Order of a Method

The order of a method is determined by how the Global truncation error varies with step size $dt$. A method is order $p$ if the global truncation error is $O(d{t}^p)$.

Method	Order	Global truncation error
Explicit Euler method	1	$O(dt)$
Midpoint method	2	$O(d{t}^2)$
Ralston method	2	$O(d{t}^2)$
Implicit Trapezoid Method	2	$O(d{t}^2)$
Fourth order Runge-Kutta	4	$O(d{t}^4)$

Higher order means that reducing the step size gives a much greater reduction in error. A method of order $p$ roughly multiplies error by $(d{t}^{\prime }/dt{)}^p$ when the step size is changed from $dt$ to $d{t}^{\prime }$.

The order is related to how many terms of the Taylor expansion are matched by the method’s increment function. See Local truncation error for the single-step version of this.

Global truncation error | Local truncation error | Order of magnitude | Order of convergence | Runge-Kutta methods