# 1.3 Order of Accuracy

## 1.3.3 Local Order of Accuracy

Suppose we are given a numerical method for solving $$u_ t = f(u,t)$$ which we write in the following form,

 $v^{n+1} = N(v^{n+1},v^ n,v^{n-1}, \ldots , {\Delta t})$ (1.48)

For simplicity, the possible dependence on $$t$$ at various $$n$$ has been omitted in the definition of $$N$$ (though it should be there). The local truncation error, $$\tau$$, is defined as,

 $\tau \equiv N(u^{n+1},u^ n,u^{n-1}, \ldots , {\Delta t}) - u^{n+1}, \label{equ:lte}$ (1.49)

and the local order of accuracy $$p$$ is,

 $|\tau | = O({\Delta t}^{p+1}) \qquad \mbox{as} \qquad {\Delta t}\rightarrow 0.$ (1.50)

Note: the local order of accuracy is defined to be one less than the order of the leading term of the local truncation error so that the local and global accuracy will be the same.

Exercise 1 What is the local order of accuracy for the forward Euler method?

Exercise 1

Exercise 2 What is the local order of accuracy for the midpoint method?

Exercise 2