Reconstruction Limiters...

Running some 2D colliding flow stuff and noticed that gudonov and ppm give very different results initially. Gudonov method looks diffusive, but correct while PPM looks funny. Not sure if this is just unresolved cooling or a potential bug in PPM or just the way the multi-D limiters are being used…

1st order reconstruction 2nd order 3rd order

I double checked that the ppm and plm methods revert to gudonov type methods when the additional shock flatteners are triggered and found one minor 'bug'. I also compared the results with no limiters at all and they are worse.

Here they are with the corrected form of the current limiters

And here is a map showing the actual value of the slope limiters used

The horizontal striping is a result of the stencil used for calculating the multidimensional limiter. First limiters are calculated using slopes in x and y. Then the multidimensional limiter for a given cell is computed as the minimum of the x and y slopes in the surrounding cells as below

y
x xy x
y

To reduce the striping I modified the multidimensional limiter to instead be the minimum of

xy xy xy
xy xy xy
xy xy xy

Here is the resulting plot of the multidimensional limiter

Finally I also added an extra dissipation that is usually used with PPM schemes that is proportional to the divergence (and is active for compressible flow).

(Left panel has the additional dissipation)

And here is a movie​

And finally I did a convergence study at 5 resolutions (left to right) of the 1st order reconstruction (bottom), ppm with diffusion (middle), and ppm without diffusion (top)

I also added some smoothing to the velocity interface to ease the shock in so that the initial velocity gradient was not a function of the resolution.

Note that the lower res PPM schemes are actually better able to resolve some features then the next higher res gudonov scheme (albeit more pixelated).

Here is the movie​

To see the modifications in action with mixing see the Colliding Flows Page

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