As expected, one can also discretize second-order derivatives. Adding the backward and forward differencing schemes and rearranging yields a second-order derivative central difference scheme: Like the first derivative, the second derivative may be approximated in a number of different ways. Alternative discretization methods (analogous forward ...

*Aug 16, 2017 · The algorithm used three data points to calculate the derivative, except at the end points, where by necessity the forward difference algorithm is used instead. If you want to use derivatives strictly formed from the central difference formula, use only the values from [1 .. #y-1], e.g.:*8. sketch the graph of the derivative from the given graph of a function. 9. given a table of function values, approximate the value of the derivative at a point using the forward difference quotient and the centered difference quotient 10. compute the value of the derivative at a point algebraically using the (limit) definition 11. The forward or backward di erence quotients for u0(x) are rst order The second centered di erence for u00(x) is second order So we need a second order approximation to u0(x) If we subtract the expansions u(x + h) = u(x) + h u0(x) + h2 2! u00(x) + h3 3! u000(x) + O(h4): u(x h) = u(x) h u0(x) + h2 2! u00(x) h3 3! u000(x) + O(h4) we get (9) (10) (11) where is the velocity (the first derivative of the position with respect to time), is the acceleration (the second derivative), is the third derivative, and so on. The Verlet algorithm [30] is the most broadly used method for integrating the trajectories of motion in MD simulations.