1-D spherical finite volume discretisation

lost.identity · October 6, 2010, 09:54

I'm trying to discretise the conservative equations using FVM for a 1-D spherical geometry (i.e. spherical symmetry).

After applying the Gaus's theorem I get the following for spatial discretisation of the diffusive terms

$\boldsymbol{\Gamma_f}\cdot\boldsymbol{A}_f$

I know that for 1-D Cartesian the face area vector is simply $\boldsymbol{A}=\boldsymbol{i}$ where i is the unit vector in the x-direction.

However, what will be the face area vector for a spherical geometry? Surely it can't just be the unit vector.

I know that the area element in spherical coordinates is given by

$d\boldsymbol{A} = r^2\sin\theta{d}\theta{d}\phi\boldsymbol{e}_r$ symbol. So in 1-D can I assume it to be just r^2?

Thanks.

david_e · October 7, 2010, 15:36

You can take it to be

, because to get the conservative formulation one usually integrates over $\theta$ and $\phi$ .

October 6, 2010, 09:54	1-D spherical finite volume discretisation	#1
lost.identity Senior Member Join Date: Apr 2009 Posts: 118 Rep Power: 17	I'm trying to discretise the conservative equations using FVM for a 1-D spherical geometry (i.e. spherical symmetry). After applying the Gaus's theorem I get the following for spatial discretisation of the diffusive terms $\boldsymbol{\Gamma_f}\cdot\boldsymbol{A}_f$ I know that for 1-D Cartesian the face area vector is simply $\boldsymbol{A}=\boldsymbol{i}$ where i is the unit vector in the x-direction. However, what will be the face area vector for a spherical geometry? Surely it can't just be the unit vector. I know that the area element in spherical coordinates is given by $d\boldsymbol{A} = r^2\sin\theta{d}\theta{d}\phi\boldsymbol{e}_r$ symbol. So in 1-D can I assume it to be just r^2? Thanks.

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October 7, 2010, 15:36		#2
david_e New Member Join Date: May 2010 Posts: 8 Rep Power: 15	You can take it to be , because to get the conservative formulation one usually integrates over $\theta$ and $\phi$ .