Move to extrapolation based boundary proximity handling

[xtalax]

Currently we use a stencil centrepoint shifting method to deal with domain edges, I propose moving to an extrapolation based approach to ensure stability, adding extra ghostpoints to pad the domain

[xtalax]

@ChrisRackauckas @ranocha thoughts?

[ChrisRackauckas]

Is the extrapolation stable? I don't remember it being so.

[xtalax]

I would use lagrange weights, so likely not. There are some modified lagrange interpolations that claim better stability, may be better to use those.

Having thought about this I think I will use a high order interpolation that also references the boundary states, though am suspicious that this will work out equivalent to just shifting the centrepoint.

Perhaps a multidimensional interpolation would be better? I would need to generalise fornberg in this case which is a significant undertaking, but doable.

Whatever the case the current implementation has caused a lot of problems and needs to go, what would you suggest as an alternative to create additional ghost points?

[ranocha]

The only provably stable WENO methods I know of are the ones of Carpenter et al.

• https://doi.org/10.1016/j.jcp.2011.01.043
• https://doi.org/10.1007/978-1-4614-5389-5_6

You won't like that you need to know something about the equations you are discretizing - so it doesn't fit the general framework here. Having said that, I think Clawpack provides some extrapolation BCs for their WENO methods (but I can't give you specific references).

[hhaensel]