Linear numerical instability in electromagnetic runs, blow up at small time step

There have been reports of various numerical instabilities from different users. A common feature is that these appear in linear simulations which include at least apar and one of phi/bpar and seem to be particularly obvious at low kperp.

Similar numerical instabilities can become unstable as the time step is reduced in nonlinear simulations.

To explore if these are related I set up GS2 to calculate the non-linear term but to exclude this from the source term. In other words, this gives us a linear simulation in which the time step may vary to try to satisfy the NL cfl condition.

We see in these simulations that we can indeed suffer from this numerical instability once the time step has dropped far enough. Interestingly this seems to depend, at least partially, on the initial time step used. This could therefore be particular difficult to avoid in NL simulations where the initial linear phase may behave well, but as the time step drops we find such numerical instabilities become more significant. One approach to stabilise this mode linearly is to use -ve fexpr values, but this can degrade our ability to capture relevant physics and change linear growth rates if taken too far.

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