Direct multiple shooting method

The multiple shooting method (English multiple shooting method) and multiple shooting method, in mathematics, a numerical method for solution of boundary value problems for ordinary differential equations. Here, the interval, at which the solution of the boundary problem is to be determined, first broken down into smaller sub-intervals, in which then a respective initial value problem is solved. With additional continuity conditions then a solution on the whole interval is determined. This method is a significant advance on the single shooting method, in particular with regard to the numerical stability.

Problem

Given a boundary value problem of the form

With the right side and the two-point boundary condition are given continuous functions and differentiable function is sought. To solve such a boundary value problem, the single shooting method proceeds as follows: Let the solution of the initial value problem

Then, the free parameter is determined so that the boundary condition

Is satisfied. To solve this vector equation is usually an iterative process, such as the Newton method, is used. For stiff initial value problems, however, small changes in initial conditions can lead to large changes in the solution, making the process is numerically unstable.

Method

The multiple shooting method used now to improve the stability of a subdivision

The interval into subintervals and calculate the solutions of a series of initial value problems

In these sub-intervals. The free parameters are determined so that the continuity conditions

And the boundary condition

Are fulfilled. Thus the composite function is defined by

Not only continuous but also differentiable, and thus a solution of the original problem. For determining the parameters is a non linear system of equations to solve by vectorial equations and unknowns, which in turn is carried out with an iterative method.