Direct Multiple Shooting Method - Multiple Shooting

Multiple Shooting

A direct multiple shooting method partitions the interval by introducing additional grid points

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The method starts by guessing somehow the values of y at all grid points tk with 0 ≤ k ≤ N − 1. Denote these guesses by yk. Let y(t; tk, yk) denote the solution emanating from the kth grid point, that is, the solution of the initial value problem

All these solutions can be pieced together to form a continuous trajectory if the values y match at the grid points. Thus, solutions of the boundary value problem correspond to solutions of the following system of N equations:

\begin{align} & y(t_1; t_0, y_0) = y_1 \\ & \qquad\qquad\vdots \\ & y(t_{N-1}; t_{N-2}, y_{N-2}) = y_{N-1} \\ & y(t_N; t_{N-1}, y_{N-1}) = y_b. \end{align}

The central N−2 equations are the matching conditions, and the first and last equations are the conditions y(ta) = ya and y(tb) = yb 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.

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