NAG Library Routine Document

D02TKF

D02TKF and its associated routines (D02TVF, D02TXF, D02TYF and D02TZF) solve the two-point boundary value problem for a nonlinear mixed order 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. Note that the error requirements apply only to the solution components

y_{1}, y_{2}, \dots, y_{n}

and that no error control is applied to derivatives of solution components. (If error control is required on derivatives then the system must be reduced in order by introducing the derivatives whose error is to be controlled as new variables. See Section 8 in D02TVF.) 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 and other details of the solution procedure, and D02TYF can be used to compute the approximate solution anywhere on the interval

[a, b]

A description of the numerical technique used in D02TKF is given in Section 3 in D02TVF.

D02TKF can also be used in the solution of a series of problems, for example in performing continuation, when the mesh used to compute the solution of one problem is to be used as the initial mesh for the solution of the next related problem. D02TXF should be used in between calls to D02TKF in this context.

See Section 8 in D02TVF for details of how to solve boundary value problems of a more general nature.

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: FFUN – SUBROUTINE, supplied by the user.External Procedure

FFUN must evaluate the functions

f_{i}

for given values

x, z (y (x))

The specification of FFUN is:

SUBROUTINE FFUN (

X, Y, NEQ, M, F)

INTEGER	NEQ, M(NEQ)
REAL (KIND=nag_wp)	X, Y(NEQ,0:*), F(NEQ)

1: X – REAL (KIND=nag_wp)Input: On entry: $x$ , the independent variable.
2: Y(NEQ, $0 : *$ ) – REAL (KIND=nag_wp) arrayInput: On entry: $Y (i, j)$ contains $y_{i}^{(j)} (x)$ , for $i = 1, 2, \dots, NEQ$ and $j = 0, 1, \dots, M (i) - 1$ .
Note: $y_{i}^{(0)} (x) = y_{i} (x)$ .
3: NEQ – INTEGERInput: On entry: the number of differential equations.
4: M(NEQ) – INTEGER arrayInput: On entry: $m_{i}$ , the order of the $i$ th differential equation, for $i = 1, 2, \dots, NEQ$ .
5: F(NEQ) – REAL (KIND=nag_wp) arrayOutput: On exit: the values of $f_{i}$ , for $i = 1, 2, \dots, NEQ$ .

FFUN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02TKF is called. Parameters denoted as Input must not be changed by this procedure.

2: FJAC – SUBROUTINE, supplied by the user.External Procedure

FJAC must evaluate the partial derivatives of

f_{i}

with respect to the elements of

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))

The specification of FJAC is:

SUBROUTINE FJAC (

X, Y, NEQ, M, DFDY)

INTEGER	NEQ, M(NEQ)
REAL (KIND=nag_wp)	X, Y(NEQ,0:), DFDY(NEQ,NEQ,0:)

1: X – REAL (KIND=nag_wp)Input: On entry: $x$ , the independent variable.
2: Y(NEQ, $0 : *$ ) – REAL (KIND=nag_wp) arrayInput: On entry: $Y (i, j)$ contains $y_{i}^{(j)} (x)$ , for $i = 1, 2, \dots, NEQ$ and $j = 0, 1, \dots, M (i) - 1$ .
Note: $y_{i}^{(0)} (x) = y_{i} (x)$ .
3: NEQ – INTEGERInput: On entry: the number of differential equations.
4: M(NEQ) – INTEGER arrayInput: On entry: $m_{i}$ , the order of the $i$ th differential equation, for $i = 1, 2, \dots, NEQ$ .
5: DFDY(NEQ,NEQ, $0 : *$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: $DFDY (i, j, k)$ must contain the partial derivative of $f_{i}$ with respect to $y_{j}^{(k)}$ , for $i = 1, 2, \dots, NEQ$ , $j = 1, 2, \dots, NEQ$ and $k = 0, 1, \dots, M (j) - 1$ . Only nonzero partial derivatives need be set.

FJAC must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02TKF is called. Parameters denoted as Input must not be changed by this procedure.

