Let's say I wanted to solve a system of ODEs using RK4, then I want to take the average value of one the solution components over some interval using some integration method like the trapezoid rule. I can't recally properly, but wouldn't I want to use some 4th order integration method, in order to preserve the accuracy of the calculation? Or am I conflating concepts? I googled variants of "matching orders of numerical methods" but didn't get anything solid. What is the concept called that I'm looking for?
2026-02-22 21:28:15.1771795695
Matching the orders of numerical solvers.
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Yes, the order of the overall calculation is (normally) controlled by the lowest order of the individual algorithms. There is little point in integrating the trajectory of an artillery shell using a fourth order method, say, RK4, if the length of the trajectory is later computed using a second order method, say, the trapezoidal rule. The computed length will only be second order accurate. Similarly, if the drag coefficient is modeled using a cubic spline, then we loose the ability to accurately estimate the error unless we use methods of order one or two.
In specific cases, this type of analysis is elementary, but the calculations might be nontrivial. In general, you can deploy Richardson's techniques to experimentally determine the order of the overall scheme or its components.