Stepsize choice

Fig. 4.1 Stability regions for Euler s method (l ) and Symplectic Euler/Verlet (right). When a harmonic oscillator is treated using these methods, the origin is unstable for Euler s method, regardless of stepsize—this means that there is no choice of scaling h which will allow us to ensure that 11 + ftA, < 1. On the other hand, the Verlet method has an interval of stability on the imaginary axis, and it is always possible to find a value of h which guarantees that hQ < 2...

An illustration of the sampling bias (i.e., due to discretization error) is shown in Eig.7.1. As the stepsize is increased, the error in sampling is increased as well, limiting the effectiveness of numerical methods. This bias can be dramatically different for different numerical methods. As we shall show, with the right choice of numerical method it is often possible to substantially reduce this error, and it is also possible to calculate (under some assumptions) the perturbation introduced by the numerical method, and to correct for its presence. 

Fig. 7.2 We plot the error in the distribution of R for U q) = (q — 1). The error in the computed distribution is a function of both the number of force evaluations and the stepsize used, with the optimal choice of stepsize for a given number of force evaluations shown by the white guideline. Due to discretization bias, choosing the maximum stepsize is not always the best strategy...

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