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series, substituting the expansions into the formula for xn+1 and comparing the
result with the Taylor series for the true local solution zn(tn+1). Notice that the b’s
form a lower triangular array, so that each ki is obtained from the previous k’s.
A popular strand of explicit Runge-Kutta methods are the Dormand-Prince pairs.
Table 6.1 lists the coefficients of the fourth- and fifth-order methods (represented by
the coefficients gi
 and gi, respectively), which are usually combined to form the
Dormand-Prince (4,5) pair. This pair is used by the SIMULINK solver ode45.
3. Linear multistep methods
The Runge-Kutta and Taylor series methods determine the value of xn+1 by means
of a function which depends only on tn, yn, and the step-size hn. Multistep methods
differ in that they use information at previous points to obtain more accuracy. Multistep
methods can be very effective, as they usually require less function evaluations
than one-step methods of equal accuracy: a number of past values can be kept in
storage as the computation proceeds. Furthermore, an estimate of the discretisation
error can often be trivially obtained [17].
Properly programmed multistep methods can efficiently provide outputs at arbitrary
points without changing the value of the step-size. The order of the method
can be selected automatically and changed dynamically. Some multistep methods
can handle ‘stiff’ equations (see section 6.1.4), and equations can be classified as ‘stiff’
or ‘nonstiff’ automatically.
Linear multistep methods can be considered as special cases of the formula:
xn+1 =
kå
i=1
ai xn+1−i + hn
kå
i=0
bi fn+1−i (6.17)
where fi = f (xi, ti) = x˙i, k is an integer, and either ak or bk is not zero. This formula
defines the general k-step method. It is linear because every fi appears linearly
in the equation; f itself does not necessarily have to be a linear function in its arguments.
Because of the requirements for past values, the multistep methods are not
self-starting; a different method must be used to calculate the starting values of x. After
equation (6.17) has been started, each step involves the calculation of xn+1 from
the known values xn, xn−1, . . . , xn−k+1, and fn, fn−1, . . . , fn−k+1.
If b0 is nonzero, the algorithm is implicit because in that case the solution xn+1
is needed to evaluate fn+1 = x˙n+1 on the right-hand side of equation (6.17). The
implicit equation must be solved at each time-step. If b0 = 0, the algorithm is explicit,
and the calculation will be straightforward.
It is customary to use a combination of two multistep methods for computing
each step of the solution: an explicit method, called predictor, followed by one or
more applications of an implicit method, which is called a corrector. The SIMULINK
solver ode113, which is based on an Adams method, is an example of a predictorcorrector
multistep method. Some important Adams solvers found in literature are
the explicit Adams-Bashforth integration method and the implicit Adams-Moulton
method.
6.1. Numerical integration methods 81
i : 1 2 3 4 5 6
b1i 1
2b2i 3 −1
12b3i 23 −16 5
24b4i 55 −59 37 −9
720b5i 1901 −2774 2616 −1274 251
1440b6i 4277 −7923 9982 −7298 2877 −475
Table 6.2: Coefficients for the Adams-Bashforth integration method
i : 1 2 3 4 5 6
b1i
 1
2b2i
 1 1
12b3i
 5 8 −1
24b4i
 9 19 −5 1
720b5i
 251 646 −264 106 −19
1440b6i
 475 1427 −798 482 −173 27
Table 6.3: Coefficients for the Adams-Moulton integration method
The k-step Adams-Bashforth formula can be written as:
xn+1 = xn + hn
kå
i=1
bki fn+1−i (6.18)
Table 6.2 lists some values bki for this method. The k-step Adams-Moulton formula
equals:
xn+1 = xn−1 + hn
k−1
å
i=0
bki
 fn+1−i (6.19)
Table 6.3 lists some values bki
 for this method. Often Adams-Bashforth is used as
predictor and Adams-Moulton is used as corrector.
4. Extrapolation methods
The predictor methods can be regarded as extrapolation methods, as they extrapolate
the value xn+1 from known previous values of x and the function evaluations f .
There exist other extrapolation methods, based on polynomial interpolation or rational
function interpolation formulas, providing alternative ODE solvers. We will not
discuss such methods will in this report, as none of the SIMULINK integrators have
　
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