Order of Accuracy

How accurate is the Euler method? To quantify this we consider a Taylor expansion of $y(t)$ around $t_n$

$$y_{n+1} = y_n + \delta t \left.\d y\over\d t\right\vert _n ... ...a t^2\over 2} \left.\d^2 y\over\d t^2\right\vert _n + \ldots$$ (1.14)

and substitute this into (1.11)

$$y_n + \delta t \left.\d y\over\d t\right\vert _n + {\delta t^2\over 2} \left.\d^2 y\over\d t^2\right\vert _n + \ldots \approx y_n - \delta t f(y_n, t_n)$$ (1.15)

$$= y_n + \delta t \left.\d y\over\d t\right\vert _n,$$ (1.16)

where we have used (1.7) to obtain the final form. Hence, we see that the term in $\delta t$ in the expansion has been correctly reproduced by the approximation, but that the higher order terms are wrong. We therefore describe the Euler method as 1st order accurate. An approximation to a quantity is $n$th order accurate if the term in $\delta t^n$ in the Taylor expansion of the quantity is correctly reproduced. The order of accuracy of a method is the order of accuracy with which the unknown is approximated. Note that the term accuracy has a slightly different meaning in this context from that which you might use to describe the results of an experiment. Sometimes the term order of accuracy is used to avoid any ambiguity. The leading order deviation is called the truncation error. Thus in (1.2.1) the truncation error is the term in $\delta t^2$.