ODEs

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: