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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3053f4d493a4bf8546c77f550f1e1264debc437e)
to solve. Three values , which lie on equidistant grid points , are related to the equation
![{\ displaystyle y_ {n-1}, y_ {n}, y_ {n + 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6567438c4e5bcbd95e674cd0adb8a5587d87f3f8)
![{\ displaystyle x_ {n-1}, x_ {n}, x_ {n + 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7521538c3fdbbdb5a0fcaae80cd86a0e3acc508a)
![{\ 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0f373e4450891c91b576ba6af9570ad4e5fe6d)
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c9917f50e04691cbed1b1b1cee150b93e815dd)
![{\ displaystyle g_ {n} = g (x_ {n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a142f0a12951070b6edb32d7a5844bdf2fd6e63f)
![{\ displaystyle s_ {n} = s (x_ {n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b31d856c3c992ab57ab64ab078aeb122d157b5f4)
![{\ displaystyle h = x_ {n + 1} -x_ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0118a65ecd99b873f7df0543fd0afb64a1e61877)
![{\ displaystyle y_ {n + 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6477fbeae2962cc55973c2298b8653cfd4f5e5d1)
Nonlinear equation
For nonlinear equations of the form
![{\ displaystyle {\ frac {d ^ {2} y} {dx ^ {2}}} = f (x, y),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d45cb45a0684257588b38663b907d394bd7a121b)
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/495b4344205f672d5d525ffb5d1f21ec2ccda5d3)
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0d34892c456ec50a2767e01129ac5f91fc237e)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d)
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:
![R (r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b97817d8756302ef44f910ec5e76346e8d4f4d)
![{\ 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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbcda35841e3e5e96731948709626600b2cdfc2d)
This equation can be brought into the form required for the Numerow method by substitution:
![{\ displaystyle u (r) = rR (r),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54f99a84b78a35ab3ae0dbbb72f8851c51921555)
![{\ displaystyle R (r) = {\ frac {u (r)} {r}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28957e20309cadee5a40e27465e44cea2f74eac6)
![{\ 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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5481e5312d969123cdfbf6da252709b1a8cb7645)
![{\ 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}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffc71d9022baa23ed3f2cafad1375cc697b0aba3)
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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb5a4a4cd4a1de4a42822072f9e970400a3a5536)
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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74a91e6b0551f8e1c2770febeaf24f17a8cba580)
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}. }](https://wikimedia.org/api/rest_v1/media/math/render/svg/34c3eb36cb35252d562006973dfa1896e7fc8979)
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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/224b90401752db41b235fbf02924ec13d6ab6341)
![{\ displaystyle g (r) = {\ frac {2m} {\ hbar ^ {2}}} (E-V _ {\ text {eff}} (r)),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/686f28a7def1c35547884eac2e82a3bd45b39a96)
![{\ displaystyle s (r) = 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88d7d78bc5fd142b4d4d205fc419adfc5f39b46f)
Derivation
The starting point is a differential equation of the form to be solved
![{\ displaystyle y '' (x) = - g (x) y (x) + s (x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30d1e8c5982304c9db81105a04e5d730cba06e67)
To derive the method, the function around the point is first developed using a Taylor series .
![y (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e871993bfd131a8b0c3591c26084cf8171a74dcd)
![x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf)
![{\ 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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/563b0e039a42ccb7eb186920a39f24217584c833)
The distance between to was defined as. The Taylor expansion in the forward or backward direction differ in the signs of the odd orders.
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf)
![{\ displaystyle h = x-x_ {0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d265eb3cfea48f0a88b98878390bd5336e5a67c)
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 :
![x](https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4)
![{\ displaystyle h = x_ {n + 1} -x_ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0118a65ecd99b873f7df0543fd0afb64a1e61877)
![{\ displaystyle y_ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c5fbb0c89590b028eba7239a8803fd0cd2e698e)
![{\ displaystyle y_ {n + 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6477fbeae2962cc55973c2298b8653cfd4f5e5d1)
![{\ 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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b23378b39d4a808f68e404b3a4010555e18fdc88)
This corresponds to a step forward . For a backward step, the Taylor expansion results with :
![H](https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a)
![{\ displaystyle -h}](https://wikimedia.org/api/rest_v1/media/math/render/svg/908e45db209f8c334c658b0e4e5e88c5e430fd56)
![{\ 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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64d18fe19c36af4dc2399d277c8b24bed2ad5653)
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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d16be145330d6b7e16b6a93d8a526f0879bb228f)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1a002eb362c9dbfe58b49826f8a2dc5f9a2839)
![{\ displaystyle y '' '' _ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e305157e924a6eedd372cf0941ffc11fd96c5479)
![{\ displaystyle y '' '' _ {n} = {\ frac {d ^ {2}} {dx ^ {2}}} (- g_ {n} y_ {n} + s_ {n}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb89bfb899f071a9942d08796ffd949ddb5fb1df)
![{\ 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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22700c33ad4e644b6ebbf35d281f76bef26c7701)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0351bec54964e441596b7140e6afb8f157bfbe)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/551c4bc944edcc288e53631bc3c5b5c9eddf6e7e)
This is the numerow method equation with an error of order .
![{\ displaystyle h ^ {6}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31dd74cddf74dfd4fde67eb5db7a259097e96024)
Individual evidence
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↑ 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 .
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↑ 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 .
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↑ 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 .