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(d) If you continue the computation beyond tM, you can continue to generate values of y. What significance, if any, do these values have?
(e) Suppose that the initial condition is changed to y(0) = 1. Repeat parts (b) and (c) for this problem.
8.4 Multistep Methods
In previous sections we have discussed numerical procedures for solving the initial value problem
y= f (t, Ó), y(t0) = Ó0> (1)
in which data at the point t = tn are used to calculate an approximate value of the solution ô(‘ï+j) at the next mesh point t = tn+r In other words, the calculated value of ô at any mesh point depends only on the data at the preceding mesh point. Such methods are called one-step methods. However, once approximate values of the solution y = ô(‘) have been obtained at a few points beyond t0, it is natural to ask whether we can make use of some of this information, rather than just the value at the last point, to calculate the value of ô(‘) at the next point. Specifically, if y at tj, y2 at t2,, yn at tn are known, how can we use this information to determine yn+1 at tn+j? Methods that use information at more than the last mesh point are referred to as multistep methods. In this section we will describe two types of multistep methods, Adams3 methods and backward differentiation formulas. Within each type one can achieve various levels of accuracy, depending on the number of preceding data points that are used. For simplicity we will assume throughout our discussion that the step size h is constant.
John Couch Adams (1819-1892), English astronomer, is most famous as codiscoverer with Joseph Leverrier of the planet Neptune in 1846. Adams was also extremely skilled at computation; his procedure for numerical integration of differential equations appeared in 1883 in a book with Francis Bashforth on capillary action.
Chapter 8. Numerical Methods
Adams Methods. Recall that
where ô() is the solution of the initial value problem (1). The basic idea of an Adams
method is to approximate ô'(t) by a polynomial Pk(t) of degree k — 1 and to use the polynomial to evaluate the integral on the right side of Eq. (2). The coefficients in Pk (t) are determined by using k previously calculated data points. For example, suppose that we wish to use a first degree polynomial P2(t) = At + B. Then we need only the two data points (tn, yn) and (tn_p yn_ 1). Since P2 is to be an approximation to ô', we require that P2(tn) = f (tn, yn) and that P2(tn-1) = f (tn-1, yn-1). Recall that we denote f (tj, yj) by fj for an integer j. Then A and B must satisfy the equations
Finally, we replace ô(Ï+1) and ô(Ï) by yn+1 and yn, respectively, and carry out some algebraic simplification. For a constant step size h we obtain
Equation (5) is the second order Adams-Bashforth formula. It is an explicit formula
We note in passing that the first order Adams-Bashforth formula, based on the polynomial P1(t) = fn of degree zero, is just the original Euler formula.
More accurate Adams formulas can be obtained by following the procedure outlined above, but using a higher degree polynomial and correspondingly more data points. For example, suppose that a polynomial P4(t) of degree three is used. The coefficients are determined from the four points (tn, yn), (tn-1, Óï-1), (Ï-2> Óï-2), and (Ï-3, y^). Substituting this polynomial for ô'(³) in Eq. (2), evaluating the integral, and simplifying the result, we eventually obtain the fourth order Adams-Bashforth formula, namely,
A variation on the derivation of the Adams-Bashforth formulas gives another set of formulas called the Adams-Moulton4 formulas. To see the difference, let us again consider the second order case. Again we use a first degree polynomial Q2(t) = at + â,
Solving for A and B, we obtain
fn— 1 tn fn tn-1
Replacing ô'(t) by P2(t) and evaluating the integral in Eq. (2), we find that
for yn+1 in terms of yn and yn_ 1 and has a local truncation error proportional to h3.
4Forest Ray Moulton (1872-1952) was an American astronomer and administrator of science. While calculating ballistics trajectories during World War I, he devised substantial improvements in the Adams formula.
8.4 Multistep Methods
but we determine the coefficients by using the points (tn, yn) and (tn+1; yn+j). Thus a and â must satisfy
atn + â = fn ’
n n (7)
atn+j + â = fn+v
and it follows that