Numerical Schemes

If $y(0)$ is known, $y(1)$ can be calculated by evaluating the Taylor series (1.1) with $h$ set to $1$. Since the Taylor series is an infinite sum, you have to truncate the series after $N$ terms, so you introduce an error that scales with the magnitude of the $N+1$'th term. This error is called the local truncation error. It scales with $h^N$.

Successive application of the series on the obtained results, gives a sequence of numbers $\langle y(0), y(h), y(2h \ldots \rangle$. If $h$ is considered a time step, this simple scheme allows you to approximate any continuous function or - as we use to call it - simulation of the variable described by that function.

The accumulation of the local truncation errors in such an approximation or simulation is called the global truncation error. To obtain more accuracy in the results, it is obvious that $h$ should be made smaller. This however results in more steps needed for the same simulated time. If you divide $h$ by two for example, the number of steps needed to obtain the same simulated time is multiplied by two with more accumulation of local error as a result. In general the global truncation error is always at most one order of magnitude larger than the local truncation error.