Truncation error (numerical integration)

Definitions

Suppose we have a differential equation $y'=f(t,y),y(a)=y_{a}$ and we wish to approximate $y(b)$.

Suppose we are approximating $y$ with:

$y_{n}=y_{n-1}+hA(t_{n-1},y_{n-1},h,f)\!$

Note: We call $A$ an increment function.

Local truncation error

The local truncation error is the error that our increment function, $A$, causes at a given iteration.

More formally, the local truncation error, $\tau _{n}$, at step $n$ is defined by:

$\tau _{n}\!$	$=|y(t_{n})-(y_{n-1}+hA(t_{n-1},y(t_{n-1}),h,f)|\!$
	$=|y(t_{n})-y_{n-1}-hA(t_{n-1},y(t_{n-1}),h,f)|\!$

Global truncation error

The global truncation error is the absolute difference between our approximation and the actual solution.

More formally, the global truncation error, $e_{n}$, is defined by:

$e_{n}\!$	$=|y(t_{n})-y_{n}|\!$
	$=|y(t_{n})-y_{n-1}-hA(t_{n-1},y_{n-1},h,f)|\!$

Relationship between local and global truncation errors

Sometimes it's possible to calculate an upper bound on the global truncation error, if we already know the local truncation error. This requires our increment function be sufficiently well-behaved.

See Also

• Truncation error

External links

• Notes on truncation errors and Runge-Kutta methods