Direct multiple shooting method

In the area of mathematics known as numerical ordinary differential equations, the direct multiple shooting method is a numerical method for the solution of boundary value problems. The method divides the interval over which a solution is sought in several smaller intervals, solve a boundary value problem in each of the smaller intervals by the single shooting method, and then iterates to fit all the solutions in the smaller intervals together to form a solution on the whole interval. This constitutes a significant improvement in stability single shooting methods.

Single shooting methods

Shooting methods can be used to solve boundary value problems like

$y''(t)=f(t,y(t),y'(t)),\quad y(t_{a})=y_{a},\quad y(t_{b})=y_{b}.$

Single shooting methods proceed as follows. Let y(t; p) denote the solution of the initial value problem

$y''(t)=f(t,y(t),y'(t)),\quad y(t_{0})=y_{0},\quad y'(t_{0})=p.$

Define the function F(p) as the difference between y(t_b; p) and the specified boundary value y_b: F(p) = y(t_b; p) − y_b. Then every solution of the boundary value problem corresponds to a root of F. This root can be solved by any root-finding method.

Multiple shooting

A direct multiple shooting method partitions the interval [t_a, t_b] by introducing additional grid points

$t_{a}=t_{0}<t_{1}<\cdots <t_{n}=t_{b}$.

The method starts by guessing somehow the values of y and y' at all grid points t_k with 0 ≤ k ≤ n − 1. Denote these guesses by y_k and p_k. Let y(t; t_k, y_k, p_k) denote the solution emanating from the kth grid point, that is, the solution of the initial value problem

$y''(t)=f(t,y(t),y'(t)),\quad y(t_{k})=y_{k},\quad y'(t_{k})=p_{k}.$

All these solutions can be pieced together if the values y and its derivative match at the grid points. Thus, solutions of the boundary value problem correspond to solutions of the following system of 2n − 1 equations:

${\begin{aligned}&y(t_{1};t_{0},y_{0},p_{0})=y_{1}\\&y'(t_{1};t_{0},y_{0},p_{0})=p_{1}\\&\qquad \qquad \vdots \\&y(t_{n-1};t_{n-2},y_{n-2},p_{n-2})=y_{n-1}\\&y'(t_{n-1};t_{n-2},y_{n-2},p_{n-2})=p_{n-1}\\&y(t_{n};t_{n-1},y_{n-1},p_{n-1})=y_{b}.\end{aligned}}$

The top 2(n−1) equations are the matching conditions, and the last equations is the condition y(t_b) = y_b from the boundary value problem. The multiple shooting method solves the boundary value problem by solving this system of equations. Typically, a modification of the Newton's method is used for the latter task.

Bock's Direct Multiple Shooting Method

(stub)

References

• Stoer, Josef; Bulirsch, Roland (2002), Introduction to Numerical Analysis (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95452-3. See Sections 7.3.5 and further.
• Bock, Hans Georg; Plitt, Karl J. (1984), "A multiple shooting algorithm for direct solution of optimal control problems", Proceedings of the 9th IFAC World Congress (PDF), Budapest{{citation}}: CS1 maint: location missing publisher (link)