NAG CL Interface
d02txc (bvp_coll_nlin_contin)
1
Purpose
d02txc allows a solution to a nonlinear twopoint boundary value problem computed by
d02tlc to be used as an initial approximation in the solution of a related nonlinear twopoint boundary value problem in a continuation call to
d02tlc.
2
Specification
The function may be called by the names: d02txc or nag_ode_bvp_coll_nlin_contin.
3
Description
d02txc and its associated functions (
d02tlc,
d02tvc,
d02tyc and
d02tzc) solve the twopoint boundary value problem for a nonlinear system of ordinary differential equations
over an interval
$\left[a,b\right]$ subject to
$p$ (
$\text{}>0$) nonlinear boundary conditions at
$a$ and
$q$ (
$\text{}>0$) nonlinear boundary conditions at
$b$, where
$p+q={\displaystyle \sum _{i=1}^{n}}{m}_{i}$. Note that
${y}_{i}^{\left(m\right)}\left(x\right)$ is the
$m$th derivative of the
$i$th solution component. Hence
${y}_{i}^{\left(0\right)}\left(x\right)={y}_{i}\left(x\right)$. The left boundary conditions at
$a$ are defined as
and the right boundary conditions at
$b$ as
where
$y=\left({y}_{1},{y}_{2},\dots ,{y}_{n}\right)$ and
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
$\left[a,b\right]$ 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
d02txc should be used between calls to
d02tlc. This avoids the overhead of a complete initialization when the setup function
d02tvc is used.
d02txc allows the solution values computed in the previous call to
d02tlc to be used as an initial approximation for the solution in the next call to
d02tlc.
You must specify the new initial mesh. The previous mesh can be obtained by a call to
d02tzc. 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 argument
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 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
Keller H B (1992) Numerical Methods for Twopoint Boundaryvalue Problems Dover, New York
5
Arguments

1:
$\mathbf{mxmesh}$ – 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:
$\mathbf{nmesh}$ – Integer
Input

On entry: the number of points to be used in the new initial mesh. It is strongly recommended that if this function is called that the suggested value (see below) for
nmesh is used. In this case the arrays
mesh and
ipmesh returned by
d02tzc can be passed to this function without any modification.
Suggested value:
$\left({n}^{*}+1\right)/2$, where
${n}^{*}$ is the number of mesh points used in the previous mesh as returned in the argument
nmesh of
d02tzc.
Constraint:
$6\le {\mathbf{nmesh}}\le \left({\mathbf{mxmesh}}+1\right)/2$.

3:
$\mathbf{mesh}\left[{\mathbf{mxmesh}}\right]$ – const double
Input

On entry: the
nmesh points to be used in the new initial mesh as specified by
ipmesh.
Suggested value:
the argument
mesh returned from a call to
d02tzc.
Constraint:
${\mathbf{mesh}}\left[{i}_{\mathit{j}}1\right]<{\mathbf{mesh}}\left[{i}_{\mathit{j}+1}1\right]$, for
$\mathit{j}=1,2,\dots ,{\mathbf{nmesh}}1$, the values of
${i}_{1},{i}_{2},\dots ,{i}_{{\mathbf{nmesh}}}$ are defined in
ipmesh.
${\mathbf{mesh}}\left[{i}_{1}1\right]$ must contain the left boundary point,
$a$, and
${\mathbf{mesh}}\left[{i}_{{\mathbf{nmesh}}}1\right]$ must contain the right boundary point,
$b$, as specified in the previous call to
d02tvc.

