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Concept# Direct multiple shooting method

Summary

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 into several smaller intervals, solves an initial value problem in each of the smaller intervals, and imposes additional matching conditions to form a solution on the whole interval. The method constitutes a significant improvement in distribution of nonlinearity and numerical stability over single shooting methods.
Shooting methods can be used to solve boundary value problems (BVP) like
in which the time points ta and tb are known and we seek
Single shooting methods proceed as follows. Let y(t; t0, y0) denote the solution of the initial value problem (IVP)
Define the function F(p) as the difference between y(tb; p) and the specified boundary value yb: F(p) = y(tb; p) − yb. Then for every solution (ya, yb) of the boundary value problem we have ya=y0 while yb corresponds to a root of F. This root can be solved by any root-finding method given that certain method-dependent prerequisites are satisfied. This often will require initial guesses to ya and yb. Typically, analytic root finding is impossible and iterative methods such as Newton's method are used for this task.
The application of single shooting for the numerical solution of boundary value problems suffers from several drawbacks.
For a given initial value y0 the solution of the IVP obviously must exist on the interval [ta,tb] so that we can evaluate the function F whose root is sought.
For highly nonlinear or unstable ODEs, this requires the initial guess y0 to be extremely close to an actual but unknown solution ya. Initial values that are chosen slightly off the true solution may lead to singularities or breakdown of the ODE solver method. Choosing such solutions is inevitable in an iterative root-finding method, however.
Finite precision numerics may make it impossible at all to find initial values that allow for the solution of the ODE on the whole time interval.

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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 into several smaller intervals, solves an initial value problem in each of the smaller intervals, and imposes additional matching conditions to form a solution on the whole interval. The method constitutes a significant improvement in distribution of nonlinearity and numerical stability over single shooting methods.

Numerical methods for ordinary differential equations

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.

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