[Simbelmynë]

Hey,

I have a question regarding the implementation of TVD schemes.

If I run the convection-diffusion equation in 1d and compare the analytical result to the results produced by standard limiters (Van Leer, Min Mod, SUPERBEE, etc.) then there is always too much upwinding. The results are of course better than standard upwind, but not good enough.

However, it seems to me that it is possible to adjust the contribution from upwind part of the derivative to the downwind part so that the results always match the exact solution. So far I have done this by trial and error.

It is clear that the matrix stability is not a good measure (too much diffusion), so I have played around with the ratio between upwind to downwind gradient in combination with the cell Peclet number. However, I have yet to find any universal input that works better than the TVD scheme in all cases (it is never worse though). Any input on this would be helpful!

The attached figures show a case with Pe=150 (based on the length of the domain) and 41 nodes.

Solid line is the analytical solution. Red is the central difference. Blue is the TVD (Van Albada limiter) and yellow is the "trial and error" case. In the final figure I have set the "trial and error" case to pure upwind.

[FMDenaro]

Have a look to the limiters proposed in these papers

B. P. Leonard. The ULTIMATE conservative difference scheme applied to unsteady one–dimensional advection. Computer Methods in Applied Mechanics and Engineering, 88:17–74, June 1991

and

https://pdfs.semanticscholar.org/0ff...12a5873279.pdf

[Simbelmynë]

Thank you. That was a rather massive investigation on the subject