Euler's Method for Integration
This is my personal favorite area which definitely is a vast subject and we'll barely scratch the surface here -Â NUMERICAL METHODS
Since, today dynamics is mostly studied on the computer, we'll see a couple of numerical integration methods. One of them is Euler's Method.
Let $$ \dot{x}=f(x,t), x(t_0)=x_0\ $$
This is approximated by making use of Taylor series approximation of the dependent variable \(x(t)\) at point \(t_{i+1}\) (Note that \(t_{i+1}-t_i=\Delta t\) ). That is
$$ x(t_{i+1})=x(t_i+\Delta t) = x(t_i) + \Delta t \dot{x}(t_i)+O(\Delta t^2) $$
Neglecting higher order terms,
$$ x_{i+1} = x_i+f(x_i,t_i)\Delta t $$
This is the simplest possible numerical integration scheme.
Location Truncation Error
The error made in a single step, the difference b/w the numerical solution after one step, \(x_1\) and the exact solution at time \(t_1=t_0+\Delta t\) is LTE.
The numerical solution is given by
$$x_1=x_0+f(x_0,t_0)\Delta x $$
For the exact solution, we use the taylor expansion
$$ x(t_0+\Delta t)=x(t_0)+\Delta t\dot{x}(t_0)+ \frac{1}{2}\Delta t^2\ddot{x}(t_0)+O(\Delta t^3) $$
Thus
$$ LTE = x(t_0+\Delta t)-x_1=\frac{1}{2}\Delta t^2\ddot{x}(t_0)+O(\Delta t^3) $$
A slightly different formulation for the local truncation error can be obtained by using the Lagrange form for the remainder term in Taylor's theorem. If \(x\) is a continuous second derivative, then there exists a \(\xi \in [t_0,t_0+\Delta t]\) such thatÂ
$$ LTE = \frac{1}{2} \Delta t^2\ddot{x}(\xi)$$
Hope you enjoyed folks. Later comes the famous Runge-Kutta Method.
