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 Numerov method can be used to solve differential equations of the form
In it, three values of taken at three equidistant points are related as follows:
where , , , and .
For nonlinear equations of the form
the method gives
This is an implicit linear multistep method, which reduces to the explicit method given above if is linear in by setting . It achieves order-4 accuracy (Hairer, Nørsett & Wanner 1993, §III.10).
In numerical physics the method is used to find solutions of the unidimensional Schrödinger equation for arbitrary potentials. An example of which is solving the radial equation for a spherically symmetric potential. In this example, after separating the variables and analytically solving the angular equation, we are left with the following equation of the radial function :
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
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:
We are given the differential equation
To derive the Numerov's method for solving this equation, we begin with the Taylor expansion of the function we want to solve, , around the point :
Denoting the distance from to by , we can write the above equation as
If we evenly discretize the space, we get a grid of points, where . By applying the above equations to this discrete space, we get a relation between the and :
Computationally, this amounts to taking a step forward by an amount . If we want to take a step backwards, we replace every with and get the expression for :
Note that only the odd powers of experienced a sign change. By summing the two equations, we derive that
We can solve this equation for by substituting the expression given at the beginning, that is . To get an expression for the factor, we simply have to differentiate twice and approximate it again in the same way we did this above:
If we now substitute this to the preceding equation, we get
This yields the Numerov's method if we ignore the term of order . It follows that the order of convergence (assuming stability) is 4.
- 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, Bibcode:1924MNRAS..84..592N, doi:10.1093/mnras/84.8.592.
- Numerov, Boris Vasil'evich (1927), "Note on the numerical integration of d2x/dt2 = f(x,t)", Astronomische Nachrichten, 230: 359–364, Bibcode:1927AN....230..359N, doi:10.1002/asna.19272301903.