The Chemistry Maths Book, Second Edition

590 Chapter 20Numerical methods

EXAMPLE 20.19Apply the Runge–Kutta methods of orders 2 and 4 to the initial

value problem in Example 20.18, with step sizeh 1 = 1 0.2.

The result of applying the recursion relations (20.55) for order 2 and (20.56) for order

4 are displayed in Table 20.11.

The table shows that the second-order method with step size h 1 = 1 0.2is already

considerably more accurate that Euler’s first-order method withh 1 = 1 0.1, and that the

fourth-order method is very much more accurate still.

0 Exercises 38–42

20.10 Systems of differential equations

A problem involving second- or higher-order differential equations can always be

reduced to one involving a system of simultaneous first-order equations. For

example, the general second-order linear equation

(20.57)

can be written as the pair of first-order equations

(20.58)

for the two functions of x, y(x)and z(x). If (20.57) is the differential equation of

an initial value problem, with initial conditionsy(x

0

) 1 = 1 y

0

andy′(x

0

) 1 = 1 z(x

0

) 1 = 1 y

1

, then

equations (20.58) are a pair of (coupled) first-order initial value problems. The

dz

dx

=− −rx pxz qxy() () ()

dy

dx

=z

dy

dx

px

dy

dx

qxy rx

2

2

++=() () ()

Table 20.11 Example of Runge–Kutta methods

order nx

n

y

n

exact error order nx

n

y

n

error

2 0 0.0 1.000000 1.000000 0.000000 4 0 0.0 1.000000 0.000000

1 0.2 1.440000 1.442806 0.002805 1 0.2 1.442800 0.000005

2 0.4 1.976800 1.983649 0.006849 2 0.4 1.983636 0.000013

3 0.6 2.631696 2.644238 0.012542 3 0.6 2.644213 0.000025

4 0.8 3.430669 3.451082 0.020413 4 0.8 3.451042 0.000040

5 1.0 4.405416 4.436563 0.031147 5 1.0 4.436502 0.000061