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
d 2 y d x 2 = − g ( x ) y ( x ) + s ( x ) . {\displaystyle {\frac {d^{2}y}{dx^{2}}}=-g(x)y(x)+s(x).}
In it, three values of y n − 1 , y n , y n + 1 {\displaystyle y_{n-1},y_{n},y_{n+1}} taken at three equidistant points x n − 1 , x n , x n + 1 {\displaystyle x_{n-1},x_{n},x_{n+1}} are related as follows:
y n + 1 ( 1 + h 2 12 g n + 1 ) = 2 y n ( 1 − 5 h 2 12 g n ) − y n − 1 ( 1 + h 2 12 g n − 1 ) + h 2 12 ( s n + 1 + 10 s n + s n − 1 ) + O ( h 6 ) , {\displaystyle y_{n+1}\left(1+{\frac {h^{2}}{12}}g_{n+1}\right)=2y_{n}\left(1-{\frac {5h^{2}}{12}}g_{n}\right)-y_{n-1}\left(1+{\frac {h^{2}}{12}}g_{n-1}\right)+{\frac {h^{2}}{12}}(s_{n+1}+10s_{n}+s_{n-1})+{\mathcal {O}}(h^{6}),}
where y n = y ( x n ) {\displaystyle y_{n}=y(x_{n})} , g n = g ( x n ) {\displaystyle g_{n}=g(x_{n})} , s n = s ( x n ) {\displaystyle s_{n}=s(x_{n})} , and h = x n + 1 − x n {\displaystyle h=x_{n+1}-x_{n}} .
Nonlinear equations For nonlinear equations of the form
d 2 y d x 2 = f ( x , y ) , {\displaystyle {\frac {d^{2}y}{dx^{2}}}=f(x,y),}
the method gives
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