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Implicit discretization of Jacobian term in FV: Construction of Coefficient matrix |
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January 19, 2017, 07:31 |
Implicit discretization of Jacobian term in FV: Construction of Coefficient matrix
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#1 |
Senior Member
Santiago Lopez Castano
Join Date: Nov 2012
Posts: 354
Rep Power: 16 |
I am currently trying to write the linear system coefficients of the following system:
But I cannot get my head around on how to come up with the coefficients for the jacobian fluxes. I mean, I see a tensor that needs to be projected along $\mathbf{n}$ and from it you do the discretization. My attempt on this enterprise is (the simplest, I think): What makes me uneasy is the fact that is a vector. Thus, Are the coefficients for each of the vector's components equal to the face-normal projected distance between the neighboring centroids? |
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January 19, 2017, 07:45 |
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#2 |
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Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,882
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I do not understand your symbolism...
1) the first term in the LHS is the time derivative of u?? 2) Since the first term is a vector, the second term in the LHS is the integral of a tensor (uu??) projected along the normal unit vector to the surface n. For example, Int [S] n.uu dSIn your equation the symbol du/dn is still a tensor (why is d(du)?). The Jabian of the flux can be derived from the differential form starting from df/dxi = (df/dui)*(dui/dxi) |
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January 19, 2017, 08:55 |
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#3 | |
Senior Member
Santiago Lopez Castano
Join Date: Nov 2012
Posts: 354
Rep Power: 16 |
Quote:
2.) The the second term is just a diffusion transformed in a flux using the green-gauss theorem. What I want is to get the coefficients of A, for the system: Part of the diagonal terms are given by the first term of LHS, but my question is how to discretize the second term. Thanks. |
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January 19, 2017, 09:28 |
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#4 |
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,882
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thus, using Gauss you have a surface integral like
Int [S] n. grad u dS why do you need to introduce a Jacobian? you can directly discretize the integral of the fluxes |
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January 20, 2017, 05:35 |
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#5 |
Senior Member
Santiago Lopez Castano
Join Date: Nov 2012
Posts: 354
Rep Power: 16 |
I referred the derivative of the vector function as jacobian. My bad.
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Tags |
finite volume method, linear system |
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