with a Hermite Algorithm
Having seen the dramatic improvement that came from switching from the forward Euler algorithm to the leapfrog, the obvious next step was to go to yet higher order algorithms. A quick look in a few books of numerical methods showed our friends that there was a bewildering choice of third- and fourth-order methods to choose from. Alice then mentioned that her thesis advisor had pointed her to an elegant and natural generalization of the leapfrog algorithm, by the name of the Hermite scheme.