Direct Multiple Shooting Method - Single Shooting Methods

Single Shooting Methods

Shooting methods can be used to solve boundary value problems (BVP) like

in which the time points t_a and t_b are known but the initial and terminal values y_a and y_b are unknown and sought.

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(t_b; p) and the specified boundary value y_b: F(p) = y(t_b; p) − y_b. Then for every solution (y_a, y_b) of the boundary value problem we have y_a=y₀ while y_b corresponds to a root of F. This root be solved by any root-finding method given that certain method-dependent prerequisites are satisfied. This often will require initial guesses to y_a and y_b. 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 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 y_a. 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.
• The nonlinearity of the ODE effectively becomes a nonlinearity of F, and requires a root-finding technique capable of solving nonlinear systems. Such methods typically converge slower as nonlinearities become more severe. The boundary value problem solver's performance suffers from this.
• Even stable and well-conditioned ODEs may make for unstable and ill-conditioned BVPs. A slight alteration of the initial value guess y₀ may generate an extremely large step in the ODEs solution y(t_b; t_a, y₀) and thus in the values of the function F whose root is sought. Non-analytic root-finding methods can seldom cope with this behaviour.