D02LAF is a routine for integrating a non-stiff system of second-order ordinary differential equations using Runge–Kutta–Nystrom techniques.
Given the initial values
x,y1,y2,…,yNEQ,y1′,y2′,…,yNEQ′ D02LAF integrates a non-stiff system of second-order differential equations of the type
from
x=T to
x=TEND using a Runge–Kutta–Nystrom formula pair. The system is defined by
FCN, which evaluates
fi in terms of
x and
y1,y2,…,yNEQ, where
y1,y2,…,yNEQ are supplied at
x.
There are two Runge–Kutta–Nystrom formula pairs implemented in this routine. The lower order method is intended if you have moderate accuracy requirements and may be used in conjunction with the interpolation routine
D02LZF to produce solutions and derivatives at user-specified points. The higher order method is intended if you have high accuracy requirements.
In one-step mode the routine returns approximations to the solution, derivative and
fi at each integration point. In interval mode these values are returned at the end of the integration range. You select the order of the method, the mode of operation, the error control and various optional inputs by a prior call to
D02LXF.
For a description of the Runge–Kutta–Nystrom formula pairs see
Dormand et al. (1986a) and
Dormand et al. (1986b) and for a description of their practical implementation see
Brankin et al. (1989).
Brankin R W, Dormand J R, Gladwell I, Prince P J and Seward W L (1989) Algorithm 670: A Runge–Kutta–Nystrom Code
ACM Trans. Math. Software 15 31–40
Dormand J R, El–Mikkawy M E A and Prince P J (1986a) Families of Runge–Kutta–Nystrom formulae
Mathematical Report TPMR 86-1 Teesside Polytechnic
Dormand J R, El–Mikkawy M E A and Prince P J (1986b) High order embedded Runge–Kutta–Nystrom formulae
Mathematical Report TPMR 86-2 Teesside Polytechnic
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
The accuracy of integration is determined by the parameters
TOL,
THRES and
THRESP in a prior call to
D02LXF. Note that only the local error at each step is controlled by these parameters. The error estimates obtained are not strict bounds but are usually reliable over one step. Over a number of steps the overall error may accumulate in various ways, depending on the system. The code is designed so that a reduction in
TOL should lead to an approximately proportional reduction in the error. You are strongly recommended to call D02LAF with more than one value for
TOL and to compare the results obtained to estimate their accuracy.
The accuracy obtained depends on the type of error test used. If the solution oscillates around zero a relative error test should be avoided, whereas if the solution is exponentially increasing an absolute error test should not be used. For a description of the error test see the specifications of the parameters
TOL,
THRES and
THRESP in routine document
D02LXF.
If D02LAF fails with
IFAIL=3 then the value of
TOL may be so small that a solution cannot be obtained, in which case the routine should be called again with a larger value for
TOL. If the accuracy requested is really needed then you should consider whether there is a more fundamental difficulty. For example:
| (a) |
in the region of a singularity the solution components will usually be of a large magnitude. D02LAF could be used in one-step mode to monitor the size of the solution with the aim of trapping the solution before the singularity. In any case numerical integration cannot be continued through a singularity, and analytical treatment may be necessary; |
| (b) |
if the solution contains fast oscillatory components, the routine will require a very small step size to preserve stability. This will usually be exhibited by excessive computing time and sometimes an error exit with IFAIL=3. The Runge–Kutta–Nystrom methods are not efficient in such cases and you should consider reposing your problem as a system of first-order ordinary differential equations and then using a routine from sub-chapter D02M–N with the Blend formulae (see D02NWF). |
D02LAF can be used for producing results at short intervals (for example, for tabulation), in two ways. By far the less efficient is to call D02LAF successively over short intervals,
t+i-1×h to
t+i×h, although this is the only way if the higher order method has been selected and precisely
not what it is intended for. A more efficient way,
only for use when the lower order method has been selected, is to use D02LAF in one-step mode. The output values of parameters
Y,
YP,
YDP,
T and
RWORK are set correctly for a call to
D02LZF to compute the solution and derivative at the required points.
This example solves the following system (the two body problem)
over the range
0,20 with initial conditions
y1=1.0-ε,
y2=0.0,
y1′=0.0 and
y2′=
1+ε
1-ε
where
ε, the eccentricity, is
0.5. The system is solved using the lower order method with relative local error tolerances
1.0D−4 and
1.0D−5 and default threshold tolerances. D02LAF is used in one-step mode (
ONESTP=.TRUE.) and
D02LZF provides solution values at intervals of
2.0.
None.
