Truncation error (numerical integration)

Truncation errors in numerical integration are of two kinds:

• local truncation errors – the error caused by one iteration, and
• global truncation errors – the cumulative error cause by many iterations.

Definitions

Suppose we have a continuous differential equation

$y'=f(t,y),\qquad y(t_{0})=y_{0},\qquad t\geq t_{0}$

and we wish to compute an approximation $y_{n}$ of the true solution $y(t_{n})$ at discrete time steps $t_{1},t_{2},\ldots ,t_{N}$. For simplicity, assume the time steps are equally spaced:

$h=t_{n}-t_{n-1},\qquad n=1,2,\ldots ,N.$

Suppose we compute the sequence $y_{n}$ with a one-step method of the form

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

The function $A$ is called the increment function, and can be interpreted as an estimate of the slope of $y_{n}$.

Local truncation error

The local truncation error is the error that our increment function, $A$, causes during a single iteration, assuming perfect knowledge of the true solution at the previous iteration.

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

$\tau _{n}=y(t_{n})-y(t_{n-1})-hA(t_{n-1},y(t_{n-1}),h,f).$

The numerical method is consistent if, for every $\varepsilon >0$, there exists an $H$ such that the local error satisfies $|\tau _{n}|<\varepsilon h$ for all $h<H$. If the increment function $A$ is differentiable, then the method is consistent if, and only if, $A(t,y,0,f)=f(t,y)$.

Furthermore, we say that the numerical method has order $p$ if for any sufficiently smooth solution of the initial value problem, there exist constants $C$ and $H$ such that $|\tau _{n}|<Ch^{p+1}$ for all $h<H$.

Global truncation error

The global truncation error is the accumulation of the local truncation error over all of the iterations, assuming perfect knowledge of the true solution at the initial time step.

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

${\begin{aligned}e_{n}&=y(t_{n})-y_{n}\\&=y(t_{n})-{\Big (}y_{0}+hA(t_{0},y_{0},h,f)+hA(t_{1},y_{1},h,f)+\cdots +hA(t_{n-1},y_{n-1},h,f){\Big )}.\end{aligned}}$

Relationship between local and global truncation errors

Sometimes it is 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.

The global truncation error satisfies the recurrence relation:

$e_{n+1}=e_{n}+h{\Big (}A(t_{n},y(t_{n}),h,f)-A(t_{n},y_{n},h,f){\Big )}+\tau _{n}.$

This follows immediately from the definitions. Now assume that the increment function in Lipschitz continuous in the second argument, that is, there exists a constant $L$ such that for all $t$ and $y_{1}$ and $y_{2}$, we have:

$|A(t,y_{1},h,f)-A(t,y_{2},h,f)|\leq L|y_{1}-y_{2}|.$

Then the global error satisfies the bound

$|e_{n}|\leq {\frac {\max _{j}\tau _{j}}{hL}}\left(\mathrm {e} ^{L(t_{n}-t_{0})}-1\right).$

It follows from the above bound for the global error that if the function $f$ in the differential equation is continuous in the first argument and Lipschitz continujous in the second argument (the condition from the Picard–Lindelöf theorem), and the increment function $A$ is continuous in all arguments and Lipschitz continuous in the second argument, then the global error tends to zero as the step size $h$ approaches zero (in other words, the numerical method converges to the exact solution).

See also

• Numerical integration
• Numerical ordinary differential equations
• Truncation error

References

• Süli, Endre; Mayers, David (2003), An Introduction to Numerical Analysis, Cambridge University Press, ISBN 0521007941.

External links

• Notes on truncation errors and Runge-Kutta methods