Numerical Methods for the approximate solution of ODE'S.

We concentrate first on the IVP and then discuss the BVP. We only consider finite difference methods (F.D.). F.D. methods invariably require that the independent variable $x$ be a discrete sequence and that $Y(x)$, derivatives of $Y(x)$, and coefficients in the equation that depend on $x$ be approximated on such a grid.

Notation let $Y(x)$ be the true solution of

$$\left\{ \begin{array}{l} Y'=f(x,Y)\\ Y(a)= Y_0. \end{array}\right.$$ (7)

let $y(x)$ represent the approximate solution and by way of notation, let

$$y(x_0)\equiv y_0, y(x_1)\equiv y_1,\ldots y(x_n)\equiv y_n.$$ (8)

let $y_h$ denote an approximation at some resolution, given by $h$ the grid spacing. If the grid is equally spaced

$$x_n=x_0+nh, n=0, 1, 2\cdots$$ (9)

Take $x_0 = a$, for simplicity, and let $N(h)$ denote the largest index $N$ for which

$$x_N\le b\qquad x_{N+1}>b$$

where $a<b$. The simplest finite difference approximation would be

$$\left\{ \begin{array}{l} y(x_{n+1})=y_n+h_nf(x_n, y_n)\\ y_0\cong Y_0, \end{array}\right.$$ where $h_n=x_{n+1}-x_n$, and $n=0, 1\ldots$.

Using an equally spaced grid the above scheme would be Euler(Forward Euler)

Perhaps the simplest most straightforward scheme. It reads

$$y_{n+1}=y_n+hf(x_n, y_n)\qquad n= 0,1, \ldots$$ (10)

$$y_0\cong Y_0$$

where $y_n$ is consistent with (8) and $x_n$ as per (9). Following Atkinson's suggestion, it is very useful to interpret the Foward Euler scheme in a variety of ways. Look at the interpretation of taking a single step: