Well discuss systematic approaches to constructing

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Unformatted text preview: s systematic approaches to constructing multistep methods in the next two sections however, let us illustrate the approach using the method of undetermined coe cients. With this technique we 1. assume a particular form of the general formula (5.1.2) by, possibly, restricting some of the coe cients and 2 2. determine the remaining coe cients of (5.1.2) such that they match terms of a Taylor's series expansion of the exact ODE solution to as high a degree as possible. Equivalently, the coe cients can be determined so that (5.1.2) produces the exact ODE solution when y(t) is a polynomial to as high a degree as possible. Here's an example. Example 5.1.1. Consider a multistep method of the form yn + 1yn;1 + 2 yn;2 = h 1fn;1 : This explicit two-step formula has three undetermined coe cients ( 1, 2, and 1) and we'll determine them so that the numerical method is exact when y(t) is an arbitrary quadratic polynomial. Since the multistep method is linear, it su ces to make the formula exact when y(t) is 1, t, and t2. If y(t) = 1 then f (t y) = 0 and (5.1.2) yields 1+ 1 + 2 = 0: (5.1.3a) When y(t) = t, f (t y) = 1 and (5.1.2) yields tn + 1(tn ; h) + 2 (tn ; 2h) = h 1: Using (5.1.3a), ; 1 ; 2 2 = 1: (5.1.3b) When y(t) = t2 , f (t y) = 2t and (5.1.2) yields t2 + 1(tn ; h)2 + 2(tn ; 2h)2 = 2h 1(tn ; h): n This may be simpli ed by (5.1.3a,b) to 1 + 4 2 = ;2 1 : (5.1.3c) The solution of (5.1.3a,b,c) is 1 =0 2 = ;1 3 1 =2 thus, the method is yn = yn;2 + 2hfn;1 n = 2 3 ::: : (5.1.4a) This scheme is called the \leap frog" scheme. It has only one function evaluation per step and, once started (with y0 and y1), is as simple as the explicit Euler method. Without having introduced a formal de nition of the local discretization error for multistep methods, let's use our experience with one-step methods to de ne it for the leap frog method as y(tn) ; y(tn;2) ; f (t y(t )): n= n;1 n;1 2h Since (5.1.4a) is exact when y(t) is a quadratic polynomial, we may either infer or a Taylor's series expansion to show that n = Ch2 y000( n) n 2 (tn;2 tn): The numerical constant C may also be determined by the method of undetermined coe cients. To do this, we select the simplest OD...
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