3: GAFUN – SUBROUTINE, supplied by the user.External Procedure

GAFUN must evaluate the boundary conditions at the left-hand end of the range, that is functions

g_{i} (z (y (a)))

for given values of

z (y (a))

The specification of GAFUN is:

SUBROUTINE GAFUN (

YA, NEQ, M, NLBC, GA)

INTEGER	NEQ, M(NEQ), NLBC
REAL (KIND=nag_wp)	YA(NEQ,0:*), GA(NLBC)

1: YA(NEQ, $0 : *$ ) – REAL (KIND=nag_wp) arrayInput: On entry: $YA (i, j)$ contains $y_{i}^{(j)} (a)$ , for $i = 1, 2, \dots, NEQ$ and $j = 0, 1, \dots, M (i) - 1$ .
Note: $y_{i}^{(0)} (a) = y_{i} (a)$ .
2: NEQ – INTEGERInput: On entry: the number of differential equations.
3: M(NEQ) – INTEGER arrayInput: On entry: $m_{i}$ , the order of the $i$ th differential equation, for $i = 1, 2, \dots, NEQ$ .
4: NLBC – INTEGERInput: On entry: the number of boundary conditions at $a$ .
5: GA(NLBC) – REAL (KIND=nag_wp) arrayOutput: On exit: the values of $g_{i} (z (y (a)))$ , for $i = 1, 2, \dots, NLBC$ .

GAFUN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02TKF is called. Parameters denoted as Input must not be changed by this procedure.

4: GBFUN – SUBROUTINE, supplied by the user.External Procedure

GBFUN must evaluate the boundary conditions at the right-hand end of the range, that is functions

{\bar{g}}_{i} (z (y (b)))

for given values of

z (y (b))

The specification of GBFUN is:

SUBROUTINE GBFUN (

YB, NEQ, M, NRBC, GB)

INTEGER	NEQ, M(NEQ), NRBC
REAL (KIND=nag_wp)	YB(NEQ,0:*), GB(NRBC)

1: YB(NEQ, $0 : *$ ) – REAL (KIND=nag_wp) arrayInput: On entry: $YB (i, j)$ contains $y_{i}^{(j)} (b)$ , for $i = 1, 2, \dots, NEQ$ and $j = 0, 1, \dots, M (i) - 1$ .
Note: $y_{i}^{(0)} (b) = y_{i} (b)$ .
2: NEQ – INTEGERInput: On entry: the number of differential equations.
3: M(NEQ) – INTEGER arrayInput: On entry: $m_{i}$ , the order of the $i$ th differential equation, for $i = 1, 2, \dots, NEQ$ .
4: NRBC – INTEGERInput: On entry: the number of boundary conditions at $b$ .
5: GB(NRBC) – REAL (KIND=nag_wp) arrayOutput: On exit: the values of ${\bar{g}}_{i} (z (y (b)))$ , for $i = 1, 2, \dots, NRBC$ .

GBFUN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02TKF is called. Parameters denoted as Input must not be changed by this procedure.

5: GAJAC – SUBROUTINE, supplied by the user.External Procedure

GAJAC must evaluate the partial derivatives of

g_{i} (z (y (a)))

with respect to the elements of

z (y (a))

(

= (y_{1} (a), y_{1}^{1} (a), \dots, y_{1}^{(m_{1} - 1)} (a), y_{2} (a), \dots, y_{n}^{(m_{n} - 1)} (a))

The specification of GAJAC is:

SUBROUTINE GAJAC (

YA, NEQ, M, NLBC, DGADY)

INTEGER	NEQ, M(NEQ), NLBC
REAL (KIND=nag_wp)	YA(NEQ,0:), DGADY(NLBC,NEQ,0:)

1: YA(NEQ, $0 : *$ ) – REAL (KIND=nag_wp) arrayInput: On entry: $YA (i, j)$ contains $y_{i}^{(j)} (a)$ , for $i = 1, 2, \dots, NEQ$ and $j = 0, 1, \dots, M (i) - 1$ .
Note: $y_{i}^{(0)} (a) = y_{i} (a)$ .
2: NEQ – INTEGERInput: On entry: the number of differential equations.
3: M(NEQ) – INTEGER arrayInput: On entry: $m_{i}$ , the order of the $i$ th differential equation, for $i = 1, 2, \dots, NEQ$ .
4: NLBC – INTEGERInput: On entry: the number of boundary conditions at $a$ .
5: DGADY(NLBC,NEQ, $0 : *$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: $DGADY (i, j, k)$ must contain the partial derivative of $g_{i} (z (y (a)))$ with respect to $y_{j}^{(k)} (a)$ , for $i = 1, 2, \dots, NLBC$ , $j = 1, 2, \dots, NEQ$ and $k = 0, 1, \dots, M (j) - 1$ . Only nonzero partial derivatives need be set.

GAJAC must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02TKF is called. Parameters denoted as Input must not be changed by this procedure.

