Numerow method

The Numerow method is a method for numerically solving ordinary differential equations contain the second-order no first order term. It is an implicit fourth order multistep method , but it can be made explicit if the differential equation is linear.

The method was developed by the Russian astronomer Boris Wassiljewitsch Numerow .

The procedure

Linear equation

The Numerow's method can be used to find differential equations of the form

${\displaystyle {\frac {d^{2}y}{dx^{2}}}=-g(x)y(x)+s(x).}$

to solve. Three values , which lie on equidistant grid points , are related to the equation ${\displaystyle y_{n-1},y_{n},y_{n+1}}$ ${\displaystyle x_{n-1},x_{n},x_{n+1}}$

${\displaystyle {\begin{aligned}y_{n+1}\left(1+{\frac {h^{2}}{12}}g_{n+1}\right)&=2y_{n}\left(1-{\frac {5h^{2}}{12}}g_{n}\right)-y_{n-1}\left(1+{\frac {h^{2}}{12}}g_{n-1}\right)\\&+{\frac {h^{2}}{12}}(s_{n+1}+10s_{n}+s_{n-1})+{\mathcal {O}}(h^{6}),\end{aligned}}}$

wherein , , , and . It can therefore be calculated on the basis of two previous values ​​and the solution can be calculated by iterating over the entire grid. ${\displaystyle y_{n}=y(x_{n})}$ ${\displaystyle g_{n}=g(x_{n})}$ ${\displaystyle s_{n}=s(x_{n})}$ ${\displaystyle h=x_{n+1}-x_{n}}$ ${\displaystyle y_{n+1}}$

Nonlinear equation

For nonlinear equations of the form

${\displaystyle {\frac {d^{2}y}{dx^{2}}}=f(x,y),}$

arises for the procedure

${\displaystyle y_{n+1}-2y_{n}+y_{n-1}={\frac {h^{2}}{12}}(f_{n+1}+10f_{n}+f_{n-1})+{\mathcal {O}}(h^{6})}$

with . This is an implicit relationship that reduces to the above explicit form when linear is in. It achieves fourth order accuracy. ${\displaystyle f_{n}=f(x_{n},y_{n})}$ ${\displaystyle f}$ ${\displaystyle y}$

application

In numerical physics, the procedure is used to find solutions to the one-dimensional Schrödinger equation for any potential. An example is solving the radial equation for a spherically symmetric potential, as it occurs in the hydrogen atom . After the variables have been separated and the angle part has been solved analytically, the radial part remains: ${\displaystyle R(r)}$

${\displaystyle {\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)-{\frac {2mr^{2}}{\hbar ^{2}}}(V(r)-E)R(r)=l(l+1)R(r).}$

This equation can be brought into the form required for the Numerow method by substitution:

${\displaystyle u(r)=rR(r),}$

${\displaystyle R(r)={\frac {u(r)}{r}},}$

${\displaystyle {\frac {dR}{dr}}={\frac {1}{r}}{\frac {du}{dr}}-{\frac {u(r)}{r^{2}}}={\frac {1}{r^{2}}}\left(r{\frac {du}{dr}}-u(r)\right),}$

${\displaystyle {\frac {d}{dr}}\left(r^{2}{\frac {dR}{dr}}\right)={\frac {du}{dr}}+r{\frac {d^{2}u}{dr^{2}}}-{\frac {du}{dr}}=r{\frac {d^{2}u}{dr^{2}}}.}$

With this substitution becomes the radial equation

${\displaystyle r{\frac {d^{2}u}{dr^{2}}}-{\frac {2mr}{\hbar ^{2}}}(V(r)-E)u(r)={\frac {l(l+1)}{r}}u(r),}$

or

${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}u}{dr^{2}}}+\left(V(r)+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}\right)u(r)=Eu(r),}$

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

${\displaystyle V_{\text{eff}}(r)=V(r)+{\frac {\hbar ^{2}}{2m}}{\frac {l(l+1)}{r^{2}}}=V(r)+{\frac {L^{2}}{2mr^{2}}},\quad L^{2}=l(l+1)\hbar ^{2}.}$

is. In order to recognize the applicability of the Numerow method, this equation can be transformed to:

