D02PWF : NAG Library, Mark 24

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

D02PWF resets the end point in an integration performed by D02PDF.

D02PWF and its associated routines (D02PDF, D02PVF, D02PXF, D02PYF and D02PZF) solve the initial value problem for a first-order system of ordinary differential equations. The routines, based on Runge–Kutta methods and derived from RKSUITE (see Brankin et al. (1991)), integrate

y^{'} = f (t, y) given y (t_{0}) = y_{0}

where

y

is the vector of

n

solution components and

t

is the independent variable.

D02PWF is used to reset the final value of the independent variable,

t_{f}

, when the integration is already underway. It can be used to extend or reduce the range of integration. The new value must be beyond the current value of the independent variable (as returned in TNOW by D02PDF) in the current direction of integration. It is much more efficient to use D02PWF for this purpose than to use D02PVF which involves the overhead of a complete restart of the integration.

If you want to change the direction of integration then you must restart by a call to D02PVF.

Brankin R W, Gladwell I and Shampine L F (1991) RKSUITE: A suite of Runge–Kutta codes for the initial value problems for ODEs SoftReport 91-S1 Southern Methodist University

On entry: the new value for

t_{f}

Constraint:

sign (TENDNU - TNOW) = sign (TEND - TSTART)

, where TSTART and TEND are as supplied in the previous call to D02PVF and TNOW is returned by the preceding call to D02PDF. TENDNU must be distinguishable from TNOW for the method and the machine precision being used.

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

On entry, an invalid input value for TENDNU was detected or an invalid call to D02PWF was made, for example without a previous call to the integration routine D02PDF. You cannot continue integrating the problem.

This example integrates a two body problem. The equations for the coordinates

(x (t), y (t))

of one body as functions of time

t

in a suitable frame of reference are

x^{''} = - \frac{x}{r^{3}}

y^{''} = - \frac{y}{r^{3}}, r = \sqrt{x^{2} + y^{2}} .

The initial conditions

\begin{array}{l} x (0) = 1 - ε, & x^{'} (0) = 0 \\ y (0) = 0, & y^{'} (0) = \sqrt{\frac{1 + ε}{1 - ε}} \end{array}

lead to elliptic motion with

0 < ε < 1

ε = 0.7

is selected and reposed as

\begin{array}{l} y_{1}^{'} = y_{3} \\ y_{2}^{'} = y_{4} \\ y_{3}^{'} = - \frac{y_{1}}{r^{3}} \\ y_{4}^{'} = - \frac{y_{2}}{r^{3}} \end{array}

over the range

[0, 6 π]

. Relative error control is used with threshold values of

1.0E−10

for each solution component and compute the solution at intervals of length

π

across the range using D02PWF to reset the end of the integration range. A high-order Runge–Kutta method (

METHOD = 3

) is also used with tolerances

TOL = 1.0E−4

and

TOL = 1.0E−5

in turn so that the solutions may be compared. The value of

π

is obtained by using X01AAF.

Note that the length of

TOL = 1.0E−4

and WORK is large enough for any valid combination of input arguments to D02PVF.

NAG Library Routine Document

D02PWF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentD02PWF