The Computable Differential Equation Lecture ... - Bruce E. Shapiro

Chapter 6
Linear Multistep Methods for
Initial Value Problems
6.1 Multistep Methods
Definition 6.1. A Linear Multistep Method for the initial value problem y ′ =
f(t, y), y(t 0 ) = y 0 is any method of the form
k∑
a j y n−j = h
j=0
k∑
b j f n−j (6.1)
where f k = f(t k , y k ), and a 0 , a 1 , . . . , a k , b 0 , b 1 , . . . , b k are constants. If b 0 = 0 the
method is explicit, and if b 0 ≠ 0 the method is implicit.
These methods are linear in the function f(t, y), as compared with Runge-Kutta
methods, which were non-linear in f. The linearity of the method does not imply
the linearity of y – in fact, there is no such restriction on y – but only the linearity
of f in the method. They are called multistep methods because they depend on
function (and possibly solution) values at up to k prior mesh points.We will also
find the following notation helpful:
Definition 6.2. The Characteristic Polynomials of a linear multistep method
are
ρ(x) =
σ(x) =
j=0
k∑
a j x k−j (6.2)
j=0
k∑
b j x k−j (6.3)
j=0
The order of a linear multistep method is particularly easy to calculate, as
we will show in the following derivation. Define the linear multistep operator
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