Approximating derivative in 2D finite volume

hunt_mat · November 9, 2023, 09:51

Suppose I have a rectangular cell with vertices at $(X_{i},Y_{j}),(X_{i+1},Y_{y}),(X_{i+1},Y_{i+1}),(X_{i},Y_{j+1})$ , If we label the sides, in the obvious way as $\{n,e,s,w\}$ . The finite volume method yields the derivatives evaluated at the cell sides, and
I know how to approximate:
$\frac{\partial\varphi}{\partial X}$

on the east, and west faces using Taylor series is a simple way. However, I am confused about how to do it on the north, and south sides. I thought about taking averages around the north(or south) edge.
$\frac{\partial\varphi}{\partial X}\Bigg|_{n}=\frac{1}{2}\left(\frac{\partial\varphi}{\partial X}\Bigg|_{j+1}+\frac{\partial\varphi}{\partial X}\Bigg|_{j}\right)$

hunt_mat · November 9, 2023, 11:48

So I think I have an answer to my own question:
I can average the derivatives to the top and bottom, so:
$\frac{\partial\varphi}{\partial X}\Bigg|_{n}=\frac{1}{2}\left(\frac{\partial\varphi}{\partial X}\Bigg|_{i,j+1}+\frac{\partial\varphi}{\partial X}\Bigg|_{i,j}\right)$
Then I use a central difference representation to obtain:
$\frac{\partial\varphi}{\partial X}\Bigg|_{i,j+1}\approx\frac{\varphi_{i+1,j+1}-\varphi_{i-1,j+1}}{2\delta X}$
$\frac{\partial\varphi}{\partial X}\Bigg|_{i,j}\approx\frac{\varphi_{i+1,j}-\varphi_{i-1,j}}{2\delta X}$
thus making:
$\frac{\partial\varphi}{\partial X}\Bigg|_{n}=\frac{\varphi_{i+1,j}+\varphi_{i+1,j+1}-\varphi_{i-1,j+1}-\varphi_{i-1,j}}{4\delta X}$
Does this look reasonable?

FMDenaro · November 9, 2023, 12:12

That depends on the choice you did for the colocation of phi. Is it at the FV center X(i+1/2),Y(j+1/2) or is cell-vertex ?

It seems a cell-center colocation, thus you need of an interpolation. But why do you need a tangential derivative on the faces? In a FV scheme it never enters into the update.

hunt_mat · November 9, 2023, 12:39

Quote:

Originally Posted by FMDenaro

That depends on the choice you did for the colocation of phi. Is it at the FV center X(i+1/2),Y(j+1/2) or is cell-vertex ?

It seems a cell-center colocation, thus you need of an interpolation. But why do you need a tangential derivative on the faces? In a FV scheme it never enters into the update.

So the i and j are centred at the cell centres. The n,e,s,w are the sides of the cell, the fluxes into and out of the cell. My stress tensor is very complicated and has all derivatives of everything, especially in Lagrangian co-ordinates.

FMDenaro · November 9, 2023, 13:17

Quote:

Originally Posted by hunt_mat

So the i and j are centred at the cell centres. The n,e,s,w are the sides of the cell, the fluxes into and out of the cell. My stress tensor is very complicated and has all derivatives of everything, especially in Lagrangian co-ordinates.

The normal component of the tensor has tangential derivative ? Then yes, use a linear interpolation.

November 9, 2023, 09:51	Approximating derivative in 2D finite volume	#1
hunt_mat Senior Member Matthew Join Date: Mar 2022 Location: United Kingdom Posts: 184 Rep Power: 4	Suppose I have a rectangular cell with vertices at $(X_{i},Y_{j}),(X_{i+1},Y_{y}),(X_{i+1},Y_{i+1}),(X_{i},Y_{j+1})$ , If we label the sides, in the obvious way as $\{n,e,s,w\}$ . The finite volume method yields the derivatives evaluated at the cell sides, and I know how to approximate: $\frac{\partial\varphi}{\partial X}$ on the east, and west faces using Taylor series is a simple way. However, I am confused about how to do it on the north, and south sides. I thought about taking averages around the north(or south) edge. $\frac{\partial\varphi}{\partial X}\Bigg\|_{n}=\frac{1}{2}\left(\frac{\partial\varphi}{\partial X}\Bigg\|_{j+1}+\frac{\partial\varphi}{\partial X}\Bigg\|_{j}\right)$

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November 9, 2023, 11:48		#2
hunt_mat Senior Member Matthew Join Date: Mar 2022 Location: United Kingdom Posts: 184 Rep Power: 4	So I think I have an answer to my own question: I can average the derivatives to the top and bottom, so: $\frac{\partial\varphi}{\partial X}\Bigg\|_{n}=\frac{1}{2}\left(\frac{\partial\varphi}{\partial X}\Bigg\|_{i,j+1}+\frac{\partial\varphi}{\partial X}\Bigg\|_{i,j}\right)$ Then I use a central difference representation to obtain: $\frac{\partial\varphi}{\partial X}\Bigg\|_{i,j+1}\approx\frac{\varphi_{i+1,j+1}-\varphi_{i-1,j+1}}{2\delta X}$ $\frac{\partial\varphi}{\partial X}\Bigg\|_{i,j}\approx\frac{\varphi_{i+1,j}-\varphi_{i-1,j}}{2\delta X}$ thus making: $\frac{\partial\varphi}{\partial X}\Bigg\|_{n}=\frac{\varphi_{i+1,j}+\varphi_{i+1,j+1}-\varphi_{i-1,j+1}-\varphi_{i-1,j}}{4\delta X}$ Does this look reasonable?

November 9, 2023, 12:12		#3
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,842 Rep Power: 73	That depends on the choice you did for the colocation of phi. Is it at the FV center X(i+1/2),Y(j+1/2) or is cell-vertex ? It seems a cell-center colocation, thus you need of an interpolation. But why do you need a tangential derivative on the faces? In a FV scheme it never enters into the update.