NAG CL Interface
d03pdc (dim1_parab_coll)
1
Purpose
d03pdc integrates a system of linear or nonlinear parabolic partial differential equations (PDEs) in one space variable. The spatial discretization is performed using a Chebyshev ${C}^{0}$ collocation method, and the method of lines is employed to reduce the PDEs to a system of ordinary differential equations (ODEs). The resulting system is solved using a backward differentiation formula method.
2
Specification
void 
d03pdc (Integer npde,
Integer m,
double *ts,
double tout,
void 
(*pdedef)(Integer npde,
double t,
const double x[],
Integer nptl,
const double u[],
const double ux[],
double p[],
double q[],
double r[],
Integer *ires,
Nag_Comm *comm),


double u[],
Integer nbkpts,
const double xbkpts[],
Integer npoly,
Integer npts,
double x[],
double acc,
double rsave[],
Integer lrsave,
Integer isave[],
Integer lisave,
Integer itask,
Integer itrace,
const char *outfile,
Integer *ind,
Nag_Comm *comm, Nag_D03_Save *saved,
NagError *fail) 

The function may be called by the names: d03pdc, nag_pde_dim1_parab_coll or nag_pde_parab_1d_coll.
3
Description
d03pdc integrates the system of parabolic equations:
where
${P}_{i,j}$,
${Q}_{i}$ and
${R}_{i}$ depend on
$x$,
$t$,
$U$,
${U}_{x}$ and the vector
$U$ is the set of solution values
and the vector
${U}_{x}$ is its partial derivative with respect to
$x$. Note that
${P}_{i,j}$,
${Q}_{i}$ and
${R}_{i}$ must not depend on
$\frac{\partial U}{\partial t}$.
The integration in time is from ${t}_{0}$ to ${t}_{\mathrm{out}}$, over the space interval $a\le x\le b$, where $a={x}_{1}$ and $b={x}_{{\mathbf{nbkpts}}}$ are the leftmost and rightmost of a userdefined set of breakpoints ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{nbkpts}}}$. The coordinate system in space is defined by the value of $m$; $m=0$ for Cartesian coordinates, $m=1$ for cylindrical polar coordinates and $m=2$ for spherical polar coordinates.
The system is defined by the functions
${P}_{i,j}$,
${Q}_{i}$ and
${R}_{i}$ which must be specified in
pdedef.
The initial values of the functions
$U\left(x,t\right)$ must be given at
$t={t}_{0}$, and must be specified in
uinit.
The functions
${R}_{i}$, for
$\mathit{i}=1,2,\dots ,{\mathbf{npde}}$, which may be thought of as fluxes, are also used in the definition of the boundary conditions for each equation. The boundary conditions must have the form
where
$x=a$ or
$x=b$.
The boundary conditions must be specified in
bndary. Thus, the problem is subject to the following restrictions:

(i)${t}_{0}<{t}_{\mathrm{out}}$, so that integration is in the forward direction;

(ii)${P}_{i,j}$, ${Q}_{i}$ and the flux ${R}_{i}$ must not depend on any time derivatives;

(iii)the evaluation of the functions ${P}_{i,j}$, ${Q}_{i}$ and ${R}_{i}$ is done at both the breakpoints and internally selected points for each element in turn, that is ${P}_{i,j}$, ${Q}_{i}$ and ${R}_{i}$ are evaluated twice at each breakpoint. Any discontinuities in these functions must therefore be at one or more of the breakpoints ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{nbkpts}}}$;

(iv)at least one of the functions ${P}_{i,j}$ must be nonzero so that there is a time derivative present in the problem;

