Numerov's method (also called Cowell'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

are related as follows:

.

Nonlinear equations

For nonlinear equations of the form

the method gives

This is an implicit linear multistep method. It achieves order-4 accuracy .

Application

In numerical physics the method is used to find solutions of the unidimensional Schrödinger equation:

This equation can be reduced to the form necessary for the application of Numerov's method with the following substitution:

And when we make the substitution, the radial equation becomes

or

which is equivalent to the one-dimensional Schrödinger equation, but with the modified effective potential

This equation we can proceed to solve the same way we would have solved the one-dimensional Schrödinger equation. We can rewrite the equation a little bit differently and thus see the possible application of Numerov's method more clearly:

Derivation

We are given the differential equation

To derive the Numerov's method for solving this equation, we begin with the Taylor expansion:

, we can write the above equation as

:

Computationally, this amounts to taking a step forward. If we want to take a step backwards:

experienced a sign change. By summing the two equations, we derive that

twice and approximate it again in the same way we did this above:

If we now substitute this to the preceding equation, we get

or

. It follows that the order of convergence (assuming stability) is 4.

References

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