void	d02tzc (Integer mxmesh, Integer nmesh, double mesh[], Integer ipmesh[], double ermx, Integer iermx, Integer ijermx, const double rcomm[], const Integer icomm[], NagError *fail)

The function may be called by the names: d02tzc or nag_ode_bvp_coll_nlin_diag.

3 Description

d02tzc and its associated functions (d02tlc, d02tvc, d02txc and d02tyc) 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, d02tvc must be called to specify the initial mesh, error requirements and other details. Then, d02tlc can be used to solve the boundary value problem. After successful computation, d02tzc can be used to ascertain details about the final mesh. d02tyc can be used to compute the approximate solution anywhere on the interval

[a, b]

using interpolation.

The functions 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

Cole J D (1968) Perturbation Methods in Applied Mathematics Blaisdell, Waltham, Mass.

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

5 Arguments

1: $mxmesh$ – Integer Input

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

Constraint: this must be identical to the value supplied for the argument mxmesh in the prior call to d02tvc.

2: $nmesh$ – Integer * Output

On exit: the number of points in the mesh last used by d02tlc.

3: $mesh [mxmesh]$ – double Output

On exit:

mesh [i - 1]

contains the

i

th point of the mesh last used by d02tlc, for

i = 1, 2, \dots, nmesh

mesh [0]

will contain

a

and

mesh [nmesh - 1]

will contain

b

. The remaining elements of mesh are not initialized.

4: $ipmesh [mxmesh]$ – Integer Output

On exit:

ipmesh [i - 1]

specifies the nature of the point

mesh [i - 1]

, for

i = 1, 2, \dots, nmesh

, in the final mesh computed by d02tlc.

$ipmesh [i - 1] = 1$: Indicates that the $i$ th point is a fixed point and was used by the solver before an extrapolation-like error test.
$ipmesh [i - 1] = 2$: Indicates that the $i$ th point was used by the solver before an extrapolation-like error test.
$ipmesh [i - 1] = 3$: Indicates that the $i$ th point was used by the solver only as part of an extrapolation-like error test.

The remaining elements of ipmesh are initialized to

−1

See Section 9 for advice on how these values may be used in conjunction with a continuation process.

5: $ermx$ – double * Output

On exit: an estimate of the maximum error in the solution computed by d02tlc, that is

ermx = \max (\frac{‖ y_{i} - v_{i} ‖}{(1.0 + ‖ v_{i} ‖)})

where

v_{i}

is the approximate solution for the

i

th solution component. If d02tlc returned successfully with

fail . code =

NE_NOERROR, ermx will be less than

tols [ijermx - 1]

in d02tvc where tols contains the error requirements as specified in Sections 3 and 5 in d02tvc.

If d02tlc returned with

fail . code =

NW_MAX_SUBINT, ermx will be greater than

tols [ijermx - 1]

in d02tvc.

If d02tlc returned any other value for fail.code then an error estimate is not available and ermx is initialized to

0.0

6: $iermx$ – Integer * Output

On exit: indicates the mesh sub-interval where the value of ermx has been computed, that is

[mesh [iermx - 1], mesh [iermx]]

If an estimate of the error is not available then iermx is initialized to

0

7: $ijermx$ – Integer * Output

On exit: indicates the component

i

(

= ijermx

) of the solution for which ermx has been computed, that is the approximation of

y_{i}

[mesh [iermx - 1], mesh [iermx]]

is estimated to have the largest error of all components

y_{i}

over mesh sub-intervals defined by mesh.

If an estimate of the error is not available then ijermx is initialized to

0

8: $rcomm [\dim]$ – const double Communication Array

Note: the dimension,

\dim

, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument rcomm in the previous call to d02tlc.

On entry: this must be the same array as supplied to d02tlc and must remain unchanged between calls.

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

9: $icomm [\dim]$ – const Integer Communication Array

Note: the dimension,

\dim

, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument icomm in the previous call to d02tlc.

On entry: this must be the same array as supplied to d02tlc and must remain unchanged between calls.

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

10: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_CONVERGENCE_SOL: The solver function did not produce any results suitable for interpolation.
NE_INT_CHANGED: On entry, $mxmesh = ⟨ value ⟩$ and $mxmesh = ⟨ value ⟩$ in d02tvc.
Constraint: $mxmesh = mxmesh$ in d02tvc.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MISSING_CALL: The solver function does not appear to have been called.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_NOT_CONVERGED: The solver function did not converge to a suitable solution.
A converged intermediate solution has been used.
Error estimate information is not available.
NW_TOO_MUCH_ACC_REQUESTED: The solver function did not satisfy the error requirements.
Information has been supplied on the last mesh used.

