# 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 the Russian astronomer Boris Vasil'evich Numerov.

## The method

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

The function is sampled in the interval [a..b] at equidistant positions . Starting from function values at two consecutive samples and the remaining function values can be calculated as

where and are the function values at the positions and is the distance between two consecutive samples.

### Nonlinear equations

For nonlinear equations of the form

the method is given by

This is an implicit linear multistep method, which reduces to the explicit method given above if *f* is linear in *y* by setting . It achieves order 4 Template:Harv.

## Application

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

The above equation can be rewritten in the form

with . If we compare this equation with the defining equation of the Numerov method we see

and thus can numerically solve the radial Schrödinger equation.

## Derivation

Start with the Taylor expansion of about a point :

Denote the distance from to by and, noting that this means , we can write the above equation as

Computationally, this amounts taking a step *forward* by an amount h. If we want to take a step *backwards*, replace every h with -h for the equation of :

Note that only the odd powers of h experienced a sign change. On an evenly spaced grid, the *n*th site on a computational grid corresponds to position if the step-size between grid points are of length (hence h should be small for the computation to be accurate). This means we have sampling points and . Taking the equations for and from continuous space to discrete space, we see that

The sum of those two equations gives

We solve this equation for and replace it by the expression which we get from the defining differential equation.

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

We replace the second derivative with the second order difference quotient and insert this into our equation for (note that we take the mixed forward and backward finite difference, **not** the double forward difference or the double backward difference)

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

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