Well, much thanks, your extensive posts on this topic have certainly got me thinking. I applied that implicit Runge-Kutta method to the Lorentz 63 system. I’m getting what looks like convergence. Here’s the x-component, y-component, z-component. As I decrease the time-step, the trajectories trace each-other further and further out in time. Am I missing something?

When you say the ‘error grows’, do you mean in time or with decreasing step-size? Does the ‘blow-up’ time of your error change as you decrease step-size?

I saw this in your comment on that other post:
There are some ranges of the independent variables for which one might can say the numerical method has converged, but is it really convergence solely because the range is restricted?

As an engineer, I’d say yes, it’s converged. I want to be able to get an arbitrarily close approximation on a finite domain (space and time). The error at infinite time doesn’t bother me.

Numerical solutions of these chaotic ODE systems will not show convergence. I have several posts on the subject, the last one is here. ( Note that almost all site-internal links are broken. ) The differences between the solution variables as the step size is reduced always grows to be of the same order as the solution variables themselves. I have used a couple of different solution methods, and different step sizes, for this MS exercise and got the same results but did not report them. The differences between the calculated solution variables and the expected MMS solutions as the step size is reduced always grows to be of the same order as the solution variables themselves.

As you know, the MMS solutions should have looked like the early-time response shown in Figure 8. When I saw that the results were not as expected, I did a few quick runs varying the step size and using higher-order methods, and got the same results every time.

I say ‘should have looked’, but as I noted in the write-up, the effects of the addition of functions of time to previously autonomous ODE systems known to exhibit chaotic response is an issue that has not been well-explored, that I know of.

I continue to have day-job consulting work and haven’t had time to get back to the blog very much at all.

Thanks for stopping by. You, George, and me comprise an unstoppable force.

To verify your method you need to observe how the error changes as you decrease the time-step. Here is an example of this sort of convergence study; you should be able to fit a line to the log transformed error, and the slope will be the order of convergence, for the integration method you are using (Euler) it should be close to 1.0.