Production term in Finite Volumes Schemes
Dear all,
I have a working scheme for LES for a transport equation of a scalar, let's say Phi, done with Finite Volumes schemes in the code I am using. I need to modify this code by adding a production term which consists substantially of two nonlinear terms, let's say k1*Phi^2 and k2*grad(Phi)*grad(Phi), where k1 and k2 are constants. I want to implement it as an implicit scheme, which should be like: A_d * Phi = A_nd * Phi + b where A_d and A_nd are the diagonal matrix and notdiagonal matrix of coefficients and b is the vector of known terms. I am confused on how to implement these new terms. Does someone have an idea or can suggest me some book where to look for? Thank you! 
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First, you have to consider the accuracy order of your FV discretization. At second order of accuracy the volume integral can be discretized simply by the mean value formula but I do not suggest that approach since it is not effective in terms of tophat filtering. Better is the multidimensional extension of the trapezoidal rule. If you have an accuracy more than two then your volume integral must be discretized congruently, for example by using shape functions of high degree so that you can integrate them analitically 
It is said that the terms in the coefficient matrix are calculated as upwind and then a second order correction is added on the RHS in a deferred way (so that they can use a blending factor I guess)...so the accuracy is second order. Therefore, I'm thinking:
1) On one side, wether using an upwindlike discretization for those terms (plus a second order correction which I would not be sure how to apply) or directly a second order scheme (ex. a central scheme). 2) I'm confused, chosen the scheme, what I should write. For example, since for each time step, a number of subiterations are performed, can I discretize the term, i.e., k1*Phi^2 in cell i as k1*Vol_i*Phi(n1)_i * Phi(n)_i, where n is the current subiteration? And what about the term grad(Phi)^2 
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Furthermore, the volume integral discretized with the mean value formula is second order of accuracy but it is not effective as filter, as you can simply see in the wavenumber space 
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Do you know some book/paper which explain this issue of the Upwind when using sgs model and the issue of the "noneffctiveness" of the filter you were mentioning (I will be required to justify my choices)? 
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k1(Phy_p)^2*Volp +k2*(grad(Phi)_p*grad(Phi)_p)*Volp which is second order too where Phy_p and grad(Phi)_p are evaluated at the cell center P and Volp is the volume of cell P. these terms of course are located in rhight hand side, i.e in b You could also implict the term k1(Phy_p)^2 while adding the quantity k1(Phy_p)*Volp in the main diagonal A_d. It can add stability if large values of Phy occurs. for k2*grad(Phi)*grad(Phi) I do not see any other way instead of expliciting this term. If you have a kind of deferred correction used in the code, it should work fine. 
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Hi leflix, consider that you are correct in term of accuracy order in physical space but for the aim of an LES, the volume average should be like a tophat filter. That means, for example in 1D, that the transfer function is in the form sin(kh/2)/kh/2. If you use simply the mean value you will find that the transfer function is the unity function...thus is not effective as filter 
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Hi Filippo, ok my contribution here was just about discretization principles. From LES concerns I do trust you !;) 
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Regards Filippo 
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