6: GBJAC – SUBROUTINE, supplied by the user.External Procedure

GBJAC must evaluate the partial derivatives of

{\bar{g}}_{i} (z (y (b)))

with respect to the elements of

z (y (b))

(

= (y_{1} (b), y_{1}^{1} (b), \dots, y_{1}^{(m_{1} - 1)} (b), y_{2} (b), \dots, y_{n}^{(m_{n} - 1)} (b))

The specification of GBJAC is:

SUBROUTINE GBJAC (

YB, NEQ, M, NRBC, DGBDY)

INTEGER	NEQ, M(NEQ), NRBC
REAL (KIND=nag_wp)	YB(NEQ,0:), DGBDY(NRBC,NEQ,0:)

1: YB(NEQ, $0 : *$ ) – REAL (KIND=nag_wp) arrayInput: On entry: $YB (i, j)$ contains $y_{i}^{(j)} (b)$ , for $i = 1, 2, \dots, NEQ$ and $j = 0, 1, \dots, M (i) - 1$ .
Note: $y_{i}^{(0)} (b) = y_{i} (b)$ .
2: NEQ – INTEGERInput: On entry: the number of differential equations.
3: M(NEQ) – INTEGER arrayInput: On entry: $m_{i}$ , the order of the $i$ th differential equation, for $i = 1, 2, \dots, NEQ$ .
4: NRBC – INTEGERInput: On entry: the number of boundary conditions at $a$ .
5: DGBDY(NRBC,NEQ, $0 : *$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: $DGBDY (i, j, k)$ must contain the partial derivative of ${\bar{g}}_{i} (z (y (b)))$ with respect to $y_{j}^{(k)} (b)$ , for $i = 1, 2, \dots, NRBC$ , $j = 1, 2, \dots, NEQ$ and $k = 0, 1, \dots, M (j) - 1$ . Only nonzero partial derivatives need be set.

GBJAC must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02TKF is called. Parameters denoted as Input must not be changed by this procedure.

7: GUESS – SUBROUTINE, supplied by the user.External Procedure

GUESS must return initial approximations for the solution components

y_{i}^{(j)}

and the derivatives

y_{i}^{(m_{i})}

, for

i = 1, 2, \dots, NEQ

and

j = 0, 1, \dots, M (i) - 1

. Try to compute each derivative

y_{i}^{(m_{i})}

such that it corresponds to your approximations to

y_{i}^{(j)}

, for

j = 0, 1, \dots, M (i) - 1

. You should not call FFUN to compute

y_{i}^{(m_{i})}

If D02TKF is being used in conjunction with D02TXF as part of a continuation process, then GUESS is not called by D02TKF after the call to D02TXF.

The specification of GUESS is:

SUBROUTINE GUESS (

X, NEQ, M, Y, DYM)

INTEGER	NEQ, M(NEQ)
REAL (KIND=nag_wp)	X, Y(NEQ,0:*), DYM(NEQ)

1: X – REAL (KIND=nag_wp)Input: On entry: $x$ , the independent variable; $x \in [a, b]$ .
2: NEQ – INTEGERInput: On entry: the number of differential equations.
3: M(NEQ) – INTEGER arrayInput: On entry: $m_{i}$ , the order of the $i$ th differential equation, for $i = 1, 2, \dots, NEQ$ .
4: Y(NEQ, $0 : *$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: $Y (i, j)$ must contain $y_{i}^{(j)} (x)$ , for $i = 1, 2, \dots, NEQ$ and $j = 0, 1, \dots, M (i) - 1$ .
Note: $y_{i}^{(0)} (x) = y_{i} (x)$ .
5: DYM(NEQ) – REAL (KIND=nag_wp) arrayOutput: On exit: $DYM (i)$ must contain $y_{i}^{(m_{i})} (x)$ , for $i = 1, 2, \dots, NEQ$ .

GUESS must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02TKF is called. Parameters denoted as Input must not be changed by this procedure.

8: WORK( $*$ ) – REAL (KIND=nag_wp) arrayCommunication Array

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

LRWORK

(see D02TVF).

On entry: this must be the same array as supplied to D02TVF and must remain unchanged between calls.

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

9: IWORK( $*$ ) – INTEGER arrayCommunication Array

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 D02TVF and must remain unchanged between calls.

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

10: 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, because for this routine the values of the output parameters may be useful even if

IFAIL \neq 0

on exit, the recommended value is

- 1

. 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).

Note: D02TKF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, an invalid call was made to D02TKF, for example, without a previous call to the setup routine D02TVF.

$IFAIL = 2$: Numerical singularity has been detected in the Jacobian used in the underlying Newton iteration. No meaningful results have been computed. You should check carefully how you have coded FJAC, GAJAC and GBJAC. If the user-supplied routines have been coded correctly then supplying a different initial approximation to the solution in GUESS might be appropriate. See also Section 8.

