4.2.6 Numerov's Method

Mostly used for a certain kind of initial value problems (IVPs) that arise when boundary value problems (BVPs) are rewritten as IVPs. Many BVP has the general form

$$\frac{d^{2}y}{dx^{2}}=-g(x)y+s(x)$$

with given boundary values $y(x_{1})$ and $y(x_{2})$.

Example: one-dimensional Poisson equation

$$\frac{d^{2}\phi}{dx^{2}}= -\rho(x)$$

with $\phi$ given at $x_{1}$ and $x_{2}$. Thus $g(x)=0$ and $s(x)=-\rho(x)$.

Let us assume we have, instead of $y(x_{1})$ and $y(x_{2})$, the full set of initial values $y(x_{1})$ and $y'(x_{1})$.

Divide the interval $[x_{1},x_{2}]$ into sub-intervals of length $\Delta x$ and at each intermediate point $x_{n}$ expand $y(x)$ into a power series. Adding the Taylor formulae for $y_{n+1}$ and $y_{n-1}$ we find

$$y_{n+1}=2y_{n}-y_{n-1}+{y_{n}}''(\Delta x)^{2}+y_{n}^{(4)} \frac{(\Delta x)^{4}}{12}+O[(\Delta x)^{6}]$$

Inserting ${y_{n}}'' = -g_{n}\, y_{n}+s_{n}$ we have

$$y_{n+1}=2y_{n}-y_{n-1}+(\Delta x)^{2}[-g_{n}y_{n}+s_{n}] +\frac{(\Delta x)^{4}}{12}y_{n}^{(4)} +O[(\Delta x)^{6}]$$

The fourth derivative $y^{(4)}$ is approximated by

$$y_{n}^{(4)} = \left. \frac{d^{2}y''}{dx^{2}} \right\vert _{n} = \left. \frac{d^... ...^{2}} \right\vert _{n} \approx \frac{1}{(\Delta x)^{2}} \delta_{n}^{2}(-gy+s) =$$

$$= \frac{1}{(\Delta x)^{2}} [-g_{n+1}y_{n+1}+2g_{n}y_{n}-g_{n-1}y_{n-1} +$$

$$\hspace{144pt}+s_{n+1}-2s_{n}+s_{n-1}]$$

In this way we find Numerov's formula

$$y_{n+1}[1+\frac{(\Delta x)^{2}}{12}g_{n+1}] = 2y_{n}[1-\frac{5}{12}(\Delta x)^{2}g_{n}]- y_{n-1}[1+\frac{(\Delta x)^{2}}{12}g_{n-1}]+$$

$$+\frac{(\Delta x)^{2}}{12} [s_{n+1}+10s_{n}+s_{n-1}]+O[(\Delta x)^{6}]$$

EXERCISE: Write a code that permits to solve a given second-order equation of motion by various algorithms. Apply the program to problems of point mechanics and explore the stabilities and accuracies of the diverse techniques.