${\displaystyle {\frac {d^{2}u}{dr^{2}}}=-{\frac {2m}{\hbar ^{2}}}(E-V_{\text{eff}}(r))u(r),}$

${\displaystyle g(r)={\frac {2m}{\hbar ^{2}}}(E-V_{\text{eff}}(r)),}$

${\displaystyle s(r)=0.}$

Derivation

The starting point is a differential equation of the form to be solved

${\displaystyle y''(x)=-g(x)y(x)+s(x).}$

To derive the method, the function around the point is first developed using a Taylor series . ${\displaystyle y(x)}$ ${\displaystyle x_{0}}$

${\displaystyle y(x_{0}\pm h)=y(x_{0})\pm hy'(x_{0})+{\frac {h^{2}}{2!}}y''(x_{0})\pm {\frac {h^{3}}{3!}}y'''(x_{0})+{\frac {h^{4}}{4!}}y''''(x_{0})\pm {\frac {h^{5}}{5!}}y'''''(x_{0})+{\mathcal {O}}(h^{6}).}$

The distance between to was defined as. The Taylor expansion in the forward or backward direction differ in the signs of the odd orders. ${\displaystyle x}$ ${\displaystyle x_{0}}$ ${\displaystyle h=x-x_{0}}$

If the space is divided into even discrete intervals, a grid of points results with . The above equation can be applied to every grid point and gives a relationship between and : ${\displaystyle x}$ ${\displaystyle h=x_{n+1}-x_{n}}$ ${\displaystyle y_{n}}$ ${\displaystyle y_{n+1}}$

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

This corresponds to a step forward . For a backward step, the Taylor expansion results with : ${\displaystyle h}$ ${\displaystyle -h}$

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

If you add both equations, the odd orders disappear and it remains

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

The second derivative can be replaced by the differential equation given at the beginning . To get an expression for the fourth derivative , the differential equation is derived twice and the second derivative is approximated: ${\displaystyle y''_{n}=-g_{n}y_{n}+s_{n}}$ ${\displaystyle y''''_{n}}$

${\displaystyle y''''_{n}={\frac {d^{2}}{dx^{2}}}(-g_{n}y_{n}+s_{n}),}$

${\displaystyle h^{2}y''''_{n}=-g_{n+1}y_{n+1}+s_{n+1}+2g_{n}y_{n}-2s_{n}-g_{n-1}y_{n-1}+s_{n-1}+{\mathcal {O}}(h^{4}).}$

Replacing the fourth derivative of the above equation with this expression gives

${\displaystyle {\begin{aligned}y_{n+1}-2y_{n}+y_{n-1}&={h^{2}}(-g_{n}y_{n}+s_{n})+{\frac {h^{2}}{12}}(-g_{n+1}y_{n+1}+s_{n+1}+2g_{n}y_{n}-2s_{n}-g_{n-1}y_{n-1}+s_{n-1})\\&+{\mathcal {O}}(h^{6}),\end{aligned}}}$

and after summarizing the terms

${\displaystyle {\begin{aligned}y_{n+1}\left(1+{\frac {h^{2}}{12}}g_{n+1}\right)-2y_{n}\left(1-{\frac {5h^{2}}{12}}g_{n}\right)+y_{n-1}\left(1+{\frac {h^{2}}{12}}g_{n-1}\right)\\={\frac {h^{2}}{12}}(s_{n+1}+10s_{n}+s_{n-1})+{\mathcal {O}}(h^{6}).\end{aligned}}}$

This is the numerow method equation with an error of order . ${\displaystyle h^{6}}$

Individual evidence

1. ↑ Numerow, Boris Wassiljewitsch (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 .
2. ↑ Numerow, Boris Wassiljewitsch (1927), "Note on the numerical integration of d ² x / d t ² = f ( x , t )", Astronomische Nachrichten , 230: 359–364, bibcode : 1927AN .... 230. .359N , doi : 10.1002 / asna.19272301903 .
3. ↑ 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 .