(v)if $m>0$ and ${x}_{1}=0.0$, which is the left boundary point, then it must be ensured that the PDE solution is bounded at this point. This can be done by either specifying the solution at $x=0.0$ or by specifying a zero flux there, that is ${\beta}_{i}=1.0$ and ${\gamma}_{i}=0.0$. See also Section 9.
The parabolic equations are approximated by a system of ODEs in time for the values of
${U}_{i}$ at the mesh points. This ODE system is obtained by approximating the PDE solution between each pair of breakpoints by a Chebyshev polynomial of degree
npoly. The interval between each pair of breakpoints is treated by
d03pdc as an element, and on this element, a polynomial and its space and time derivatives are made to satisfy the system of PDEs at
${\mathbf{npoly}}1$ spatial points, which are chosen internally by the code and the breakpoints. In the case of just one element, the breakpoints are the boundaries. The userdefined breakpoints and the internally selected points together define the mesh. The smallest value that
npoly can take is one, in which case, the solution is approximated by piecewise linear polynomials between consecutive breakpoints and the method is similar to an ordinary finite element method.
In total there are $\left({\mathbf{nbkpts}}1\right)\times {\mathbf{npoly}}+1$ mesh points in the spatial direction, and ${\mathbf{npde}}\times \left(\left({\mathbf{nbkpts}}1\right)\times {\mathbf{npoly}}+1\right)$ ODEs in the time direction; one ODE at each breakpoint for each PDE component and (${\mathbf{npoly}}1$) ODEs for each PDE component between each pair of breakpoints. The system is then integrated forwards in time using a backward differentiation formula method.
4
References
Berzins M (1990) Developments in the NAG Library software for parabolic equations Scientific Software Systems (eds J C Mason and M G Cox) 59–72 Chapman and Hall
Berzins M and Dew P M (1991) Algorithm 690: Chebyshev polynomial software for ellipticparabolic systems of PDEs ACM Trans. Math. Software 17 178–206
Zaturska N B, Drazin P G and Banks W H H (1988) On the flow of a viscous fluid driven along a channel by a suction at porous walls Fluid Dynamics Research 4
5
Arguments

1:
$\mathbf{npde}$ – Integer
Input

On entry: the number of PDEs in the system to be solved.
Constraint:
${\mathbf{npde}}\ge 1$.

2:
$\mathbf{m}$ – Integer
Input

On entry: the coordinate system used:
 ${\mathbf{m}}=0$
 Indicates Cartesian coordinates.
 ${\mathbf{m}}=1$
 Indicates cylindrical polar coordinates.
 ${\mathbf{m}}=2$
 Indicates spherical polar coordinates.
Constraint:
${\mathbf{m}}=0$, $1$ or $2$.

3:
$\mathbf{ts}$ – double *
Input/Output

On entry: the initial value of the independent variable $t$.
On exit: the value of
$t$ corresponding to the solution values in
u. Normally
${\mathbf{ts}}={\mathbf{tout}}$.
Constraint:
${\mathbf{ts}}<{\mathbf{tout}}$.

4:
$\mathbf{tout}$ – double
Input

On entry: the final value of $t$ to which the integration is to be carried out.

5:
$\mathbf{pdedef}$ – function, supplied by the user
External Function

pdedef must compute the values of the functions
${P}_{i,j}$,
${Q}_{i}$ and
${R}_{i}$ which define the system of PDEs. The functions may depend on
$x$,
$t$,
$U$ and
${U}_{x}$ and must be evaluated at a set of points.
The specification of
pdedef is:
void 
pdedef (Integer npde,
double t,
const double x[],
Integer nptl,
const double u[],
const double ux[],
double p[],
double q[],
double r[],
Integer *ires,
Nag_Comm *comm)



1:
$\mathbf{npde}$ – Integer
Input

On entry: the number of PDEs in the system.

2:
$\mathbf{t}$ – double
Input

On entry: the current value of the independent variable $t$.

3:
$\mathbf{x}\left[\mathit{dim}\right]$ – const double
Input

On entry: contains a set of mesh points at which ${P}_{i,j}$, ${Q}_{i}$ and ${R}_{i}$ are to be evaluated. ${\mathbf{x}}\left[0\right]$ and ${\mathbf{x}}\left[{\mathbf{nptl}}1\right]$ contain successive usersupplied breakpoints and the elements of the array will satisfy ${\mathbf{x}}\left[0\right]<{\mathbf{x}}\left[1\right]<\cdots <{\mathbf{x}}\left[{\mathbf{nptl}}1\right]$.

4:
$\mathbf{nptl}$ – Integer
Input

On entry: the number of points at which evaluations are required (the value of ${\mathbf{npoly}}+1$).

