[ERIKY]

I'm currently working on an assignment which is about using Central Difference(CDS), QUICK, Upwind, and MUSCL scheme (using flux limiter) to solve the convection-diffusion equation. however when I compile the MATLAB code there seems to be some error, I do not understand why the result of CDS after Peclet number like 10 and 100 looks so horrible. Although I knew that the scheme is invalid for Peclet number over 2, it should be at least looking similar to the result under Peclet number of 2. is the structure of the code wrong or the error part or sth else? Plus, the QUICK scheme seems cannot converge, but I do not know what went wrong....

Also, does anyone know how to apply the flux limiter to the code.... I understand the theory just do not know how to transfer it into code...

The problem description and the code will be attached, genuine thankful for the assistance.

[FMDenaro]

You should have studied the features of the CDS scheme when the cell Pe number is greater than 2. Have you first developed your homework in 1D? There are a lot of textbook where you can compare the results.
The limiter is a non linear function of the state variables that applies on the flux between two cells. You can have a look to the textbook of Ferziger, Peric and Street.

[FMDenaro]

Originally Posted by LandonT.F.

Can you show the sources where I can get those textbooks please? At least, one would be very welcome.

https://link.springer.com/book/10.10...-3-319-99693-6

[aerosayan]

Originally Posted by ERIKY

I do not understand why the result of CDS after Peclet number like 10 and 100 looks so horrible. Although I knew that the scheme is invalid for Peclet number over 2, it should be at least looking similar to the result under Peclet number of 2.

High Peclet numbers indicate that the convective effects dominate the flow instead of diffusive effects. Thus, when you use CDS, the physics is not well represented in the maths.

Considering a 1D flow from left to right ...

Basically, CDS takes data both from the left and right cells, and considers them to be equally important , thus it represents the physics wrong, as clearly, the importance of the upwind cell (left cell) is more than the downwind cell (right cell).

So, it fails with high Peclet numbers. What you need, is an upwind scheme.

For FEM based CFD, Galerkin formulation also has a similar issue, where the the Galerkin discretization assumes equal importance for the left and right cells. Thus methods like SUPG (Streamline Upwind Petrov Galerkin) method were introduced to mimick what upwind schemes do in FVM CFD.

Basically, both FEM and FVM based CFD solves the same problem in the same way, by introducing upwind schemes, so the physics is represented well.