Numerov's method

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Numerov's method is a numerical method to solve ordinary differential equations of second order in which the first-order term does not appear. It is a fourth-order linear multistep method. The method is implicit, but can be made explicit if the differential equation is linear.

Numerov's method was developed by Boris Vasil'evich Numerov.

1 The method
- 1.1 Nonlinear equations
2 Application
3 Derivation
4 References
5 External links

[edit] The method

The Numerov method can be used to solve differential equations of the from

$\left ( \frac{d^2}{dx^2} + k(x) \right ) y(x) = 0$

The function $y (x)$ is sampled in the interval [a..b] at equidistant positions $x n$ . Starting from function values at two consecutive samples $x n - 1$ and $x n$ the remaining function values can be calculated as

$y_{n+1} = \frac {\left( 2-\frac{5 h^2}{6} k_n \right) y_n - \left( 1+\frac{h^2}{12}k_{n-1} \right)y_{n-1}}{1+\frac{h^2}{12}k_{n+1}}$

where $k n = k (x n)$ and $y n = y (x n)$ are the function values at the positions $x n$ and $h = x n - x n - 1$ is the distance between two consecutive samples.

[edit] Nonlinear equations

The Numerov method applies more generally. It can solve any equation of the form

$\frac{d^2}{dx^2} y(x) = f(x,y(x)).$

For this nonlinear equation, the method is given by

$y_{n+1} = 2y_n - y_{n-1} + \tfrac{1}{12} h^2 (f_{n+1} + 10f_n + f_{n-1}).$

This is an implicit linear multistep method, which reduces to the explicit method given above if the function f is linear in y. It achieves order 4 (Hairer, Nørsett & Wanner 1993, §III.10).

[edit] Application

In numerical physics the method is used to find solutions of the radial Schrödinger Equation for arbitrary potentials.

$\left [ -{\hbar^2 \over 2\mu} \left ( \frac{1}{r} {\partial^2 \over \partial r^2} r - {l(l+1) \over r^2} \right ) + V(r) \right ] R(r) = E R(r)$

The above equation can be rewritten in the form

$\left [ {\partial^2 \over \partial r^2} - {l(l+1) \over r^2} + { 2\mu \over \hbar^2} \left( E - V(r)\right) \right ] u(r) = 0$

with $u (r) = r R (r)$ . If we compare this equation with the defining equation of the Numerov method we see

$k(x) = \frac{2\mu}{\hbar^2} \left(E - V(x) \right) - \frac{l(l+1)}{x^2}$

and thus can numerically solve the radial Schrödinger equation.

[edit] Derivation

Starting from the Taylor expansion for $y (x n)$ we get for the two adjacent sampling points

$y_{n+1} = y(x_n+h) = y(x_n) + hy'(x_n) + \frac{h^2}{2}y''(x_n) + \frac{h^3}{6}y'''(x_n) + \frac{h^4}{24}y''''(x_n) + \mathcal{O} (h^5)$

$y_{n-1} = y(x_n-h) = y(x_n) - hy'(x_n) + \frac{h^2}{2}y''(x_n) - \frac{h^3}{6}y'''(x_n) + \frac{h^4}{24}y''''(x_n) - \mathcal{O} (h^5)$

The sum of those two equations gives

$y_{n-1} + y_{n+1} = 2y_n + {h^2}y''_n + \frac{h^4}{12}y''''_n - \mathcal{O} (h^6)$

We solve this equation for $y'' n$ and replace it by the expression $y'' n = - k n y n$ which we get from the defining differential equation.

$k_n y_n = \frac{1}{h^2} \left(2y_n-y_{n-1} - y_{n+1} + \frac{h^4}{12}y''''_n \right) - \mathcal{O} (h^4)$

We take the second derivative of our defining differential equation and get

$y''''(x) = - \frac{d^2}{d x^2} \left[ k(x) y(x) \right]$

where we can replace the differentiation with the differences quotient and inset this into our equation for $k n y n$

$k_n y_n = \frac{1}{h^2} \left(2y_n-y_{n-1} - y_{n+1} - \frac{h^4}{12} \frac{k_{n-1} y_{n-1} -2 k_{n} y_{n} + k_{n+1} y_{n+1}}{h^2}\right) - \mathcal{O} (h^4)$

We neglect all terms of the order $\mathcal{O}(h^4)$ and gather all terms for $y n$ and get

$\left(1+\frac{h^2}{12}k_{n+1}\right)y_{n+1} = \left(2-\frac{h^2(12-2)}{12}k_{n}\right)y_{n} - \left(1+\frac{h^2}{12}k_{n-1}\right)y_{n-1}$

$y_{n+1} = \frac {\left( 2-\frac{5 h^2}{6} k_n \right) y_n - \left( 1+\frac{h^2}{12}k_{n-1} \right)y_{n-1}}{1+\frac{h^2}{12}k_{n+1}}$

[edit] References

Hairer, Ernst; Nørsett, Syvert Paul & Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0 .
This book includes the following references:
- Numerov, Boris Vasil'evich (1924), “A method of extrapolation of perturbations”, Monthly Notices of the Royal Astronomical Society 84: 592–601 .
- Numerov, Boris Vasil'evich (1927), “Note on the numerical integration of d²x/dt² = f(x,t)”, Astronomische Nachrichten 230: 359–364 .