5:
$\mathbf{u}\left[{\mathbf{npde}}\times {\mathbf{nptl}}\right]$ – const double
Input

Note: the $\left(i,j\right)$th element of the matrix $U$ is stored in ${\mathbf{u}}\left[\left(j1\right)\times {\mathbf{npde}}+i1\right]$.
On entry: ${\mathbf{u}}\left[\left(\mathit{j}1\right)\times {\mathbf{npde}}+\mathit{i}1\right]$ contains the value of the component ${U}_{\mathit{i}}\left(x,t\right)$ where $x={\mathbf{x}}\left[\mathit{j}1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nptl}}$.

6:
$\mathbf{ux}\left[{\mathbf{npde}}\times {\mathbf{nptl}}\right]$ – const double
Input

Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{ux}}\left[\left(j1\right)\times {\mathbf{npde}}+i1\right]$.
On entry: ${\mathbf{ux}}\left[\left(\mathit{j}1\right)\times {\mathbf{npde}}+\mathit{i}1\right]$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}\left(x,t\right)}{\partial x}$ where $x={\mathbf{x}}\left[\mathit{j}1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nptl}}$.

7:
$\mathbf{p}\left[\mathit{dim}\right]$ – double
Output
Note: the dimension,
dim, of the array
p
is
${\mathbf{npde}}\times {\mathbf{npde}}\times {\mathbf{nptl}}$.
where ${\mathbf{P}}\left(i,j,k\right)$ appears in this document, it refers to the array element ${\mathbf{p}}\left[\left(k1\right)\times {\mathbf{npde}}\times {\mathbf{npde}}+\left(j1\right)\times {\mathbf{npde}}+i1\right]$.
On exit: ${\mathbf{p}}\left[{\mathbf{npde}}\times {\mathbf{npde}}\times \left(\mathit{k}1\right)+{\mathbf{npde}}\times \left(\mathit{j}1\right)+\left(\mathit{i}1\right)\right]$ must be set to the value of ${P}_{\mathit{i},\mathit{j}}\left(x,t,U,{U}_{x}\right)$ where $x={\mathbf{x}}\left[\mathit{k}1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$, $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{k}=1,2,\dots ,{\mathbf{nptl}}$.

8:
$\mathbf{q}\left[{\mathbf{npde}}\times {\mathbf{nptl}}\right]$ – double
Output

Note: the $\left(i,j\right)$th element of the matrix $Q$ is stored in ${\mathbf{q}}\left[\left(j1\right)\times {\mathbf{npde}}+i1\right]$.
On exit: ${\mathbf{q}}\left[\left(\mathit{j}1\right)\times {\mathbf{npde}}+\mathit{i}1\right]$ must be set to the value of ${Q}_{\mathit{i}}\left(x,t,U,{U}_{x}\right)$ where $x={\mathbf{x}}\left[\mathit{j}1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nptl}}$.

9:
$\mathbf{r}\left[{\mathbf{npde}}\times {\mathbf{nptl}}\right]$ – double
Output

Note: the $\left(i,j\right)$th element of the matrix $R$ is stored in ${\mathbf{r}}\left[\left(j1\right)\times {\mathbf{npde}}+i1\right]$.
On exit: ${\mathbf{r}}\left[\left(\mathit{j}1\right)\times {\mathbf{npde}}+\mathit{i}1\right]$ must be set to the value of ${R}_{\mathit{i}}\left(x,t,U,{U}_{x}\right)$ where $x={\mathbf{x}}\left[\mathit{j}1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nptl}}$.

10:
$\mathbf{ires}$ – Integer *
Input/Output

On entry: set to $1$ or $1$.
On exit: should usually remain unchanged. However, you may set
ires to force the integration function to take certain actions as described below:
 ${\mathbf{ires}}=2$
 Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
 ${\mathbf{ires}}=3$
 Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{ires}}=3$ when a physically meaningless input or output value has been generated. If you consecutively set ${\mathbf{ires}}=3$, d03pdc returns to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.

11:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
pdedef.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
d03pdc you may allocate memory and initialize these pointers with various quantities for use by
pdedef when called from
d03pdc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: pdedef should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
d03pdc. If your code inadvertently
does return any NaNs or infinities,
d03pdc is likely to produce unexpected results.

