Is this numerical techique for solving ODEs widely known?

[m4r35n357]

Whilst studying symplectic integrators (as a hobby!) I accidentally stumbled on http://www.maia.ub.edu/~angel/taylor/taylor.pdf, which contains a link to GPL source code for the method described. I found it fascinating, especially since searching around the topic (Taylor Series Methods) revealed that outside the immediate field the method is seemingly ignored by a large part of the literature.

I have been tinkering with the sources and have generated some amazing plots with this, I hope it is interesting to at least one more person!

The process dates back to about the 50s I think, but many more recent educational courses and resources do not mention it. This is not because of a lack of computing power in recent history, because I have been doing 32nd order simulations with 236 bit precision on a Raspberry Pi . . .

 

[m4r35n357]

Bumping this because I have found what looks like a prime example of general unawareness here. The authors use a variety of (low order, approximated) methods to analyze a rather complex set of ODEs, but bemoan the dependence of its behaviour on simulation parameters e.g step size.

For some values of b, the results depend drastically on the step-size, the initial conditions, and the numerical methods used. So, the available efficient numerical methods for ODEs, implemented in different software packages, might give unexpectedly different results for the same parameter values and initial conditions, while fixed-step-size schemes (such as the standard Runge-Kutta method RK4, or the predictor-corrector LIL method (see Appendix and [Danca(2006)], utilized in this paper) generally give more accurate results, although in some cases these are strongly dependent on the step-sizes.

I am just beginning an investigation to attempt to reproduce the results therein, armed with an arbitrary order, arbitrary precision solver using exact derivatives (automatic differentiation). I can post my code here if my credentials are in doubt ;)

BTW if the term "clean numerical simulation" is unfamiliar, here is one of several papers by its main proponent.

 

[m4r35n357]

Hmm, this rabbit hole gets deeper!

Whilst studying the paper in my previous post I stumbled across this one, which pretty much kills the "clean numerical simulation" idea IMO (gives the impression of being written in 2003, but contains references up to 2009, including a comment from Lorenz himself).