# Order of local error when integrating ODE with discontinous derivatives

I'm working with ODEs, $$\dot{x} = f(x, t),$$ where the (higher) derivatives of the right-hand side have discontinuities. In particular, $$f(x, t)$$ is obtained by interpolation of discrete samples, and the number of continious derivatives thus depend on the order of interpolation.

I know from Theorem II.3.1 on page 157 of Hairer, Nørsett & Wanner that if a Runge-Kutta method is of order $$p$$, then the local error is bounded by $$C h^{p+1}$$ if all partial derivatives of $$f(x,t)$$ up to order $$p$$ exist and are continuous.

Hence, I know that if I for example use cubic spline interpolation to obtain $$f(x, t)$$, then only the first and second derivatives are continuous, and hence I cannot expect fourth-order accuracy from a fourth-order Runge-Kutta method. In practice, though, I do see fourth-order convergence when $$f(x, t)$$ is obtained by cubic spline interpolation, while if $$f(x, t)$$ is obtained with linear interpolation (where not even the first derivative is continuous) I get second-order convergence from fourth-order Runge-Kutta.

Hairer, Nørsett & Wanner only tells me that I cannot expect a method of order $$p$$ to work as advertised unless I have $$p$$ continous partial derivatives, but they don't tell me how large the error will be. I've done some numerical experiments, but I'm curious to see if there exists some more rigorous literature on this point.

My question is therefore if there exists any (reasonably accessible) theory that I can use to reason about this.

• On a side note, if you know the abscissae of the discontinuities, you can use them to split the original integration intervals into multiple subintervals where $f$ is as smooth as needed, and thus maintain the original order of accuracy. Mar 20, 2023 at 10:01
• Thanks, I'm actually doing some research on that topic now. I'm mainly looking for some background material and further reading. And again, Hairer, Nørsett and Wanner also mention what you suggest (pp. 196-198 in the 2nd edition), but I can't find any details about the errors you pick up when stepping across the discontinuity.
– Tor
Mar 20, 2023 at 13:03
• The question is how often your trajectory $x(t)$ crosses the places where some derivative of $f$ is discontinuous. If that happens only a fixed number of times, regardless of the size of the time step, then an adapted version of Gronwall's lemma may be able to guarantee the original convergence order. If you're switching back and forth areas where $f$ is smooth with every time step, then you lose convergence orders. Mar 20, 2023 at 19:41
• In my particular case, the number of discontinuity crossings remain approximately constant with changing timesteps. I'll have a look at Gronwall and see if I get any smarter. Thanks. I can also provide additional details of what precisely I'm trying to do, if that will help.
– Tor
Mar 20, 2023 at 19:47