6:
$\mathbf{bndary}$ – function, supplied by the user
External Function

bndary must compute the functions
${\beta}_{i}$ and
${\gamma}_{i}$ which define the boundary conditions as in equation
(3).
The specification of
bndary is:
void 
bndary (Integer npde,
double t,
const double u[],
const double ux[],
Integer ibnd,
double beta[],
double gamma[],
Integer *ires,
Nag_Comm *comm)



1:
$\mathbf{npde}$ – Integer
Input

On entry: the number of PDEs in the system.

2:
$\mathbf{t}$ – double
Input

On entry: the current value of the independent variable $t$.

3:
$\mathbf{u}\left[{\mathbf{npde}}\right]$ – const double
Input

On entry:
${\mathbf{u}}\left[\mathit{i}1\right]$ contains the value of the component
${U}_{\mathit{i}}\left(x,t\right)$ at the boundary specified by
ibnd, for
$\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.

4:
$\mathbf{ux}\left[{\mathbf{npde}}\right]$ – const double
Input

On entry:
${\mathbf{ux}}\left[\mathit{i}1\right]$ contains the value of the component
$\frac{\partial {U}_{\mathit{i}}\left(x,t\right)}{\partial x}$ at the boundary specified by
ibnd, for
$\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.

5:
$\mathbf{ibnd}$ – Integer
Input

On entry: specifies which boundary conditions are to be evaluated.
 ${\mathbf{ibnd}}=0$
 bndary must set up the coefficients of the lefthand boundary, $x=a$.
 ${\mathbf{ibnd}}\ne 0$
 bndary must set up the coefficients of the righthand boundary, $x=b$.

6:
$\mathbf{beta}\left[{\mathbf{npde}}\right]$ – double
Output

On exit:
${\mathbf{beta}}\left[\mathit{i}1\right]$ must be set to the value of
${\beta}_{\mathit{i}}\left(x,t\right)$ at the boundary specified by
ibnd, for
$\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.

7:
$\mathbf{gamma}\left[{\mathbf{npde}}\right]$ – double
Output

On exit:
${\mathbf{gamma}}\left[\mathit{i}1\right]$ must be set to the value of
${\gamma}_{\mathit{i}}\left(x,t,U,{U}_{x}\right)$ at the boundary specified by
ibnd, for
$\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.

8:
$\mathbf{ires}$ – Integer *
Input/Output

On entry: set to $1$ or $1$.
On exit: should usually remain unchanged. However, you may set
ires to force the integration function to take certain actions as described below:
 ${\mathbf{ires}}=2$
 Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
 ${\mathbf{ires}}=3$
 Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{ires}}=3$ when a physically meaningless input or output value has been generated. If you consecutively set ${\mathbf{ires}}=3$, d03pdc returns to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.

9:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
bndary.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
d03pdc you may allocate memory and initialize these pointers with various quantities for use by
bndary when called from
d03pdc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: bndary should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
d03pdc. If your code inadvertently
does return any NaNs or infinities,
d03pdc is likely to produce unexpected results.

7:
$\mathbf{u}\left[{\mathbf{npde}}\times {\mathbf{npts}}\right]$ – double
Input/Output

Note: the $\left(i,j\right)$th element of the matrix $U$ is stored in ${\mathbf{u}}\left[\left(j1\right)\times {\mathbf{npde}}+i1\right]$.
On entry: if
${\mathbf{ind}}=1$ the value of
u must be unchanged from the previous call.
On exit: ${\mathbf{u}}\left[\left(j1\right)\times {\mathbf{npde}}+i1\right]$ will contain the computed solution at $t={\mathbf{ts}}$.

8:
$\mathbf{nbkpts}$ – Integer
Input

On entry: the number of breakpoints in the interval $\left[a,b\right]$.
Constraint:
${\mathbf{nbkpts}}\ge 2$.

9:
$\mathbf{xbkpts}\left[{\mathbf{nbkpts}}\right]$ – const double
Input

On entry: the values of the breakpoints in the space direction. ${\mathbf{xbkpts}}\left[0\right]$ must specify the lefthand boundary, $a$, and ${\mathbf{xbkpts}}\left[{\mathbf{nbkpts}}1\right]$ must specify the righthand boundary, $b$.
Constraint:
${\mathbf{xbkpts}}\left[0\right]<{\mathbf{xbkpts}}\left[1\right]<\cdots <{\mathbf{xbkpts}}\left[{\mathbf{nbkpts}}1\right]$.

