approximation to the second derivative: Verlet integration in the form used as the Störmer method uses this equation to obtain the next position vector from the previous two without using the velocity as Discretisation error The time symmetry inherent in the method reduces the level of local errors introduced into the integration by the discretization by removing all odd-degree terms, here the terms in of degree three. The local error is quantified by inserting the exact values into the iteration and computing the Taylor expansions at time of the position vector in different time directions: where is the position, the velocity, the acceleration, and the jerk (third derivative of the position with respect to the time). Adding these two expansions gives We can see that the first- and third-order terms from the Taylor expansion cancel out, thus making the Verlet integrator an order more accurate than integration by simple Taylor expansion alone. Caution should be applied to the fact that the acceleration here is computed from the exact solution, , whereas in the iteration it is computed at the central iteration point, . In computing the global error, that is the distance between exact solution and approximation sequence, those two terms do not cancel exactly, influencing the order of the global error. A simple example To gain insight into the relation of local and global errors, it is helpful to examine simple examples where the exact solution, as well as the approximate solution, can be expressed in explicit formulas. The standard example for this task is the exponential function. Consider the linear differential equation with a constant . Its exact basis solutions are and . The Störmer method applied to this differential equation leads to a linear recurrence relation or It can be solved by finding the roots of its characteristic polynomial . These are The basis solutions of the linear recurrence are and . To compare them with the exact solutions, Taylor expansions are computed: The quotient of this series with the one of the exponential starts with , so From there it follows that for the first basis solution the error can be computed as That is, although the local discretization error is of order 4, due to the second order of the differential equation the global error is of order 2, with a constant that grows exponentially in time. Starting the iteration Note that at the start