Truncation error

Summary

Truncation error is the error made by replacing an infinite or exact mathematical process by a finite approximation, even when all arithmetic is exact. Examples are cutting a Taylor series after finitely many terms, replacing a derivative by a difference quotient, or an integral by a finite sum. It usually shrinks as the number of terms grows or the step size $h$ goes to $0$, often like a power $h^p$. For ODE methods, the error made in one step is the local truncation error. It is the counterpart of round-off error, which comes from finite-precision arithmetic.