  Unconditionally stable time stepping schemes are useful and often practically
necessary for advancing parabolic operators in multi-scale systems. However,
serious accuracy problems may emerge when taking time steps that far exceed the
explicit stability limits. In our previous work, we compared the accuracy and
performance of advancing parabolic operators in a thermodynamic MHD model using
an implicit method and an explicit super time-stepping (STS) method. We found
that while the STS method outperformed the implicit one with overall good
results, it was not able to damp oscillatory behavior in the solution
efficiently, hindering its practical use. In this follow-up work, we evaluate
an easy-to-implement method for selecting a practical time step limit (PTL) for
unconditionally stable schemes. This time step is used to `cycle' the
operator-split thermal conduction and viscosity parabolic operators. We test
the new time step with both an implicit and STS scheme for accuracy,
performance, and scaling. We find that, for our test cases here, the PTL
dramatically improves the STS solution, matching or improving the solution of
the original implicit scheme, while retaining most of its performance and
scaling advantages. The PTL shows promise to allow more accurate use of
unconditionally stable schemes for parabolic operators and reliable use of STS
methods.
