NAG Library Routine Document

D02TXF

+− 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

1 Purpose

D02TXF allows a solution to a nonlinear two-point boundary value problem computed by D02TKF to be used as an initial approximation in the solution of a related nonlinear two-point boundary value problem in a continuation call to D02TKF.

2 Specification

SUBROUTINE D02TXF (

MXMESH, NMESH, MESH, IPMESH, RWORK, IWORK, IFAIL)

INTEGER	MXMESH, NMESH, IPMESH(MXMESH), IWORK(*), IFAIL
REAL (KIND=nag_wp)	MESH(MXMESH), RWORK(*)

3 Description

D02TXF and its associated routines (D02TKF, D02TVF, D02TYF and D02TZF) solve the two-point boundary value problem for a nonlinear system of ordinary differential equations

\begin{array}{rcl} y_{1}^{(m_{1})} (x) & = & f_{1} (x, y_{1}, y_{1}^{(1)}, \dots, y_{1}^{(m_{1} - 1)}, y_{2}, \dots, y_{n}^{(m_{n} - 1)}) \\ y_{2}^{(m_{2})} (x) & = & f_{2} (x, y_{1}, y_{1}^{(1)}, \dots, y_{1}^{(m_{1} - 1)}, y_{2}, \dots, y_{n}^{(m_{n} - 1)}) \\ ⋮ \\ y_{n}^{(m_{n})} (x) & = & f_{n} (x, y_{1}, y_{1}^{(1)}, \dots, y_{1}^{(m_{1} - 1)}, y_{2}, \dots, y_{n}^{(m_{n} - 1)}) \end{array}

over an interval

[a, b]

subject to

p

(

> 0

) nonlinear boundary conditions at

a

and

q

(

> 0

) nonlinear boundary conditions at

b

, where

p + q = \sum_{i = 1}^{n} m_{i}

. Note that

y_{i}^{(m)} (x)

is the

m

th derivative of the

i

th solution component. Hence

y_{i}^{(0)} (x) = y_{i} (x)

. The left boundary conditions at

a

are defined as

g_{i} (z (y (a))) = 0, i = 1, 2, \dots, p,

and the right boundary conditions at

b

{\bar{g}}_{j} (z (y (b))) = 0, j = 1, 2, \dots, q,

where

y = (y_{1}, y_{2}, \dots, y_{n})

and

z (y (x)) = (y_{1} (x), y_{1}^{(1)} (x), \dots, y_{1}^{(m_{1} - 1)} (x), y_{2} (x), \dots, y_{n}^{(m_{n} - 1)} (x)) .

First, D02TVF must be called to specify the initial mesh, error requirements and other details. Then, D02TKF can be used to solve the boundary value problem. After successful computation, D02TZF can be used to ascertain details about the final mesh. D02TYF can be used to compute the approximate solution anywhere on the interval

[a, b]

using interpolation.

If the boundary value problem being solved is one of a sequence of related problems, for example as part of some continuation process, then D02TXF should be used between calls to D02TKF. This avoids the overhead of a complete initialization when the setup routine D02TVF is used. D02TXF allows the solution values computed in the previous call to D02TKF to be used as an initial approximation for the solution in the next call to D02TKF.

You must specify the new initial mesh. The previous mesh can be obtained by a call to D02TZF. It may be used unchanged as the new mesh, in which case any fixed points in the previous mesh remain as fixed points in the new mesh. Fixed and other points may be added or subtracted from the mesh by manipulation of the contents of the array parameter IPMESH. Initial values for the solution components on the new mesh are computed by interpolation on the values for the solution components on the previous mesh.

The routines are based on modified versions of the codes COLSYS and COLNEW (see Ascher et al. (1979) and Ascher and Bader (1987)). A comprehensive treatment of the numerical solution of boundary value problems can be found in Ascher et al. (1988) and Keller (1992).

4 References

Ascher U M and Bader G (1987) A new basis implementation for a mixed order boundary value ODE solver SIAM J. Sci. Stat. Comput. 8 483–500

Ascher U M, Christiansen J and Russell R D (1979) A collocation solver for mixed order systems of boundary value problems Math. Comput. 33 659–679

Ascher U M, Mattheij R M M and Russell R D (1988) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations Prentice–Hall

Keller H B (1992) Numerical Methods for Two-point Boundary-value Problems Dover, New York

5 Parameters

1: MXMESH – INTEGERInput

On entry: the maximum number of points allowed in the mesh.

Constraint: this must be identical to the value supplied for the parameter MXMESH in the prior call to D02TVF.

2: NMESH – INTEGERInput

On entry: the number of points to be used in the new initial mesh.

Suggested value:

(n^{*} + 1) / 2

, where

n^{*}

is the number of mesh points used in the previous mesh as returned in the parameter NMESH of D02TZF.

Constraint:

6 \leq NMESH \leq (MXMESH + 1) / 2

3: MESH(MXMESH) – REAL (KIND=nag_wp) arrayInput

On entry: the NMESH points to be used in the new initial mesh as specified by IPMESH.

