Stabilized Many Stage Runge-Kutta methods instead of Local/Multirate Time Stepping

Question

Locally refined meshes are often inevitable for accurate, yet feasible computations. In the context of time-dependent PDEs, however, this comes at the cost that (due to the CFL condition) reducing the mesh width $\Delta x$ implies also that the timestep $\Delta t$ has to be reduced.

To alleviate this problem, many effort has been spend on local/multi-rate timestepping, see for instance [1], [2], [3].

Relatively recently, Brian Vermiere proposed a family of Runge-Kutta methods that uses in refined part of the mesh explicit optimized/stabilized Runge-Kutta methods with higher number of stages, while in coarse regions Runge-Kutta methods with fewer stage evaluations are used.

Unfortunately, there are no details given in [4] regarding implementation and numerical analysis.

Therefore, I am looking for similar publications, where Runge-Kutta methods with different number of stages instead of locally reduced timesteps are used to compute locally refined problems.

Answer 1

What you are looking for are stabilized Runge-Kutta methods or the family of Runge-Kutta-Chebyshev methods. This technical report is a good start, and a growing list of references are listed in the OrdinaryDiffEq.jl documentation. The ROCK methods have been the most successful subset of this family.

There are implementations in the DifferentialEquations.jl software suite. This has the most comprehensive set of optimizations and benchmarking around it, trying to figure out if these ESRK methods are potentially viable. There's some pieces confirming that they are indeed not efficient on non-stiff equations. The Filament PDE benchmarks demonstrate ROCK2 as the most efficient method for moderate tolerances on a nonlinear PDE, which gives them a bit of hope.

Though on most other stiff ODE benchmarks in the benchmark suite, we see that in many cases the ESRK methods are simply unstable. Many chemical reaction networks simply need more stability properties than what they provide. Remember, ESRK is looking at RKC for linear stability. In the theory of stiff ODEs, there are much stronger definitions of stability than A-stability, such as L-stability and B-stability for nonlinear stability. The ESRK methods do not have good properties for these stronger definitions of stability, and that's probably a nice avenue for future research. Until then, the "amount" of stability that ESRK provides is simply not enough to stabilize the "very stiff" chemical reaction networks, but seems very well suited for PDE semi-discretizations.