[PMueller]

Hi there,

I am currently studying, learning, and working on MUSCL-type schemes to solve the Euler equations.
So far, everything is functioning well; however, I keep questioning one detail on the linear reconstruction of the approximate solution and the associated slope limiting process.

Specifically, it’s about the choice of variables employed in the reconstruction and slope limiting process.
In all the standard literature the first mentioned straightforward approach for systems of conservations laws involves reconstructing the conservative variables. The improved cell-interface values are the utilized to evaluate the numerical flux, either using an upwind or a centered scheme. Subsequently, the new time level is obtained by using a conservative update
dUj/dt = -1/dx*[Fj+1/2 – Fj-1/2]
with an appropriate time integration method.
Now I am aware that the slope limiting is commonly applied to the primitive variables instead of the conserved ones which, for the Euler equations, is helping to ensure a positive reconstruction of density and pressure. In the book by Toro (Riemann Solvers and Numerical Methods for Fluid Dynamics) this also addressed. A scheme based on the reconstruction of primitive variables is presented, but the time update is done by using the primitive formulation of the equations.

My Question is now, the following:
‘Is it possible to perform the reconstruction and slope limiting in terms of the primitive variables, while still advancing the solution in time using the conservative formulation?’
The Scheme would the look like:
1. Transform the conservative cell-centered average to primitive cell-centered average
2. Reconstruct the primitive variables in each cell and limit the slopes (e.g. minmod) to obtain the improved cell-interface values based in terms of the primitive variables
3. Transform the primitive cell-interface values to conservative cell-interface values
4. Calculate the numerical Flux based on the conservative cell-interface values
5. Update the conservative variables using the conservative finite volume update (mentioned above)

I am aware of the extra computational effort by transforming the variables each time integration step twice.

I have looked up numerous CFD textbooks, research papers, and forum discussions; however, I have not encountered a direct mention of the “primitive variable reconstruction combined with a conservative update” approach. This leaves me somewhat uncertain as to whether this method is considered too straightforward to be explicitly documented, if I may have simply overlooked its discussion, or if it is fundamentally incorrect or not regarded as 'best practice' within the field. Some numerical tests regarding the 1D Euler-Equations seem to work just fine with this approach…

I would greatly appreciate any insights or clarifications on this matter.
Thank you very much in advance for your help!

Cheers,
Paul =)

[arjun]

I used this approach in this


https://youtu.be/So0XSFjTzag?si=ol97eAJrraciTKH9

[PMueller]

Hello Arjun,

thank you very much for your answer!

Just to clarify , you are referring to the approach, where the reconstruction is based on the primitive Variables, which are then transformed in to the conserved ones, right?

Do you happen to have any textbook, or is it really just 'trivial'?

The Video you shared is awesome. Respect for doing that yourself! I’m really looking forward to getting to that level as well.

[arjun]

Originally Posted by PMueller

Hello Arjun,



Just to clarify , you are referring to the approach, where the reconstruction is based on the primitive Variables, which are then transformed in to the conserved ones, right?

.

Yes. After the solution the variables are also constructed to primitive variables. Then they are interpolated to faces and then transport variables were created to solve Riemann problem.