Arcadia197
commented
2 years ago However, I did not take into account, that we would need the positions for the estimation of the velocities as well. This cannot be saved from the footpints, therefore making higher order linear multistep methods not workable. However, there exist Runge Kutta methods with more optimal time steps chosen, from which I implemented a third and fourth order method. These still do not show any error improvement, but give a slight performance boost.
In theory even better chosen methods could be chosen, where the first step does not start at n (which corresponds to n+1 for the backward integration), but rather at n+1 (n+2, n+3), however, this would require completely rewriting the whole theory behind runge kutta methods together with error bounds and is by far out of the scope of this work
RK methods are fine, they are well known and work nice. But since we need intermediate estimates of the velocity, this makes not that much sense. We already have an adam-bashfords method with predictor and corrector working, why don't we update it this way to third order with an adam-bashfords or just backwards differentiation formula. This should be waaaay cheaper and achieve the same level of accuracy.