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 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.
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.
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.
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 nth 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)
We solve for to get
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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