10:
$\mathbf{npoly}$ – Integer
Input

On entry: the degree of the Chebyshev polynomial to be used in approximating the PDE solution between each pair of breakpoints.
Constraint:
$1\le {\mathbf{npoly}}\le 49$.

11:
$\mathbf{npts}$ – Integer
Input

On entry: the number of mesh points in the interval $\left[a,b\right]$.
Constraint:
${\mathbf{npts}}=\left({\mathbf{nbkpts}}1\right)\times {\mathbf{npoly}}+1$.

12:
$\mathbf{x}\left[{\mathbf{npts}}\right]$ – double
Output

On exit: the mesh points chosen by
d03pdc in the spatial direction. The values of
x will satisfy
${\mathbf{x}}\left[0\right]<{\mathbf{x}}\left[1\right]<\cdots <{\mathbf{x}}\left[{\mathbf{npts}}1\right]$.

13:
$\mathbf{uinit}$ – function, supplied by the user
External Function

uinit must compute the initial values of the PDE components
${U}_{\mathit{i}}\left({x}_{\mathit{j}},{t}_{0}\right)$, for
$\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and
$\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
The specification of
uinit is:
void 
uinit (Integer npde,
Integer npts,
const double x[],
double u[],
Nag_Comm *comm)



1:
$\mathbf{npde}$ – Integer
Input

On entry: the number of PDEs in the system.

2:
$\mathbf{npts}$ – Integer
Input

On entry: the number of mesh points in the interval $\left[a,b\right]$.

3:
$\mathbf{x}\left[{\mathbf{npts}}\right]$ – const double
Input

On entry: ${\mathbf{x}}\left[\mathit{j}1\right]$, contains the values of the $\mathit{j}$th mesh point, for $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.

4:
$\mathbf{u}\left[{\mathbf{npde}}\times {\mathbf{npts}}\right]$ – double
Output

Note: the $\left(i,j\right)$th element of the matrix $U$ is stored in ${\mathbf{u}}\left[\left(j1\right)\times {\mathbf{npde}}+i1\right]$.
On exit: ${\mathbf{u}}\left[\left(\mathit{j}1\right)\times {\mathbf{npde}}+\mathit{i}1\right]$ must be set to the initial value ${U}_{\mathit{i}}\left({x}_{\mathit{j}},{t}_{0}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.

5:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
uinit.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
d03pdc you may allocate memory and initialize these pointers with various quantities for use by
uinit when called from
d03pdc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: uinit should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
d03pdc. If your code inadvertently
does return any NaNs or infinities,
d03pdc is likely to produce unexpected results.

14:
$\mathbf{acc}$ – double
Input

On entry: a positive quantity for controlling the local error estimate in the time integration. If
$E\left(i,j\right)$ is the estimated error for
${U}_{i}$ at the
$j$th mesh point, the error test is:
Constraint:
${\mathbf{acc}}>0.0$.

15:
$\mathbf{rsave}\left[{\mathbf{lrsave}}\right]$ – double
Communication Array

If
${\mathbf{ind}}=0$,
rsave need not be set on entry.
If
${\mathbf{ind}}=1$,
rsave must be unchanged from the previous call to the function because it contains required information about the iteration.

16:
$\mathbf{lrsave}$ – Integer
Input

On entry: the dimension of the array
rsave.
Constraint:
${\mathbf{lrsave}}\ge 11\times {\mathbf{npde}}\times {\mathbf{npts}}+50+\mathit{nwkres}+\mathit{lenode}$.

17:
$\mathbf{isave}\left[{\mathbf{lisave}}\right]$ – Integer
Communication Array

If
${\mathbf{ind}}=0$,
isave need not be set on entry.
If
${\mathbf{ind}}=1$,
isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular:
 ${\mathbf{isave}}\left[0\right]$
 Contains the number of steps taken in time.
 ${\mathbf{isave}}\left[1\right]$
 Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.
 ${\mathbf{isave}}\left[2\right]$
 Contains the number of Jacobian evaluations performed by the time integrator.
 ${\mathbf{isave}}\left[3\right]$
 Contains the order of the last backward differentiation formula method used.
 ${\mathbf{isave}}\left[4\right]$
 Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a backsubstitution using the $LU$ decomposition of the Jacobian matrix.

