Generalizations of Forward Euler by its Different Interpretations

1. Geometric Interpretation. See Figure 0.1.3.

Figure 3: Geometrical interpretation of a single-step using Forward Euler scheme

$$\frac{D y}{h}=Y'(x_0)=f(x_0, Y_0)$$

$Y(x_1)-Y(x_0)\approx D y=hY'(x_0)\Rightarrow Y(x_1)\approx Y(x_0)+hf(x_0, Y(x_0)),$

Repeating the argument for $[x_1, x_2], [x_2, x_3]\cdots$

2. Taylor Series

$Y(x_{n+1})=Y(x_n)+ hY'(x_n)+\underbrace{\frac{h^2}{2}Y''(\xi_n)}_{T_n= \frac{h^2}{2}Y''(\xi_n)}$

3. Numerical Differentiation

$$\frac{Y(x_{n+1})-Y(x_n)}{h}\approx Y'(x_n)=f(x_n, Y(x_n))$$

$\therefore Y(x_{n+1})\cong Y'(x_n)+hf(x_n, Y(x_n))$

4. Numerical Integration. See Figure 0.1.3

Integrate $Y'(x)=f(x,y)$

Over $[x_n, x_{n+1}]$

Figure 4: Quadrature interpretation of single-step using Forward Euler

$Y_{n+1}=Y_n+\underbrace{\int^{x_{n+1}}_{x_n}f(t,Y(t))dt}_{\mbox {L.H.S }\approx hf (x_n, Y_n)}$

Remark. Interpretations (2) and (4) above form the basis of a set of methods that are progressively more accurate.

$$\left\{ \begin{array}{ll} (2) \mbox{ generalizes to what are ... ...re known at MULTI STEP} & \mbox {METHODS.} \end{array}\right.$$

Interpretation (3) $\rightarrow$ doesn't lead to many possibilities, but leads to a way to solve stiff equations (to be discussed later). It also leads to an academically interesting case, the Midpoint method, which is ideal to introduce the concept of instability in the context of the approximate solution of ODE's.

$\Box$