TABLE OF CONTENTS Numerical integration is needed to calculate the solution x( t + ? t) at time t + ? t from knowledge of previous time points. The local truncation error (LTE) is the error introduced by the solution at x( t + ? t) assuming that the previous time points are exact. Thus the total error in the solution x( t + ? t) is determined by LTE and the build-up of error at previous time points (i.e. its propagation through the solution). The stability characteristics of the integration algorithm are a function of how this error propagates.
A numerical integration algorithm is either explicit or implicit. In an explicit integration algorithm the integral of a function f, from t to t + ? t, is obtained without using f( t + ? t). An example of explicit integration is the forward Euler method:
| (C.1) |  |
In an implicit integration algorithm f( x( t + ? t), t + ? t) is required to calculate the solution at x( t + ? t). Examples are, the backward Euler method, i.e.
| (C.2) |  |
and the trapezoidal rule, i.e.
| (C.3) |  |
There are various ways of developing numerical integration algorithms, such as manipulation of Taylor series expansions or the use of numerical solution by polynomial approximation. Among the wealth of material from...
