Suppose we take the backward-difference approximation
, and expand
in Taylor series
about
. This yields:
So the difference scheme approximates the continuous time equation to
an accuracy that depends on
, the step size. Thus we expect that
the discretization will do a better job as
gets small.
Performing the same analysis for the trapezoid rule yields:
So we say that the trapezoid rule is second-order accurate in
.