Numerical Solution of Initial Value Problems

Some of the key concepts associated with the numerical solution of IVPs are the Local Truncation Error, the Order and the Stability of the Numerical Method. We should also be able to distinguish explicit techniques from implicit ones. In the following, these concepts will be introduced through simple examples.

We are interested in the numerical solution of the IVP

$$\frac{dy}{dt} = f(y,t), \:\:\: y(t=0) = y_0.$$ (5)

In particular, if $f(y,t)\equiv g(y)$, the IVP above is called autonomous and if g(y) = ky where k is a constant, the IVP is linear. We assume that a unique solution exists and denote that solution by y^e (t). So, for from now on, y(t) refers to the numerically computed solution, which at the best is only an approximation to y^e(t).