More on ODEs, MMS and Ill-Posed IVPs
First for the nomenclature: ODEs means Ordinary Differential Equations, MMS means the Method of Manufactured Solutions, and IVPs means Initial Value Problems.
This previous post provided some information on these subjects. So far as I know, that post presented the first results for application of MMS to ill-posed IVPs. That post suggested that for the numerical solution methods used therein, the original Lorenz system of 1963 has yet to be correctly solved.
I have some additional results, a summary of which is:
I think the Lorenz system has not yet been accurately integrated by any numerical solution methods. Higher-order methods plus, at the same time, higher precision representation of numbers will give results that might appear to be solutions. But, calculations for sufficiently long time spans will show that errors always increase.
I’ve uploaded a file here.
Models Methods Software 
Dan,
I’m away from my normal computer or I’d do this myself. Do the Jacobian eigenvalues have negative real parts for your choice of parameters throughout the period of the manufactured solution? I’m pretty sure 1.8(b) has positive real parts, but not sure about 1.8(a).
If you want, I’ll try to replicate your results with my little implicit Radau scheme and see if we get the same sort of behaviour.
Josh
Josh,
Thanks for the remainder. I haven’t done that yet and need to get back to it. The parameters given in (1.8b) are the classic values for exhibiting chaotic response. I did integrate the equations using (1.8a) out to 200 LTUs without the MMS functions as I noted in the write-up. The results were completely as expected; rock solid approach to an equilibrium state. And the differences between two runs with different step sizes decreased as the time increased. Decreased to the number of good digits expected with double precision arithmetic.
When you get back to your regular machine, replicating my results is a very good idea.
Part of the problem is that the chaotic response is associated with the original un-modified Lorenz system. Adding the source terms, which are functions of the independent variable, changes the equation system at a fundamental level relative to temporal chaotic response theory. I remain surprised that the growth occurs although the results with MMS show the expected outcome for limited time periods.
I haven’t yet looked at other classic ODE systems showing chaotic response. I’m not sure what to do if those systems don’t have a region of parameter space that doesn’t show chaotic response.
This write-up has some details on computability that’s relevant to convergence analysis for Lorenz ’63. Mentions that the theoretical limit for computability on a 16 digit precision machine is T ~ 48 (this was using a very high-order accurate finite element approach, O[(delta t)^30]). Also references some work on extended precision stuff (haven’t got to that yet, but figured you might be interested).
Favorite quote (I’m sure you’ll agree):
As we shall see, obtaining a sequence of converging approximations for the solution of the Lorenz system is non-trivial.
Here’s their result:
With 16 digits of precision, this limit occurs at T ≈ 50. With 420 digits, the limit occurs at T ≈ 1050. We compute a converging sequence of solutions up to order p = 200 and precision ǫ = 10−420 to obtain a solution of the Lorenz system accurate on the time interval [0, 1000]. We further demonstrate that the computability of the Lorenz system is given by T ∼ 2.5 nǫ , where nǫ = log10 ǫ is the number of significant digits.
They have a site with code and results: lorenzsystem.net/