We will start off our investigation of numerical techniques by solving ordinary differential equations (ODEs). Our first method is a relatively direct discretization of Newton's second law. This is called the Verlet method, and is an efficient way to find solutions (i.e. a vector function x(t)) given a force that depends on time and position (and appropriate initial data). The method has generalizations, but we will think about it only in the context of equations of motion. As a first method, it has the advantage of being straightforward, while relying in its derivation on ideas that can be extended to develop more involved numerical differential equation solvers.

In the more general context that follows the Verlet method, we note that any ODE can be re-written in the form: f′(x) = G(x, f(x)) for vector-valued function G and vector-valued target function f(x). The punch-line will be that if we are given initial conditions f(0), we can use Runge–Kutta methods to find values approximating f(x) on a grid. First, we'll set up a few ODEs of interest, then introduce the process of discretization and the (special) Runge–Kutta family of methods that can be used to solve the discretized problem.