4:
$\mathbf{ipmesh}\left[{\mathbf{mxmesh}}\right]$ – const Integer
Input

On entry: specifies the points in
mesh to be used as the new initial mesh. Let
$\left\{{i}_{j}:j=1,2,\dots ,{\mathbf{nmesh}}\right\}$ be the set of array indices of
ipmesh such that
${\mathbf{ipmesh}}\left[{i}_{j}1\right]=1\text{or}2$ and
$1={i}_{1}<{i}_{2}<\cdots <{i}_{{\mathbf{nmesh}}}$. Then
${\mathbf{mesh}}\left[{i}_{j}1\right]$ will be included in the new initial mesh.
If ${\mathbf{ipmesh}}\left[{i}_{j}1\right]=1$, ${\mathbf{mesh}}\left[{i}_{j}1\right]$ will be a fixed point in the new initial mesh.
If ${\mathbf{ipmesh}}\left[k1\right]=3$ for any $k$, ${\mathbf{mesh}}\left[k1\right]$ will not be included in the new mesh.
Suggested value:
the argument
ipmesh returned in a call to
d02tzc.
Constraints:
 ${\mathbf{ipmesh}}\left[\mathit{k}1\right]=1$, $2$ or $3$, for $\mathit{k}=1,2,\dots ,{i}_{{\mathbf{nmesh}}}$;
 ${\mathbf{ipmesh}}\left[0\right]={\mathbf{ipmesh}}\left[{i}_{{\mathbf{nmesh}}}1\right]=1$.

5:
$\mathbf{rcomm}\left[\mathit{dim}\right]$ – double
Communication Array

Note: the dimension,
$\mathit{dim}$, 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.

6:
$\mathbf{icomm}\left[\mathit{dim}\right]$ – Integer
Communication Array

Note: the dimension,
$\mathit{dim}$, 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.

7:
$\mathbf{fail}$ – 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 $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_CONVERGENCE_SOL

The solver function did not produce any results suitable for remeshing.
 NE_INT

An element of
ipmesh was set to
$1$ before
nmesh elements containing
$1$ or
$2$ were detected.
${\mathbf{ipmesh}}\left[i\right]\ne 1$, $1$, $2$ or $3$ for some $i$.
On entry, ${\mathbf{ipmesh}}\left[0\right]=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ipmesh}}\left[0\right]=1$.
On entry, ${\mathbf{nmesh}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nmesh}}\ge 6$.
You have set the element of
ipmesh corresponding to the last element of
mesh to be included in the new mesh as
$\u2329\mathit{\text{value}}\u232a$, which is not
$1$.
 NE_INT_2

On entry, ${\mathbf{nmesh}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{mxmesh}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nmesh}}\le \left({\mathbf{mxmesh}}+1\right)/2$.
 NE_INT_CHANGED

On entry,
${\mathbf{mxmesh}}=\u2329\mathit{\text{value}}\u232a$ and
${\mathbf{mxmesh}}=\u2329\mathit{\text{value}}\u232a$ in
d02tvc.
Constraint:
${\mathbf{mxmesh}}={\mathbf{mxmesh}}$ 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_MESH_ERROR

The first element of array
mesh does not coincide with the lefthand end of the range previously specified.
First element of
mesh:
$\u2329\mathit{\text{value}}\u232a$; lefthand of the range:
$\u2329\mathit{\text{value}}\u232a$.
The last point of the new mesh does not coincide with the right hand end of the range previously specified.
Last point of the new mesh: $\u2329\mathit{\text{value}}\u232a$; righthand end of the range: $\u2329\mathit{\text{value}}\u232a$.
 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.
 NE_NOT_STRICTLY_INCREASING

The entries in
mesh are not strictly increasing.
7
Accuracy
Not applicable.
8
Parallelism and Performance
d02txc is not threaded in any implementation.
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 argument 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.
10
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
d02tlc,
d02tvc,
d02tyc and
d02tzc, 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
$\left(r,\theta ,z\right)$, in which the
$\theta $ component of the corresponding velocity field behaves like
${r}^{n}$, the governing equations are
with boundary conditions
where
$s$ is the magnetic field strength, and
$\gamma $ is the Rossby number.
Some solutions of interest are for
$\gamma =1$, small
$n$ and
$s\to 0$. An added complication is the infinite range, which we approximate by
$\left[0,L\right]$. We choose
$n=0.2$ and first solve for
$L=60.0,s=0.24$ using the initial approximations
$f\left(x\right)={x}^{2}{e}^{x}$ and
$g\left(x\right)=1.0{e}^{x}$, 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
d02txc prior to a continuation call, that is only every second point of the preceding mesh is used.
The equations are first mapped onto
$\left[0,1\right]$ to yield
10.1
Program Text
10.2
Program Data
10.3
Program Results