Implicit discretization of Jacobian term in FV: Construction of Coefficient matrix

Santiago · January 19, 2017, 07:31

I am currently trying to write the linear system coefficients of the following system:

$\frac{d\widehat{\mathbf{u}}}{\Delta t} - \underbrace{\frac{1}{|\Omega|}\int_{\partial \Omega}\frac{\partial \, d\widehat{\mathbf{u}}}{\partial \mathbf{n}}\,d\sigma}_{J(x)} = R(\mathbf{u})$

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):

$\left(\frac{\partial \, d\widehat{\mathbf{u}}}{\partial \mathbf{n}}\right)_e = \frac{d\widehat{\mathbf{u}}_P-d\widehat{\mathbf{u}}_E}{(\mathbf{x}_P-\mathbf{x}_E)\cdot\mathbf{n}}$

What makes me uneasy is the fact that $d\widehat{\mathbf{u}}$ 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?

FMDenaro · January 19, 2017, 07:45

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)

Santiago · January 19, 2017, 08:55

Quote:

Originally Posted by FMDenaro

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)

1.) No it's not the time derivative of $\hat{u}$ , $d \hat{u}$ it's just a vector. Besides the equation is already discrete in time.

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:

$[A][d\hat{\mathbf{u}}] = [R(\mathbf{u})]$

Part of the diagonal terms are given by the first term of LHS, but my question is how to discretize the second term.

Thanks.

FMDenaro · January 19, 2017, 09:28

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

Santiago · January 20, 2017, 05:35

I referred the derivative of the vector function as jacobian. My bad.

Sent from my GT-I8190L using CFD Online Forum mobile app

January 19, 2017, 07:31	Implicit discretization of Jacobian term in FV: Construction of Coefficient matrix	#1
Santiago 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: $\frac{d\widehat{\mathbf{u}}}{\Delta t} - \underbrace{\frac{1}{\|\Omega\|}\int_{\partial \Omega}\frac{\partial \, d\widehat{\mathbf{u}}}{\partial \mathbf{n}}\,d\sigma}_{J(x)} = R(\mathbf{u})$ 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): $\left(\frac{\partial \, d\widehat{\mathbf{u}}}{\partial \mathbf{n}}\right)_e = \frac{d\widehat{\mathbf{u}}_P-d\widehat{\mathbf{u}}_E}{(\mathbf{x}_P-\mathbf{x}_E)\cdot\mathbf{n}}$ What makes me uneasy is the fact that $d\widehat{\mathbf{u}}$ 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		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,897 Rep Power: 73	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)

January 19, 2017, 09:28		#4
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,897 Rep Power: 73	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

January 20, 2017, 05:35		#5
Santiago 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. Sent from my GT-I8190L using CFD Online Forum mobile app