Overview

We begin our discussion of numerical methods for initial value ordinary differential equations (ODEs) with an introduction of the most basic concepts involved. To illustrate these concepts, we use three simple discretization methods: forward Euler, backward Euler (also called implicit Euler), and trapezoidal. The problem to be solved is written, as before, in the general form

with y(0) = c given. You can think of this at first as a scalar ODE most of what we are going to discuss generalizes to systems directly, and we will highlight occasions where the size of the system is important.

We will assume sufficient smoothness and boundedness on f( t, y) so as to guarantee a unique existence of a solution y( t) with as many bounded derivatives as referred to in the sequel. This assumption will be relaxed in Section 3.7.

3.1 A Simple Method: Forward Euler

To approximate (3.1), we first discretize the interval of integration by a mesh

and let h _n = t _n ? t _{n ?1} be the nth step size. We then construct approximations

with y _n an intended approximation of y( t _n).

In the case of an initial value problem (IVP) we know y ₀ and may proceed to integrate the ODE in steps, where on each step n (1 ? n ? N) we know an approximation y _{n ?1}