# What is state of the art for the Shooting Method?

I am interested in examples where the Shooting Method has been used to find solutions to systems of ordinary differential equations that are either

• reasonably large systems, or
• the search algorithm in the shooting parameters is somewhat prohibitive because of the nature of the solutions, or
• both of the above.

Any references, descriptions, recent progress, folklore, in the ballpark would be of interest. Feel free to interpret "reasonably large" subjectively if necessary.

## 2 Answers

I don't know about state-of-the-art and I'm not sure if this is the kind of thing you were looking for...however in my two first papers I've used the shooting method in a parameter space that was originally too big (4 and 6 dimensions if I recall correctly) and the problem was that with randomly chosen parameters the numerical solver would not reach the other end of the domain and so I could not use a root-finding algorithm to search for the correct initial conditions.

The problem was there were unstable directions in the ODE and thus even with the correct initial conditions, the numerical noise would grow so large that you would not reach the other side.

My solution was to find more natural variables to use (using an algebraic similarity solution that satisfies the boundary conditions) and to rewrite the system in terms of the new variables. In the new variables the similarity solution is a fixed point and one can reach this fixed point only via its stable manifold, which had a lower dimension than the original space (in my case...). This allowed the root-finding algorithm to kick in and find a solution.

OK, This was a little vague. Here are the two papers (shameless plug):

(Added later:)

Recently I've been working on another problem that has highly unstable directions and there I use the collocation method, which (AFAIK) basically amounts so splitting the domain into many smaller part, doing shooting on each part, and trying to get the pieces to match up. If the problem is linear, this is a simple linear problem, if the problem isn't linear you need a non-linear root finder. I didn't write the code for the collocation, Matlab does it for me...look up BVP4C or BVP5C.

In writing this answer, I looked for "collocation method" online and found very little that seemed relevant. So I can only refer you to the Matlab function. perhaps someone else can find a reference that is relevant here.

Another shameless plug ... Coworkers and I used the Evans function formalism, which is a variant of the shooting method to deal with unstable directions (probably the same problem as mentioned by yfarjoun), on a boundary value problem of the form $y'(t) = (\lambda A_1 + A_2(t)) y(t)$ with $y$ specified as $t \to \pm \infty$. This is very similar to a Sturm-Liouville problem except that the differential operator is not self-adjoint. The application we're interested in is to do stability analysis of a travelling wave of a 2d reaction-diffusion equation. The main problem is that $y(t)$ is a fairly big vector with up to about 200 entries.

For details, please ask or see the paper at http://arxiv.org/abs/0805.1706 and references therein.