Shooting procedure

from Wikipedia, the free encyclopedia

The shooting method , even simple shooting method ( English (single) shooting method ), is a numerical method to boundary value problems of ordinary differential equations to be solved. The basic idea of ​​the method is to reduce the problem to the solution of an initial value problem .

The procedure is reminiscent of zeroing in artillery , a method of hitting a distant target with a projectile. The projectile is fired with a certain initial pitch. You vary this initial slope until you hit the target. Hence the name shooting procedure .


The boundary value problem of the second order with the function sought and on the right

is reformulated into an initial value problem

The second, unknown initial value can be freely selected. The initial value problem is integrated depending on the parameter c until the condition at the other edge is met. The solution of the initial value problem can be done with a numerical method, e.g. B. Runge-Kutta can be solved. depends on the initial value . Define a function F for this

This often non-linear system of equations can be solved numerically, for example with the Newton method or the bisection method . The solution of the initial value problem is a solution of the boundary value problem if and only if F has a zero in c :

In practice, for reasons of stability, the so-called multi - target variant of the shooting method is used, in which solutions are calculated piece by piece in sub-intervals of a grid , from which the solution is then composed.


  • J. Stoer, R. Bulirsch : Introduction to Numerical Analysis. Springer, New York 1980
  • A. Willers: Methods of practical analysis. 2nd Edition. De Gruyter, Berlin 1950
  • L. Collatz: Numerical treatment of differential equations. Springer, Berlin 1951
  • M. Hermann: Numerics of ordinary differential equations. Initial and marginal value problems , Oldenbourg Wissenschaftsverlag Munich and Vienna 2004, ISBN 3-486-27606-9