Super Time Stepping

Explicit integration of diffusive physics requires taking a parabolic time-step, which is proportional to , where is the linear resolution of the calculation. This parabolic time-step is often much smaller than the hyperbolic time-step, thus dramatically increasing the required compute time when running Athena++ with diffusive physics enabled.

Super-time-stepping offers an avenue to alleviate some of the penalty associated with an explicit scheme. When super-time-stepping is enabled, diffusive physics is treated in an operator-split fashion using the RKL algorithm (see Meyer et al. 2014). Athena++ has available the first-order accurate RKL1 integrator (sts_integrator=rkl1) and the second order accurate RKL2 integrator (sts_integrator=rkl2). These integrators enable us to take the hyperbolic time-step inside the main integration loop, while diffusive physics is integrated using super-time-steps. A super-time-step is comprised of s stages where

Above, a and b are constants that depend on STS integrator choice. , is the super-time-step, equal to the hyperbolic time-step in the RKL1 integrator, or half the hyperbolic time-step in the RKL2 integrator. Thus, one can see that for the RKL1 integrator, one operator split super-time-step is needed to integrate diffusive physics forward by one hyperbolic time-step. Similarly, for the RKL2 integrator, two operator split super-time-steps are needed (one taken before main integration, one after) to integrate diffusive physics forward by one hyperbolic time-step.

The larger the ratio between the hyperbolic and parabolic time-step, the larger the value of s. The speedup associated with super-time-stepping is proportional to s.

Configuration

Super-time-stepping can only be applied to problems with diffusive physics. To enable super-time-stepping, configure Athena++ with the -sts option.

> python configure.py -sts

Input File

If super-time-stepping is enabled, at least one of the following input parameters for diffusion processes must be set in the input file under the <problem> block:

    <problem>
    nu_iso      = 0.01       # isotropic viscosity coefficient
    nu_aniso    = 0.0        # anisotropic viscosity coefficient
    kappa_iso   = 0.01       # isotropic thermal conduction coefficient
    kappa_aniso = 0.0        # anisotropic thermal conduction coefficient
    eta_ohm     = 0.01       # Ohmic resistivity coefficient
    eta_ad      = 0.0        # Ambipolar diffusion coefficient

There are several optional parameters that can be set in the input file under the <time> block:

    <time>
    sts_integrator = rkl1   # sts integration algorithm
    dt_diagnostics = -1     # extra timestep diagnostic info
    sts_max_dt_ratio = -1   # max dt to parabolic dt ratio

• sts_integrator (optional) : sts integration algorithm. Choices:

  • rkl1 : first-order accurate RKL1 algorithm
  • rkl2 (default) : second-order accurate RKL2 algorithm

• dt_diagnostics (optional) : output extra timestep diagnostic info to stdout (at every ncycle_out interval). Choices:

  • -1 (default) : disabled
  • 0 : list dt_hyperbolic, dt_parabolic, and dt_user (if active) and their respective ratios to the overall main integrator dt. With STS enabled, the following metric is also output: stage_ratio = (dt*nstages)/(dt_parabolic*(nstages + nstages_sts)), where nstages corresponds to the main multistage explicit time-integrator.
  • n = positive integer : same behavior as 0, but output extra diagnostics at every STS stage for proof-of-progress if -sts was used during Configuring.

• sts_max_dt_ratio (optional) : limit time-step if the ratio of overall dt to dt_parabolic exceeds this value (-1 = infinity = default)

Example

Ohmic diffusion of a Gaussian magnetic field:

> python configure.py -b -sts --prob=resist --eos=isothermal

Convergence and Performance

Error convergence and performance diagnostics for RKL1 super-time-stepping vs. explicit integration in the Ohmic diffusion example problem:

Notes

• For diffusion problems with very large dt to dt_parabolic ratios, super-time-stepping accuracy begins to diminish. Although very problem/physics/resolution dependent, we advise users to enroll sts_max_dt_ratio in cases where such behavior is seen. In our experience, sts_max_dt_ratio=100 adequately throttles super-time-stepping, however, we strongly encourage users to test and use caution.
• If enrolling user-defined source terms with STS, note that flux contains only non-diffusive fluxes.
• When using orbital advection with STS, orbital_advection/OAorder=2 and time/sts_integrator=rkl2 are recommended.