18:
$\mathbf{lisave}$ – Integer
Input

On entry: the dimension of the array
isave.
Constraint:
${\mathbf{lisave}}\ge {\mathbf{npde}}\times {\mathbf{npts}}+24$.

19:
$\mathbf{itask}$ – Integer
Input

On entry: specifies the task to be performed by the ODE integrator.
 ${\mathbf{itask}}=1$
 Normal computation of output values u at $t={\mathbf{tout}}$.
 ${\mathbf{itask}}=2$
 One step and return.
 ${\mathbf{itask}}=3$
 Stop at first internal integration point at or beyond $t={\mathbf{tout}}$.
Constraint:
${\mathbf{itask}}=1$, $2$ or $3$.

20:
$\mathbf{itrace}$ – Integer
Input

On entry: the level of trace information required from
d03pdc and the underlying ODE solver.
itrace may take the value
$1$,
$0$,
$1$,
$2$ or
$3$.
 ${\mathbf{itrace}}=1$
 No output is generated.
 ${\mathbf{itrace}}=0$
 Only warning messages from the PDE solver are printed.
 ${\mathbf{itrace}}>0$
 Output from the underlying ODE solver is printed. This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.
If ${\mathbf{itrace}}<1$, $1$ is assumed and similarly if ${\mathbf{itrace}}>3$, $3$ is assumed.
The advisory messages are given in greater detail as
itrace increases.

21:
$\mathbf{outfile}$ – const char *
Input

On entry: the name of a file to which diagnostic output will be directed. If
outfile is
NULL the diagnostic output will be directed to standard output.

22:
$\mathbf{ind}$ – Integer *
Input/Output

On entry: indicates whether this is a continuation call or a new integration.
 ${\mathbf{ind}}=0$
 Starts or restarts the integration in time.
 ${\mathbf{ind}}=1$
 Continues the integration after an earlier exit from the function. In this case, only the argument tout should be reset between calls to d03pdc.
Constraint:
${\mathbf{ind}}=0$ or $1$.
On exit: ${\mathbf{ind}}=1$.

23:
$\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).

24:
$\mathbf{saved}$ – Nag_D03_Save *
Communication Structure

saved must remain unchanged following a previous call to a
Chapter D03 function and prior to any subsequent call to a
Chapter D03 function.

25:
$\mathbf{fail}$ – 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_ACC_IN_DOUBT

Integration completed, but a small change in
acc is unlikely to result in a changed solution.
${\mathbf{acc}}=\u2329\mathit{\text{value}}\u232a$.
 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_FAILED_DERIV

In setting up the ODE system an internal auxiliary was unable to initialize the derivative. This could be due to your setting
${\mathbf{ires}}=3$ in
pdedef or
bndary.
 NE_FAILED_START

acc was too small to start integration:
${\mathbf{acc}}=\u2329\mathit{\text{value}}\u232a$.
 NE_FAILED_STEP

Error during Jacobian formulation for ODE system. Increase
itrace for further details.
Repeated errors in an attempted step of underlying ODE solver. Integration was successful as far as
ts:
${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
Underlying ODE solver cannot make further progress from the point
ts with the supplied value of
acc.
${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$,
${\mathbf{acc}}=\u2329\mathit{\text{value}}\u232a$.
 NE_INCOMPAT_PARAM

On entry, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{xbkpts}}\left[0\right]=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{m}}\le 0$ or ${\mathbf{xbkpts}}\left[0\right]\ge 0.0$
 NE_INT

ires set to an invalid value in call to
pdedef or
bndary.
On entry, ${\mathbf{ind}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ind}}=0$ or $1$.
On entry, ${\mathbf{itask}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{itask}}=1$, $2$ or $3$.
On entry, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{m}}=0$, $1$ or $2$.
On entry, ${\mathbf{nbkpts}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nbkpts}}\ge 2$.
On entry, ${\mathbf{npde}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{npde}}\ge 1$.
On entry, ${\mathbf{npoly}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $1\le {\mathbf{npoly}}\le 49$.
On entry, ${\mathbf{npoly}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{npoly}}\le 49$.
On entry, ${\mathbf{npoly}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{npoly}}\ge 1$.
On entry, on initial entry ${\mathbf{ind}}=1$.
Constraint: on initial entry ${\mathbf{ind}}=0$.
 NE_INT_2

