\partial \phi} \right ) ^* \left ( \phi_P - \phi_P^* \right ) </math> <br><br />
therefore<br><br />
<math> S_C = S^* - \left ( \frac {\partial S}{\partial \phi} \right ) ^* \phi_P^*</math> <br><br />
<math> S_P = \left( \frac {\partial S}{\partial \phi} \right ) ^* </math> <br><br />
where <math>\left ( \frac {\partial S}{\partial \phi} \right ) ^* </math> is the gradient of S evaluated at <math>\phi_P^*</math>. <br><br />
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==Example==<br />
As an illustrative example, consider <math> S = -T^3 + 10 \,</math>. Following Picard's method, we have <br><br />
<math>\left( \frac {\partial S}{\partial \phi} \right ) = -3T^2 </math> <br><br />
<math> S_C = -T_P^{*3} +10 + 3T_P^{*2}T_P^* = 2T_P^{*3} +10 </math> <br><br />
<math> S_P = -3T_P^{*2} </math> <br><br />
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==References==<br />
#{{reference-book|author=Patankar, S.V.|year=1980|title=Numerical Heat Transfer and Fluid Flow|rest=ISBN 0070487405, Hemisphere Publishing Corporation, USA.}}<br />
#{{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])}} <br />
#{{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}}</div>Continillohttps://www.cfd-online.com/Wiki/Flux_limitersFlux limiters2007-02-22T10:19:12Z<p>Continillo: /* References */</p>
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<div>Most of us might have seen the behaviour of numerical schemes in order to capture shocks and discontinuity that arises in hyperbolic equations. Physically, these equations model the convective fluid flow. It has been observed that low-order schemes are usually stable but quite dissipative in nature around the points of discontinuity/shocks. On the other hand higher-order numerical schemes are unstable in nature and show oscillations in the vicinity of discontinuity. One can have a better understanding of such behaviour by analysing the modified equation of these schemes.<br />
<br />
Now as said above one can not have high order accuracy without oscillations. The objective is to have highly accurate and stable oscillation free schemes. This kind of schemes are known as high resolution schemes. In 1984 Harten proposed the constuction and gave TVD criteria for such scheme. Here we talk only about using flux limiters function to construct high resolution schemes. The idea is to tune the numerical flux of high order and low order scheme using the flux limiter function in such a way that the resulting scheme gives a high order accuracy in the smooth region of flow and sticks with first order of accuracy in the vicinity of socks/discontinuities as follows: <br />
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Define the numerical flux fuction of high resolution conservative scheme as<br />
<br />
:<math>F_{j+ \frac{1}{2}} = L_{j+\frac{1}{2}} + \phi(H_{j+ \frac{1}{2}} - L_{j+ \frac{1}{2}})</math><br />
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where <math>L, H</math> are the numerical flux of conservative low order and high order schemes respectively.<br />
and <math>\phi</math> is a function of smoothness parameter <math>\theta</math> usually defined as <br />
<br />
:<math>\theta_j = \frac{U_j -U_{j-1}}{U_{j+1} - U{j}}</math><br />
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What remains next is to define the limiter fuction <math>\phi(\theta)</math> in such a way that it satisfies at least the following properties:<br />
<br />
* remains positive <math>\forall \theta</math>,<br />
* satisfies <math> \phi(1) = 1</math><br />
* passes through a perticular region known as TVD region associted with the underlying scheme in order to guarantee stability of the scheme.<br />
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==References==<br />
<br />
* C. B. Laney, "Computational Gas Dynamics"<br />
* E. F. Toro, "Riemann Solvers and Numerical Methods for Fluid Dynamics"<br />
* R. J. Leveque, "Numerical Methods for Conservation Laws"</div>Continillo