7 Accuracy

Not applicable.

8 Parallelism and Performance

d02tzc is not threaded in any implementation.

9 Further Comments

Note that:

if d02tlc returned $fail . code =$ NE_NOERROR, NW_MAX_SUBINT or NW_NOT_CONVERGED then it will always be the case that $ipmesh [0] = ipmesh [nmesh - 1] = 1$ ;
if d02tlc returned $fail . code =$ NE_NOERROR or NW_MAX_SUBINT then it will always be the case that $ipmesh [i - 1] = 3$ , for $i = 2, 4, \dots, nmesh - 1$ (even $i$ ) and $ipmesh [i - 1] = 1$ or $2$ , for $i = 3, 5, \dots, nmesh - 2$ (odd $i$ );
if d02tlc returned $fail . code =$ NW_NOT_CONVERGED then it will always be the case that $ipmesh [i - 1] = 1$ or $2$ , for $i = 2, 3, \dots, nmesh - 1$ .

If d02tzc returns

fail . code =

NE_NOERROR, then examination of the mesh may provide assistance in determining a suitable starting mesh for d02tvc in any subsequent attempts to solve similar problems.

If the problem being treated by d02tlc is one of a series of related problems (for example, as part of a continuation process), then the values of ipmesh and mesh may be suitable as input arguments to d02txc. Using the mesh points not involved in the extrapolation error test is usually appropriate. ipmesh and mesh should be passed unchanged to d02txc but nmesh should be replaced by

(nmesh + 1) / 2

If d02tzc returns

fail . code =

NW_NOT_CONVERGED or NW_TOO_MUCH_ACC_REQUESTED, nothing can be said regarding the quality of the mesh returned. However, it may be a useful starting mesh for d02tvc in any subsequent attempts to solve the same problem.

If d02tlc returns

fail . code =

NW_MAX_SUBINT, this corresponds to the solver requiring more than mxmesh mesh points to satisfy the error requirements. If mxmesh can be increased and the preceding call to d02tlc was not part, or was the first part, of a continuation process then the values in mesh may provide a suitable mesh with which to initialize a subsequent attempt to solve the same problem. If it is not possible to provide more mesh points then relaxing the error requirements by setting

tols [ijermx - 1]

to ermx might lead to a successful solution. It may be necessary to reset the other components of tols. Note that resetting the tolerances can lead to a different sequence of meshes being computed and hence to a different solution being computed.

10 Example

The following example is used to illustrate the use of fixed mesh points, simple continuation and numerical approximation of a Jacobian. See also d02tlc, d02tvc, d02txc and d02tyc, for the illustration of other facilities.

Consider the Lagerstrom–Cole equation

y^{''} = (y - y y^{'}) / ε

with the boundary conditions

y (0) = α y (1) = β,

(1)

where

ε

is small and positive. The nature of the solution depends markedly on the values of

α, β

. See Cole (1968).

We choose

α = - \frac{1}{3}, β = \frac{1}{3}

for which the solution is known to have corner layers at

x = \frac{1}{3}, \frac{2}{3}

. We choose an initial mesh of seven points

[0.0, 0.15, 0.3, 0.5, 0.7, 0.85, 1.0]

and ensure that the points

x = 0.3, 0.7

near the corner layers are fixed, that is the corresponding elements of the array ipmesh are set to

1

. First we compute the solution for

ε = 1.0e−4

using in guess the initial approximation

y (x) = α + (β - α) x

which satisfies the boundary conditions. Then we use simple continuation to compute the solution for

ε = 1.0e−5

. We use the suggested values for nmesh, ipmesh and mesh in the call to d02txc prior to the continuation call, that is only every second point of the preceding mesh is used and the fixed mesh points are retained.

Although the analytic Jacobian for this system is easy to evaluate, for illustration the procedure fjac uses central differences and calls to ffun to compute a numerical approximation to the Jacobian.

d02tz: FL CL CPP AD

NAG CL Interfaced02tzc (bvp_​coll_​nlin_​diag)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
d02tzc (bvp_coll_nlin_diag)