# Verlet integration - first iteration

1. Jun 20, 2012

### CactusPie

I'm trying to make a simple simulation of a projectile motion with air drag. I have already implemented it using Euler and fourth order Runge Kutta methods. I am however unsure about Verlet integration. The equation goes as follows:

$$r(t + Δt) = 2r(t) - r(t-Δt) + a(t)Δt^2$$

I don't really know what value should I use during first iteration for

$$r(t-Δt)$$

2. Jun 20, 2012

### D H

Staff Emeritus
Verlet is one of many "multistep" integration techniques that need past data. As you noted, that past data doesn't exist on the first step. Some other technique, ideally one with comparable error characteristics, needs to be used to bootstrap a multistep integrator. For verlet integration, just use a quadratic estimation for that very first step,
$$r(t+\Delta t) = r(t) + v(t)\Delta t + \frac 1 2 a(t)\Delta t^2$$
This has a global error of order 2, which is the same as verlet.

From the second step onward you can use verlet integration.

3. Jun 20, 2012

### CactusPie

Thank you very much, that really helped!