About the flux in FVM

Tommy Chen · March 25, 2015, 16:57

Currently I am reading some example code of 1D euler equation with FVM.

I wonder is that in FVM, the state at the boundary of a cell should be equal to the center value ? or it has to be interpolate using a slope computed by neighbored cells so that it is not equal to the cell center?

FMDenaro · March 25, 2015, 17:03

Quote:

Originally Posted by Tommy Chen

Currently I am reading some example code of 1D euler equation with FVM.

I wonder is that in FVM, the state at the boundary of a cell should be equal to the center value ? or it has to be interpolate using a slope computed by neighbored cells so that it is not equal to the cell center?

the time update is provided in terms of volume average, therefore you have to expresse a functional relation between average and pointwise value for computing the flux. For example, the Burgers equation is solved for the average u_bar but the flux is u^2/2. Therefore you need a reconstruction u=u(u_bar). Assuming u=u_bar in the center of the FV is a second order accurate reconstruction. Higher order reconstruction is more complex

Tommy Chen · March 25, 2015, 17:07

Quote:

Originally Posted by FMDenaro

the time update is provided in terms of volume average, therefore you have to expresse a functional relation between average and pointwise value for computing the flux. For example, the Burgers equation is solved for the average u_bar but the flux is u^2/2. Therefore you need a reconstruction u=u(u_bar). Assuming u=u_bar in the center of the FV is a second order accurate reconstruction. Higher order reconstruction is more complex

So the state could be either treated equal to the cell center value everywhere inside the cell, or computed by a reconstruction approach， depending on the requirement of accuracy. Did I understand it correctly ?

FMDenaro · March 25, 2015, 17:12

Quote:

Originally Posted by Tommy Chen

So the state could be either treated equal to the cell center value everywhere inside the cell, or computed by a reconstruction approach， depending on the requirement of accuracy. Did I understand it correctly ?

yes, correct

Tommy Chen · March 25, 2015, 21:44

Quote:

Originally Posted by FMDenaro

yes, correct

Thank you !

Very clear explanation , really helpful !

sbaffini · March 26, 2015, 16:36

Also consider that u=u_bar is second order in the cell center only. If, on a cell face, you compute the velocity as u_f = u_bar (the average value in the cell), you ar at most first order accurate. If you pick the wrong cell for such interpolation (i.e., downwind) you are not even first order.

March 25, 2015, 16:57	About the flux in FVM	#1
Tommy Chen Member Tommy Chen Join Date: Mar 2011 Location: University of Michigan Posts: 96 Rep Power: 15	Currently I am reading some example code of 1D euler equation with FVM. I wonder is that in FVM, the state at the boundary of a cell should be equal to the center value ? or it has to be interpolate using a slope computed by neighbored cells so that it is not equal to the cell center?

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March 26, 2015, 16:36		#6
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,151 Blog Entries: 29 Rep Power: 39	Also consider that u=u_bar is second order in the cell center only. If, on a cell face, you compute the velocity as u_f = u_bar (the average value in the cell), you ar at most first order accurate. If you pick the wrong cell for such interpolation (i.e., downwind) you are not even first order.