GSL ODE-solvers' status

[Tuomo Keskitalo <Tuomo dot Keskitalo at iki dot fi>]

Dear all,

Patrick found this recent blog post by Christopher Rackauckas, who is 
the developer of DifferentialEquations.jl, which contains a wide 
selection of ODE-solvers for Julia:

http://www.stochasticlifestyle.com/comparison-differential-equation-solver-suites-matlab-r-julia-python-c-fortran/

Christopher also has given an informative talk at the Julia Developers 
conference:

https://www.juliabloggers.com/video-introduction-to-differentialequations-jl/

I'm happy to see that there has been at least some progress on ODE 
solving methods! Unfortunately, I am no longer really up-to-date with 
this field. If someone has the opportunity to work on GSL ODE-solvers, I 
try to point towards potential places of improvement. The methods 
suggested by Cristoph in his presentation are:

On explicit side:
- Bogacki-Shampine 3/2 method
- Tsitouras 5/4 Runge-Kutta method
- Verner 7/6 Runge-Kutta method
On implicit side:
- Rosenbrock 2/3 method
- Radau IIA Runge-Kutta variable order method
- CVODE_BDF. GSL's msbdf is based on this method.

More information on these are surely available in 
DifferentialEquations.jl source codes.

Also, for time stepping, there appears to be some PI-controller based 
method for adjusting step size, which is very interesting.

Finally, Cristoph does not talk in favor of extrapolation methods, but I 
have previously wondered how much better bsimp would fare, if the 
internal iteration would be modified so that it continues only up to 
user given tolerances, instead of near machine precision.

It would be interesting to see if these (old and) new methods would 
yield significant improvements compared to those existing now in 
ode-initval2, or would improvements remain marginal?

As for GSL ode-initval2 framework: Room for improvement exists in better 
co-operation of GSL stepper and control routines. For multistep methods, 
it is not good that method order is adjusted internally by stepper and 
step size is adjusted independently by control routines.

Final tip for all ODE-solver users: Benchmark different steppers for 
your problem. Depending on the problem and your tolerance requirements, 
you may get significant improvements just by changing the stepper.

BR,
Tuomo

--
Tuomo.Keskitalo@iki.fi
http://iki.fi/tuomo.keskitalo