CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Forums > General Forums > Main CFD Forum

conservative finite differences and finite volumes

Register Blogs Community New Posts Updated Threads Search

Reply
 
LinkBack Thread Tools Search this Thread Display Modes
Old   January 23, 2014, 16:26
Default conservative finite differences and finite volumes
  #1
Senior Member
 
Joachim
Join Date: Mar 2012
Location: Paris, France
Posts: 145
Rep Power: 15
Joachim is on a distinguished road
Hey everyone!

I have a question regarding finite volume / difference methods. If someone could explain this to me, that would be really cool.

Let's say you have a structured grid. If you use generalizes coordinates, can you say that a conservative finite difference scheme IS a finite volume scheme? (computing the fluxes at the midpoints, etc).

Basically, can you use the methods for cartesian finite volumes schemes with finite difference approximations in generalized coordinates?

(U1n+1(i,j)) - U1n(i,j))/dt = F1(i+1/2,j) - F1(i-1/2,j) + G1(i,j+1/2) - G1(i,j-1/2)

where U1, F1 and G1 are defined using generalized coordinates, etc.

sorry if the question is not super clear!

Thanks!

Joachim
Joachim is offline   Reply With Quote

Old   January 23, 2014, 16:41
Default
  #2
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,849
Rep Power: 73
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
FD is a method for discretizing the pointwise form of NS equations, conversely FV is a method for discretizing the integral form of the NS equations ...
Therefore, the methods are definitely different in general
FMDenaro is offline   Reply With Quote

Old   January 23, 2014, 16:46
Default
  #3
Senior Member
 
Joachim
Join Date: Mar 2012
Location: Paris, France
Posts: 145
Rep Power: 15
Joachim is on a distinguished road
hmm, you sometime end up with the same equations after using both methods though...
My question: if you try to solve the equation in generalized coordinates:

dU1/dt + dF1/dxsi = 0

then, using finite volumes, you would end up with

dU1/dt + F1(i+1/2,j) - F1(i-1/2,j) = 0

(assuming the midpoint rule)
since delta_xsi = 1. Your volume would appear as the jacobian of the cell in the equation. Then you can get the fluxes using finite differences in generalized coordinates. Can you then still say that this approach is finite volume?
Joachim is offline   Reply With Quote

Old   January 23, 2014, 17:10
Default
  #4
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,849
Rep Power: 73
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
Quote:
Originally Posted by Joachim View Post
hmm, you sometime end up with the same equations after using both methods though...
My question: if you try to solve the equation in generalized coordinates:

dU1/dt + dF1/dxsi = 0

then, using finite volumes, you would end up with

dU1/dt + F1(i+1/2,j) - F1(i-1/2,j) = 0

(assuming the midpoint rule)
since delta_xsi = 1. Your volume would appear as the jacobian of the cell in the equation. Then you can get the fluxes using finite differences in generalized coordinates. Can you then still say that this approach is finite volume?

the only case in which FV and FD produces the same algebraic equation is for linear equation discretized with second order central scheme, otherwise you get different equations.

Then, the integral equation writes as

d/dt Int [V] U dV + Int [BV] n.F dS = 0

in physical space. You can use any type of grid and write this equation in a FV manner directly in the physical space. It retains its phycial meaing of conservation equation

Conversely,

dU/dt + Div.F dS = 0

need a transformation into the computational space but that does not correspond to solve a physical integral equation in the transformed space
FMDenaro is offline   Reply With Quote

Old   January 23, 2014, 17:12
Default
  #5
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,849
Rep Power: 73
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
just as note, you can looking for some similar posts on CFD Online
FMDenaro is offline   Reply With Quote

Old   January 23, 2014, 17:14
Default
  #6
Senior Member
 
Joachim
Join Date: Mar 2012
Location: Paris, France
Posts: 145
Rep Power: 15
Joachim is on a distinguished road
so solving the integral form of the equations written in generalized coordinates does not make the scheme finite volume?
Joachim is offline   Reply With Quote

Old   January 23, 2014, 17:30
Default
  #7
Senior Member
 
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,849
Rep Power: 73
FMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura aboutFMDenaro has a spectacular aura about
Quote:
Originally Posted by Joachim View Post
so solving the integral form of the equations written in generalized coordinates does not make the scheme finite volume?
In my opinion it has no sense to think that the system of coordinates can make the method finite volume or something else...
have a look to the dedicated chapter in the Ferziger & Peric book
FMDenaro is offline   Reply With Quote

Old   January 23, 2014, 17:32
Default
  #8
Senior Member
 
Joachim
Join Date: Mar 2012
Location: Paris, France
Posts: 145
Rep Power: 15
Joachim is on a distinguished road
I have it right there...the don't use general coordinates at all for finite volume methods.
I thought that solving the integral form of the equations would make a finite volume scheme, not the coordinates...
Joachim is offline   Reply With Quote

Reply


Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off
Trackbacks are Off
Pingbacks are On
Refbacks are On


Similar Threads
Thread Thread Starter Forum Replies Last Post
Comparison: Finite Volume Method vs. Analytic Method m-fry Main CFD Forum 1 April 20, 2010 15:40
Natural convection - Inlet boundary condition max91 CFX 1 July 29, 2008 21:28
Multigrid for finite differences Enda Bigarelli Main CFD Forum 4 November 6, 2001 13:04
Finite differences Zdravko Stojanovic Main CFD Forum 8 March 24, 2001 23:31
U.F.D. :Finite Differences on Unstructure grids STROHM Main CFD Forum 4 August 6, 1998 16:25


All times are GMT -4. The time now is 16:25.