On entry, ${\mathbf{lisave}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{lisave}}\ge \u2329\mathit{\text{value}}\u232a$.
On entry, ${\mathbf{lrsave}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{lrsave}}\ge \u2329\mathit{\text{value}}\u232a$.
 NE_INT_3

On entry, ${\mathbf{npts}}=\u2329\mathit{\text{value}}\u232a$, ${\mathbf{nbkpts}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{npoly}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{npts}}=\left({\mathbf{nbkpts}}1\right)\times {\mathbf{npoly}}+1$.
 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.
Serious error in internal call to an auxiliary. Increase
itrace for further details.
 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_CLOSE_FILE

Cannot close file $\u2329\mathit{\text{value}}\u232a$.
 NE_NOT_STRICTLY_INCREASING

On entry, $\mathit{i}=\u2329\mathit{\text{value}}\u232a$, ${\mathbf{xbkpts}}\left[\mathit{i}1\right]=\u2329\mathit{\text{value}}\u232a$, $\mathit{j}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{xbkpts}}\left[\mathit{j}1\right]=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{xbkpts}}\left[0\right]<{\mathbf{xbkpts}}\left[1\right]<\cdots <{\mathbf{xbkpts}}\left[{\mathbf{nbkpts}}1\right]$.
 NE_NOT_WRITE_FILE

Cannot open file $\u2329\mathit{\text{value}}\u232a$ for writing.
 NE_REAL

On entry, ${\mathbf{acc}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{acc}}>0.0$.
 NE_REAL_2

On entry, ${\mathbf{tout}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{tout}}>{\mathbf{ts}}$.
On entry, ${\mathbf{tout}}{\mathbf{ts}}$ is too small:
${\mathbf{tout}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
 NE_SING_JAC

Singular Jacobian of ODE system. Check problem formulation.
 NE_TIME_DERIV_DEP

Flux function appears to depend on time derivatives.
 NE_USER_STOP

In evaluating residual of ODE system,
${\mathbf{ires}}=2$ has been set in
pdedef or
bndary. Integration is successful as far as
ts:
${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
7
Accuracy
d03pdc controls the accuracy of the integration in the time direction but not the accuracy of the approximation in space. The spatial accuracy depends on the degree of the polynomial approximation
npoly, and on both the number of breakpoints and on their distribution in space. In the time integration only the local error over a single step is controlled and so the accuracy over a number of steps cannot be guaranteed. You should therefore test the effect of varying the accuracy argument,
acc.
8
Parallelism and Performance
d03pdc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d03pdc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
d03pdc is designed to solve parabolic systems (possibly including elliptic equations) with secondorder derivatives in space. The argument specification allows you to include equations with only firstorder derivatives in the space direction but there is no guarantee that the method of integration will be satisfactory for such systems. The position and nature of the boundary conditions in particular are critical in defining a stable problem.
The time taken depends on the complexity of the parabolic system and on the accuracy requested.
10
Example
The problem consists of a fourthorder PDE which can be written as a pair of secondorder ellipticparabolic PDEs for
${U}_{1}\left(x,t\right)$ and
${U}_{2}\left(x,t\right)$,
where
$1\le x\le 1$ and
$t\ge 0$. The boundary conditions are given by
The initial conditions at
$t=0$ are given by
The absence of boundary conditions for
${U}_{2}\left(x,t\right)$ does not pose any difficulties provided that the derivative flux boundary conditions are assigned to the first PDE
(4) which has the correct flux,
$\frac{\partial {U}_{1}}{\partial x}$. The conditions on
${U}_{1}\left(x,t\right)$ at the boundaries are assigned to the second PDE by setting
${\beta}_{2}=0.0$ in equation
(3) and placing the Dirichlet boundary conditions on
${U}_{1}\left(x,t\right)$ in the function
${\gamma}_{2}$.
10.1
Program Text
10.2
Program Data
None.
10.3
Program Results