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

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.

• The term you are looking for is "multirate Runge-Kutta". Sep 29 at 9:56
• Do you have an example where multirate Runge-Kutta does not use different timesteps for the stiffer components, but more stage evaluations? I can only find resources where different timesteps are used. Sep 29 at 10:03
• "Using different timesteps" can always be equivalently written as "using more stage evaluations", so this is really a matter of your point of view rather than an actual difference. But you may be interested in this paper of mine, where the goal is not multirate but specifically using RK methods with different stability properties. Sep 29 at 11:16
• Take a look at Table 2 of this paper for instance, to see how stages/steps can be made equivalent. Sep 29 at 11:21
• You touched this issue slightly in the first paper you mentioned with a blending approach, and this paper takes a close look at what happens at the interface. Oct 6 at 7:53

## 1 Answer

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.

• Thnaks for the answer, but I am in particular interested inHyperbolic Conservation Laws instead of parabolic equations, so Runge-Kutta Chebyshev is not exactly what I am looking for. Furthermore, I am especially interested in spatially partitioned methods and try to find some resources for this. Oct 6 at 6:41
• You could construct an SSP-optimal ROCK expansion method. I don't know of anyone who has done it, so it would be a great research project. Oct 7 at 18:01