CS 340 – Introduction to Scientific Computing
Section 6.4: Multistep Methods
Monday, April 19, 2005
Definition 1 (m-step multistep method)
An m-step multistep method for solving the initial-value problem
dy
dx =f(x, y), a ≤x≤b, y(x0) = y0
uses past values of yand f(x, y)to construct a polynomial that approximates f, integrates the polynomial
to approximate the solution y, and extrapolates it’s values. The number of past points that are used sets
the degree of the polynomial and is therefore responsible for the truncation error. The order of the global
error of the method is equal to number of previous terms of the formula, which is also equal to one more
than the degree of the interpolating polynomial.
Thus has a difference equation for finding the approximation yi+1 at the mesh point xi+1 using the previous
mvalues (mis an integer greater than 1) is given by:
yi+1 =aiyi+ai−1yi−1+· · · +ai−(m−1) yi−(m−1)
+h[bif(xi, yi) + · · · +bi−(m−1) f(xi−(m−1), yi−(m−1) )]
with the PREVIOUSLY SPECIFIED values
y0, y1, y2,· · · , ym−1
which are often generated by some higher order method such as Runge-Kutta-Fehlberg. This method has
global error O(hm).
Method 1 (Adams-Bashford Second Order Method (Linear Interpolating Polynomial))
yi+1 =yi+h
2[3f(xi, yi)−f(xi−1, yi−1)]
given y0and y1to start. This method has global truncation error O(h2)and local truncation error O(h3).
Method 2 (Adams-Bashford Third Order Method (Quadratic Interpolating Polynomial))
yi+1 =yi+h
12[23f(xi, yi)−16f(xi−1, yi−1)+5f(xi−2,yi−2)]
given y0,y1and y2to start. This method has global truncation error O(h3)and local truncation error
O(h4).
Method 3 (Adams-Bashford Fourth Order Method (Cubic Interpolating Polynomial))
yi+1 =yi+h
24[55f(xi, yi)−59f(xi−1, yi−1) + 37f(xi−2, yi−2)−9f(xi−3, yi−3)]
given y0,y1,y2and y3to start. This method has global truncation error O(h4)and local truncation error
O(h5).
