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case, the solutions of the ODE converge as time proceeds, and numerical integration
errors do not increase with time. Nonlinear differential equations may be unstable in
some regions and stable in others. For multiple coupled ODEs, the situation will be
even more complex. One should always be aware of possible instabilities of dynamic
systems when assessing numerical results.
Numerical integration methods introduce two types of errors: discretisation errors
and round-off errors. Discretisation errors are a property of the numerical integration
method, while round-off errors occur due to the finite number of digits used
in the calculations (hence they are a property of the computer and the program that
is used). In general, reducing the step-size hn will decrease the discretisation error.
76 Chapter 6. Analytical tools
x
t
Figure 6.3: Family of solutions of a stable ODE
hn
Total error
Discretisation
error
Round-off
error
Error
min
hn
Min.
error
Figure 6.4: Discretisation error, round-off error, and total error as a function of step-size
The total error will also decrease with hn, until a point is reached where the round-off
error is starting to become dominant. This has been illustrated in figure 6.4.
These errors may cause the numerical solution to become unstable, even when
the ODE itself is stable. An example of this numerical instability will be shown later
in figure 6.5, which demonstrates a system where the numerical error of the Euler
method is amplified in each step.
The order of a numerical integration method is defined in terms of the local discretisation
error dn, obtained when the method is applied to problems with smooth
solutions. A method is said to be of order p if a number C exists so that:
|dn| Chn
p+1 (6.4)
6.1. Numerical integration methods 77
C may depend on the derivatives of the function which defines the differential equation
and on the length of the interval over which the solution is sought, but it should
be independent of the step number n and the step-size hn.
6.1.3 Main categories of numerical integration methods
Refs.[17] and [18] recognise four general categories of step-by-step integration methods.
Based upon these references, we will provide a brief overview of these methods
in this section. All equations can easily be converted to vector notations when evaluating
sets of coupled ODEs. More information can also be found in ref.[33].
1. Taylor series methods
A smooth solution x(t) of equation (6.2) can be approximated by a Taylor series expansion:
x(t + hn) = x(t) + hnx˙(t) + hn
2
2!
¨ x(t) + . . . (6.5)
Provided it is possible to calculate higher-order time-derivatives of x, a numerical
method of order p can be obtained by using:
xn+1 = xn + hnx˙n + hn
2
2!
¨ xn + . . . + hn
p
p!
dpx
dtp
(6.6)
The derivatives x˙n, x¨n, etc. can be expressed in terms of the partial derivatives of the
function f from the state equation (6.2). The first neglected term provides an estimate
of the local discretisation error and can be used to select an appropriate step-size.
An example of a Taylor series method is the Euler method from equation (6.3),
which neglects all time-derivatives of order two and higher. Hence, the order of the
Euler method equals p = 1. Higher-order Taylor series methods involve increasingly
complex computations of the higher-order derivatives, so the general applicability of
these methods is rather limited.
2. Runge-Kutta methods
Runge-Kutta methods approximate Taylor series methods without evaluating timederivatives
beyond the first. The higher-order derivatives are replaced by a number
of evaluations of the function f . Modern Runge-Kutta algorithms typically include
techniques for estimating the discretisation error in order to control the step-size.
The Runge-Kutta methods require only one solution value xn in order to compute
xn+1, which makes them self-starting.
In order to gain a little more insight in the relation between the Taylor series and
Runge-Kutta methods, we will now first take a look at the derivation of a second
order Runge-Kutta method. The second-order method uses two function evaluations
per step [17]:
k1 = hn f (xn, tn)
k2 = hn f (xn + bk1, tn + ahn) (6.7)
78 Chapter 6. Analytical tools
The second step is a fractional step based on k1. a and b are two as yet unknown
coefficients. The two function evaluations are combined to make a complete step,
which involves two additional unknown coefficients g1 and g2:
xn+1 = xn + g1k1 + g2k2 (6.8)
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