Shooting procedure

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 .

Procedure

The boundary value problem of the second order with the function sought and on the right ${\displaystyle \,y(t)}$ ${\displaystyle f}$

${\displaystyle y''(t)=f(t,y(t),y'(t)),\quad y(t_{1})=a,\quad y(t_{2})=b}$

is reformulated into an initial value problem

${\displaystyle y''(t)=f(t,y(t),y'(t)),\quad y(t_{1})=a,\quad y'(t_{1})=c}$

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 ${\displaystyle c}$ ${\displaystyle \quad y(t_{2})=b}$ ${\displaystyle y(t\,;c)}$ ${\displaystyle y(t\,;c)}$ ${\displaystyle c}$

${\displaystyle F(c):=y(t_{2},c)-b\quad }$

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 :

${\displaystyle 0=y(t_{2},c)-b\quad \Leftrightarrow \quad y(t_{2},c)=b}$

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. ${\displaystyle \Delta =\left\{t_{1}=x_{0},x_{1},\dots ,x_{n-1},x_{n}=t_{2}\right\}}$ ${\displaystyle [a,b]}$

literature

• 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