$IFAIL = 3$: The nonlinear iteration has failed to converge. At no time during the computation was convergence obtained and no meaningful results have been computed. You should check carefully how you have coded procedures FJAC, GAJAC and GBJAC. If the procedures have been coded correctly then supplying a better initial approximation to the solution in GUESS might be appropriate. See also Section 8.

$IFAIL = 4$: The nonlinear iteration has failed to converge. At some earlier time during the computation convergence was obtained and the corresponding results have been returned for diagnostic purposes and may be inspected by a call to D02TZF. Nothing can be said regarding the suitability of these results for use in any subsequent computation for the same problem. You should try to provide a better mesh and initial approximation to the solution in GUESS. See also Section 8.

$IFAIL = 5$: The expected number of sub-intervals required exceeds the maximum number specified by the argument MXMESH in the setup routine D02TVF. Results for the last mesh on which convergence was obtained have been returned. Nothing can be said regarding the suitability of these results for use in any subsequent computation for the same problem. An indication of the error in the solution on the last mesh where convergence was obtained can be obtained by calling D02TZF. The error requirements may need to be relaxed and/or the maximum number of mesh points may need to be increased. See also Section 8.

7 Accuracy

The accuracy of the solution is determined by the parameter TOLS in the prior call to D02TVF (see Sections 3 and 8 in D02TVF for details and advice). Note that error control is applied only to solution components (variables) and not to any derivatives of the solution. An estimate of the maximum error in the computed solution is available by calling D02TZF.

8 Further Comments

If D02TKF returns with

IFAIL = 2

3

4

5

and the call to D02TKF was a part of some continuation procedure for which successful calls to D02TKF have already been made, then it is possible that the adjustment(s) to the continuation parameter(s) between calls to D02TKF is (are) too large for the problem under consideration. More conservative adjustment(s) to the continuation parameter(s) might be appropriate.

9 Example

The following example is used to illustrate the treatment of a high-order system, control of the error in a derivative of a component of the original system, and the use of continuation. See also D02TVF, D02TXF, D02TYF and D02TZF, for the illustration of other facilities.

Consider the steady flow of an incompressible viscous fluid between two infinite coaxial rotating discs. See Ascher et al. (1979) and the references therein. The governing equations are

\begin{array}{rcl} \frac{1}{\sqrt{R}} f^{'''} + f f^{'''} + g g^{'} & = & 0 \\ \frac{1}{\sqrt{R}} g^{''} + f g^{'} - f^{'} g & = & 0 \end{array}

subject to the boundary conditions

f (0) = f^{'} (0) = 0, g (0) = Ω_{0}, f (1) = f^{'} (1) = 0, g (1) = Ω_{1},

where

R

is the Reynolds number and

Ω_{0}, Ω_{1}

are the angular velocities of the disks.

We consider the case of counter-rotation and a symmetric solution, that is

Ω_{0} = 1, Ω_{1} = - 1

. This problem is more difficult to solve, the larger the value of

R

. For illustration, we use simple continuation to compute the solution for three different values of

R

(

= 10^{6}, 10^{8}, 10^{10}

). However, this problem can be addressed directly for the largest value of

R

considered here. Instead of the values suggested in Section 5 in D02TXF for NMESH, IPMESH and MESH in the call to D02TXF prior to a continuation call, we use every point of the final mesh for the solution of the first value of

R

, that is we must modify the contents of IPMESH. For illustrative purposes we wish to control the computed error in

f^{'}

and so recast the equations as

\begin{array}{rcl} y_{1}^{'} & = & y_{2} \\ y_{2}^{'''} & = & - \sqrt{R} (y_{1} y_{2}^{''} + y_{3} y_{3}^{'}) \\ y_{3}^{''} & = & \sqrt{R} (y_{2} y_{3} - y_{1} y_{3}^{'}) \end{array}

subject to the boundary conditions

y_{1} (0) = y_{2} (0) = 0, y_{3} (0) = Ω, y_{1} (1) = y_{2} (1) = 0, y_{3} (1) = - Ω, Ω = 1 .

For the symmetric boundary conditions considered, there exists an odd solution about

x = 0.5

. Hence, to satisfy the boundary conditions, we use the following initial approximations to the solution in GUESS:

\begin{array}{rcl} y_{1} (x) & = & - x^{2} (x - \frac{1}{2}) {(x - 1)}^{2} \\ y_{2} (x) & = & - x (x - 1) (5 x^{2} - 5 x + 1) \\ y_{3} (x) & = & - 8 Ω {(x - \frac{1}{2})}^{3} . \end{array}

NAG Library Routine DocumentD02TKF

+− 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 Document

D02TKF