Multiple shooting method for two-point boundary value problems

Abstract

The common techniques for solving two-point boundary value problemscan be classified as either "shooting" or "finite difference"methods. Central to a shooting method is the ability to integratethe differential equations as an initial value problem with guessesfor the unknown initial values. This ability is not required with afinite difference method, for the unknowns are considered to be thevalues of the true solution at a number of interior mesh points.Each method has its advantages and disadvantages. One seriousshortcoming of shooting becomes apparent when, as happensaltogether too often, the differential equations are so unstablethat they "blow up" before the initial value problem can becompletely integrated. This can occur even in the face of extremelyaccurate guesses for the initial values. Hence, shooting seems tooffer no hope for some problems. A finite difference method doeshave a chance for it tends to keep a firm hold on the entiresolution at once. The purpose of this note is to point out acompromising procedure which endows shooting-type methods with thisparticular advantage of finite difference methods. For suchproblems, then, all hope need not be abandoned for shootingmethods. This is desirable because shooting methods are generallyfaster than finite difference methods. The organization is as follows: I. The two-point boundary value problem is stated in quite generalform. II. A particular shooting method is described which is designed tosolve the problem in this form. III. The two-point boundary value problem is then restated in sucha way that: (a) the restatement still falls within the general form, and (b) the shooting method now has a better chance of success when theequations are unstable. © 1962, ACM. All rights reserved.