Numerov's method

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.

The method

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

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

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

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

where f_n = f(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.

Nonlinear equations

For nonlinear equations of the form

$\frac{d^2}{dt^2} y = f(t,y)$

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).

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

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

and thus can numerically solve the radial Schrödinger equation.

Derivation

Starting from the Taylor expansion for y(x_n) we get for the two sampling points adjacent to x_n

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

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

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 = − f_ny_n which we get from the defining differential equation.

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

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

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

We replace the second derivative $\frac{d^2}{d x^2}$ with the second order difference quotient and insert this into our equation for f_ny_n

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

We solve for y_{n + 1} to get

$y_{n+1} = \frac {\left( 2-\frac{5 h^2}{6} f_n \right) y_n - \left( 1+\frac{h^2}{12}f_{n-1} \right)y_{n-1}}{1+\frac{h^2}{12}f_{n+1}} + \mathcal{O} (h^6).$

This yields Numerov's method if we ignore the term of order h⁶. It follows that the order of convergence (assuming stability) is 4.

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 .

External links

• Lecture notes: Computerphysik und Numerik - by Jan Krieger
• Lecture notes of Werner Scholz - At Vienna University of Technology
• Lecture notes of Alexander Wagner