Shooting Method Tutorial.doc

Shooting Method Tutorial Gary Parker July 9, 2006 Let f = f(x) denote a function to be determined by the boundary value problem (1a,b,c) This problem has an exact solution: (2) The shooting method can be used to find this solution numerically. Of course, a numerical method is not necessary to solve (1). But the shooting method also works for nonlinear boundary value problems for which there is no closed-form solution. Before introducing the shooting method it is useful to review the Newton-Raphson method for solving algebraic equations iteratively. Let R(s) be some specified function of s (e.g. R(s) = 1-e-s). We wish to solve the equation (3) Let s be a guess for the root. We can get an improved guess snew by defining (4) and solving the linearized form based on Taylor expansion (5) to obtain the relation (6) Equation (6) can now be solved iteratively for s. To implement the solution of (1a,b,c) by the shooting method we first define (7) so that (1a) reduces to two first-order equations, (8a,b) We could solve this equation by marching from x = 0 using e.g. the Euler step method if both boundary conditions (1b) and (1c) were specified at x = 0. Boundary condition (1c), however, is specified at x = 1. To overcome this problem, we replace (1c) with the condition (9) where the right value of s is the one that yields the value f = 1 at x = 1. We don’t know what s should be yet, so we treat it as a free variable. Thus f and g are not only functions of x, but also of s; (10a,b) The boundary value problem (1) now becomes (11a,b,c,d) where s is to be determined. Now (11a,b) can be solved numerically using e.g. the Euler step method. Let (12) where N is an appropriately large positive integer, and let xn = (n-1)(x, n = 1..N + 1, so that x1 = 0 and xN+1 = 1. Further letting fn and gn denote the values of f and g at xn, (11a,b) discretize to (13a,b) where since