Numerical Methods for Ordinary Differential Equations

We are solving the differential equation . There are several methods for this: Euler method Reverse-Euler Method Trapezoidal rule (ODEs) Multi-Step Methods Runge-Kutta Methods

Order

Suppose the numerical method is Then we define the order of it to be the largest integer such that: where is the step size.

So we assume that all our approximations were correct, and substitute those into the formula for . Then we Taylor expand, and hope for cancellations. Note that we know the derivatives of by chain rule: