[Users] McLachlan with shift advection but centred stencils

[Ian Hinder]

On 11 Apr 2016, at 20:49, Erik Schnetter <schnetter at cct.lsu.edu> wrote:

> Roland
> 
> This is currently no implemented. There are currently two choices for
> calculating "upwind" derivatives (i.e. the shift-related spatial part
> of the Lie derivatives):
> 
> If[splitUpwindDerivsKranc,
>   Upwind[dir_, var_, idx_] := dir PDua[var,idx] + Abs[dir] PDus[var,idx],
>   Upwind[dir_, var_, idx_] := dir PDu[var,idx]];
> 
> By setting
> 
> Upwind[var_,dir_,idx_] = PD[var,idx];
> 
> you can change these "upwind" derivatives to be identical to regular
> derivatives. This happens at the level of Mathematica, before the
> equations are passed to Kranc, so that you will need one less ghost
> zone.
> 
> I would be happy to add a respective parameter to the Kranc script.
> For performance reasons this can't be a run-time parameter, though.
> 
> Note that this is always possible with the BSSN equations -- this is
> not a question of stability. We are using upwind derivatives since
> Peter Diener (?) found out that these lead to more accurate binary
> black hole evolutions, compared to centred differences.

If you do use centred differences, then make sure you also use some artificial dissipation.  The dissipativity of the upwind difference operators seems to be enough to stabilise the scheme, but with centred operators, I found that the scheme appeared unstable until I added some dissipation..

-- 
Ian Hinder
http://members.aei.mpg.de/ianhin

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