Does Runge Kutta methods run well with variable $h$?

Question

When using Runge Kutta methods for real time simulations there is a problem with the constant step length $h$ since the operating systems often interrupt the simulation and the main loop therefore has a variable length.

Is it relevant to use a slightly variable $h$ being the real time for the calculation in the loop? Are there other methods better suited for real time simulations?

Answer 1

No, the great thing about Runge-Kutta is that even though it appears to be a third-order method, by a "miracle" of the algebra it is correct to fourth order.

If you let $h$ vary and use naively use the same formulas, the algorithm gets errors on order of $h^2$ and so you might as well be using the basic Euler method. If instead you properly correct for the varying $h$ you can restore it to third order but the fortuitous cancellation that gives you one free order will never recur.

Answer 2

Operating system interrupting execution should be irrelevant if you use a real-time operating system, right?

There are methods of varying $h$ in Runge-Kutta but they are for estimating the error term for adaptive step-size control, such as RKF45 or DOPRI methods. They solve a different problem.

Answer 3

You need to do full steps of the one-step method with a fixed $h$. However, between the steps the value of $h$ can vary. If you can guarantee that in every step $h<\bar h$ with some constant $\bar h$ then the error will still be bounded by $O(\bar h^4)$. So if in the real-time simulation there is a sleep time of $\underline{h}$ then other processes may cause the actual time that the evaluation resumes is larger. You compute $h$ using the actual time elapsed since the last step. Assuming that this delay is a bounded quantity, this gives the mentioned $\bar h$ so that $\underline{h}\le h\le \bar h$.

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The advanced way to solve this problem is to compute the next ODE integration step with an $h$ that guarantees the needed accuracy of the simulation, where one can also use an adaptive step size strategy, Richardson extrapolation or embedded methods are often used. For the real time display you then interpolate the values for the time of the frame that is displayed, in the most simple case use linear interpolation. Then compute the next integration step when it becomes necessary. As you can use larger step sizes in the numerical ODE integration, the evaluation of the simulation dynamic has to occur less often, which takes less computational effort and may make the animation smoother.