Shooting method

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

The process is reminiscent of the zeroing in of artillery, a method to meet a distant target with a projectile. The projectile is fired with a certain initial slope. This initial slope is varied until it hits the target. Therefore, the name comes shooting method.

Method

The second-order boundary value problem with studied function and right side

Is reformulated as an initial value problem

The second, unknown initial value is arbitrary. The initial value problem as long integrated in dependence on the parameter C, until the condition is satisfied at the other edge. The solution of the initial value problem can be solved by a numerical method, such as Runge- Kutta. depends on the initial value. Define a function F to

This often non-linear system of equations can be solved numerically, for example using Newton's method or the bisection method. The solution of the initial value problem is just a solution of the boundary value problem, if F has a zero in C:

In practice, one used for stability reasons, the so-called multi-objective variant of the firing process, are calculated at the piecemeal solutions into sub-intervals of a lattice, from which then make up the solution.