Suggested value: the parameter MESH returned from a call to D02TZF.

Constraint:

MESH (i_{j}) < MESH (i_{j + 1})

, for

j = 1, 2, \dots, NMESH - 1

, the values of

i_{1}, i_{2}, \dots, i_{NMESH}

are defined in IPMESH.

MESH (i_{1})

must contain the left boundary point,

a

, and

MESH (i_{NMESH})

must contain the right boundary point,

b

, as specified in the previous call to D02TVF.

4: IPMESH(MXMESH) – INTEGER arrayInput

On entry: specifies the points in MESH to be used as the new initial mesh. Let

\{i_{j} : j = 1, 2, \dots, NMESH\}

be the set of array indices of IPMESH such that

IPMESH (i_{j}) = 1 ​ or ​ 2

and

1 = i_{1} < i_{2} < \dots < i_{NMESH}

. Then

MESH (i_{j})

will be included in the new initial mesh.

IPMESH (i_{j}) = 1

MESH (i_{j})

will be a fixed point in the new initial mesh.

IPMESH (k) = 3

for any

k

, then

MESH (k)

will not be included in the new mesh.

Suggested value: the parameter IPMESH returned in a call to D02TZF.

Constraints:

$IPMESH (k) = 1$ , $2$ or $3$ , for $k = 1, 2, \dots, i_{NMESH}$ ;
$IPMESH (1) = IPMESH (i_{NMESH}) = 1$ .

5: RWORK( $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array RWORK must be at least

LRWORK

(see D02TVF).

On entry: this must be the same array as supplied to D02TKF and must remain unchanged between calls.

On exit: contains information about the solution for use on subsequent calls to associated routines.

6: IWORK( $*$ ) – INTEGER arrayInput/Output

Note: the dimension of the array IWORK must be at least

LIWORK

(see D02TVF).

On entry: this must be the same array as supplied to D02TKF and must remain unchanged between calls.

On exit: contains information about the solution for use on subsequent calls to associated routines.

7: IFAIL – INTEGERInput/Output

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).

6 Error Indicators and Warnings

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:

$IFAIL = 1$: An invalid call to D02TXF was made, for example without a previous successful call to the solver routine D02TKF, or, on entry, an invalid value for NMESH, MESH or IPMESH was detected.

7 Accuracy

Not applicable.

8 Further Comments

For problems where sharp changes of behaviour are expected over short intervals it may be advisable to:

–	cluster the mesh points where sharp changes in behaviour are expected;
–	maintain fixed points in the mesh using the parameter IPMESH to ensure that the remeshing process does not inadvertently remove mesh points from areas of known interest.

In the absence of any other information about the expected behaviour of the solution, using the values suggested in Section 5 for NMESH, IPMESH and MESH is strongly recommended.

9 Example

This example illustrates the use of continuation, solution on an infinite range, and solution of a system of two differential equations of orders

3

and

2

. See also D02TKF, D02TVF, D02TYF and D02TZF, for the illustration of other facilities.

Consider the problem of swirling flow over an infinite stationary disk with a magnetic field along the axis of rotation. See Ascher et al. (1988) and the references therein. After transforming from a cylindrical coordinate system

(r, θ, z)

, in which the

θ

component of the corresponding velocity field behaves like

r^{- n}

, the governing equations are

\begin{array}{lcl} f^{'''} + \frac{1}{2} (3 - n) f f^{''} + n {(f^{'})}^{2} + g^{2} - s f^{'} & = & γ^{2} \\ g^{''} + \frac{1}{2} (3 - n) f g^{'} + (n - 1) g f^{'} - s (g - 1) & = & 0 \end{array}

with boundary conditions

f (0) = f^{'} (0) = g (0) = 0, f^{'} (\infty) = 0, g (\infty) = γ,

where

s

is the magnetic field strength, and

γ

is the Rossby number.

Some solutions of interest are for

γ = 1

, small

n

and

s \to 0

. An added complication is the infinite range, which we approximate by

[0, L]

. We choose

n = 0.2

and first solve for

L = 60.0, s = 0.24

using the initial approximations

f (x) = - x^{2} e^{- x}

and

g (x) = 1.0 - e^{- x}

, which satisfy the boundary conditions, on a uniform mesh of

21

points. Simple continuation on the parameters

L

and

s

using the values

L = 120.0, s = 0.144

and then

L = 240.0, s = 0.0864

is used to compute further solutions. We use the suggested values for NMESH, IPMESH and MESH in the call to D02TXF prior to a continuation call, that is only every second point of the preceding mesh is used.

The equations are first mapped onto

[0, 1]

to yield

\begin{array}{lcl} f^{'''} & = & L^{3} (γ^{2} - g^{2}) + L^{2} s g^{'} - L (\frac{1}{2} (3 - n) f f^{''} + n {(g^{'})}^{2}) \\ g^{''} & = & L^{2} s (g - 1) - L (\frac{1}{2} (3 - n) f g^{'} + (n - 1) f^{'} g) . \end{array}

NAG Library Routine DocumentD02TXF