Consistency of interpolation in terms of flux at node sides

[mertcan]

Hi, first of all I am aware of the fact that QUICK SCHEME used in computational fluid dynamics has consistent slope at the sides of node elements (for instance at the left side of node 3 in my attachment same slopes exist) as you can see in my picture/attachment. But I must express that I can prove (differentiating the second order interpolation polynom at side point of nodes) slopes at the left side of node(like in picture) are equal in QUICK SCHEME thus it is consistent but I know there are another schemes like VAN LEER or VAN ALBADA SCHEME which are non linear and I can NOT prove how those SCHEMES may be consistent in terms of slope at the left side of node like QUICK SCHEME. At the centre of length (length between node 2 and node 3 in my attachment) which means left side of node 3 QUICK SCHEME always ensure the consistency of slope and I can prove but HOW DO WE KNOW THAT VAN LEER VAN ALBADA SCHEMES MAY ENSURE THE CONSISTENCY OF slope at the left side of node 3?? How can we PROVE it?

For instance in order to prove QUICK SCHEME is consistent in terms of slope at the left point of node element I can use the continuous second order polynomial interpolation and differentiate it. BUT in VAN LEER or VAN ALBADA SCHEME I do not know the continuous form of polynomial interpolation thus I can not differentiate like in the case of QUICK SCHEME...

 

[bigfooted]

Do you mean van Leer's MUSCL scheme? Anyway, to determine if a scheme is flux-conserving, proceed as follows:
Consider a one-dimensional discretized domain with N+1 nodes from 0..N and N cells from 1..N. Consider nodes "i-1" and "i" inside the domain. Compute the flux entering this element and the flux exiting this element. Then do the same for the nodes "i" and "i+1".
If the flux exiting cell i is the same as the flux entering cell i+1, it is a global flux-conserving scheme.
Hope this helps.

These are very good summer school lectures: http://www2.mpia-hd.mpg.de/~dullemon/lectures/fluiddynamics08/
check out chapter 4.

 

[mertcan]

Do you mean van Leer's MUSCL scheme? Anyway, to determine if a scheme is flux-conserving, proceed as follows:
Consider a one-dimensional discretized domain with N+1 nodes from 0..N and N cells from 1..N. Consider nodes "i-1" and "i" inside the domain. Compute the flux entering this element and the flux exiting this element. Then do the same for the nodes "i" and "i+1".
If the flux exiting cell i is the same as the flux entering cell i+1, it is a global flux-conserving scheme.
Hope this helps.

These are very good summer school lectures: http://www2.mpia-hd.mpg.de/~dullemon/lectures/fluiddynamics08/
check out chapter 4.

I am curious about the a circumstance that is : when we apply finite difference to equation, the differenced equation may not be suitable for TVD(total variance diminishing) and open to some oscillations, so to prevent the oscillation we use flux limiters as a multiplication form, but when we modify our differenced equation with flux limiter how do we know that we still preserve the same truncation error? Attaching flux limiter to differenced equation may be decrease or increase truncation error?? How do we measure that??