努梅罗夫方法

可由努梅罗夫方法求解的微分方程形式为

${\displaystyle \left({\frac {d^{2}}{dx^{2}}}+f(x)\right)y(x)=0}$ 

求出函数 ${\displaystyle y(x)}$  在区间 ${\displaystyle [a,b]}$  上等距格点上的值，从连续的两个格点上的函数值 ${\displaystyle x_{n-1}}$  和 ${\displaystyle x_{n}}$  开始，其他的函数值可由

${\displaystyle y_{n+1}={\frac {\left(2-{\frac {5h^{2}}{6}}f_{n}\right)y_{n}-\left(1+{\frac {h^{2}}{12}}f_{n-1}\right)y_{n-1}}{1+{\frac {h^{2}}{12}}f_{n+1}}}}$ 

算得。

其中， ${\displaystyle f_{n}=f(x_{n})}$  和 ${\displaystyle y_{n}=y(x_{n})}$  为在格点 ${\displaystyle x_{n}}$  上的函数值，${\displaystyle h=x_{n}-x_{n-1}}$ 为格点间距。

对于非线性方程，

${\displaystyle {\frac {d^{2}}{dt^{2}}}y=f(t,y)}$ 

则非线性方程的努梅罗夫方法为

${\displaystyle y_{n+1}=2y_{n}-y_{n-1}+{\tfrac {1}{12}}h^{2}(f_{n+1}+10f_{n}+f_{n-1}).}$ 

该式为隐式的线性多步方法。当 ${\displaystyle f}$  是 ${\displaystyle y}$  的线性函数时，该式变为显式方法，精度为4阶（Hairer，Nørsett & Wanner 1993，§III.10）。

在物理中用于数值求解任意势场中径向薛定谔方程：

${\displaystyle \left[-{\hbar ^{2} \over 2\mu }\left({\frac {1}{r}}{\partial ^{2} \over \partial r^{2}}r-{l(l+1) \over r^{2}}\right)+V(r)\right]R(r)=ER(r)}$ 

此式可重写为

${\displaystyle \left[{\partial ^{2} \over \partial r^{2}}-{l(l+1) \over r^{2}}+{2\mu \over \hbar ^{2}}\left(E-V(r)\right)\right]u(r)=0}$ 

其中 ${\displaystyle u(r)=rR(r)}$ . 与Numerov方法求解的方程形式做比较，

${\displaystyle f(x)={\frac {2\mu }{\hbar ^{2}}}\left(E-V(x)\right)-{\frac {l(l+1)}{x^{2}}}}$ 

这样，我们可以数值求解薛定谔方程。

从 ${\displaystyle y(x_{n})}$  的泰勒展开开始， 我们可求 ${\displaystyle x_{n}}$  的相接邻点上的函数值

${\displaystyle y_{n+1}=y(x_{n}+h)=y(x_{n})+hy'(x_{n})+{\frac {h^{2}}{2!}}y''(x_{n})+{\frac {h^{3}}{3!}}y'''(x_{n})+{\frac {h^{4}}{4!}}y''''(x_{n})+{\frac {h^{5}}{5!}}y'''''(x_{n})+{\mathcal {O}}(h^{6})}$ 

${\displaystyle y_{n-1}=y(x_{n}-h)=y(x_{n})-hy'(x_{n})+{\frac {h^{2}}{2!}}y''(x_{n})-{\frac {h^{3}}{3!}}y'''(x_{n})+{\frac {h^{4}}{4!}}y''''(x_{n})-{\frac {h^{5}}{5!}}y'''''(x_{n})+{\mathcal {O}}(h^{6})}$ 

上两式之和为

${\displaystyle y_{n-1}+y_{n+1}=2y_{n}+{h^{2}}y''_{n}+{\frac {h^{4}}{12}}y''''_{n}+{\mathcal {O}}(h^{6})}$ 

用所求微分方程的定义式 ${\displaystyle y''_{n}=-f_{n}y_{n}}$  替换掉 ${\displaystyle y''_{n}}$ ，

${\displaystyle h^{2}f_{n}y_{n}=2y_{n}-y_{n-1}-y_{n+1}+{\frac {h^{4}}{12}}y''''_{n}+{\mathcal {O}}(h^{6})}$ 

对所求微分方程的定义式 ${\displaystyle y''_{n}=-f_{n}y_{n}}$  取二次微分

${\displaystyle y''''(x)=-{\frac {d^{2}}{dx^{2}}}\left[f(x)y(x)\right]}$ 

将其代入到四阶微分项中，并把二阶导 ${\displaystyle {\frac {d^{2}}{dx^{2}}}\left[f(x)y(x)\right]}$  替换为 ${\displaystyle f_{n}y_{n}}$  的二阶差分公式 ${\displaystyle {\frac {f_{n-1}y_{n-1}-2f_{n}y_{n}+f_{n+1}y_{n+1}}{h^{2}}}}$ 

${\displaystyle h^{2}f_{n}y_{n}=2y_{n}-y_{n-1}-y_{n+1}-{\frac {h^{4}}{12}}{\frac {f_{n-1}y_{n-1}-2f_{n}y_{n}+f_{n+1}y_{n+1}}{h^{2}}}+{\mathcal {O}}(h^{6})}$ 

求解 ${\displaystyle y_{n+1}}$  可得

${\displaystyle y_{n+1}={\frac {\left(2-{\frac {5h^{2}}{6}}f_{n}\right)y_{n}-\left(1+{\frac {h^{2}}{12}}f_{n-1}\right)y_{n-1}}{1+{\frac {h^{2}}{12}}f_{n+1}}}+{\mathcal {O}}(h^{6}).}$ 

忽略掉 ${\displaystyle {\mathcal {O}}(h^{6})}$  就可以得到努梅罗夫方法，最终收敛阶数为4(假定稳定)。

• Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard, Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, 1993, ISBN 978-3-540-56670-0 .
  This book includes the following references:
  • Numerov, Boris Vasil'evich, A method of extrapolation of perturbations, Monthly Notices of the Royal Astronomical Society, 1924, 84: 592–601 .
  • Numerov, Boris Vasil'evich, Note on the numerical integration of d²x/dt² = f(x,t), Astronomische Nachrichten, 1927, 230: 359–364 .

• Lecture notes: Computerphysik und Numerik （页面存档备份，存于互联网档案馆） - by Jan Krieger
• Lecture notes of Werner Scholz - At Vienna University of Technology
• Lecture notes of Alexander Wagner （页面存档备份，存于互联网档案馆）