# NAG Library Routine Document

## 1Purpose

d03pdf/d03pda 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.
d03pda is a version of d03pdf that has additional arguments in order to make it safe for use in multithreaded applications (see Section 5).

## 2Specification

### 2.1Specification for d03pdf

Fortran Interface
 Subroutine d03pdf ( npde, m, ts, tout, u, npts, x, acc, ind,
 Integer, Intent (In) :: npde, m, nbkpts, npoly, npts, lrsave, lisave, itask, itrace Integer, Intent (Inout) :: isave(lisave), ind, ifail Real (Kind=nag_wp), Intent (In) :: tout, xbkpts(nbkpts), acc Real (Kind=nag_wp), Intent (Inout) :: ts, u(npde,npts), rsave(lrsave) Real (Kind=nag_wp), Intent (Out) :: x(npts) External :: pdedef, bndary, uinit
#include nagmk26.h
 void d03pdf_ (const Integer *npde, const Integer *m, double *ts, const double *tout, void (NAG_CALL *pdedef)(const Integer *npde, const double *t, const double x[], const Integer *nptl, const double u[], const double ux[], double p[], double q[], double r[], Integer *ires),void (NAG_CALL *bndary)(const Integer *npde, const double *t, const double u[], const double ux[], const Integer *ibnd, double beta[], double gamma[], Integer *ires),double u[], const Integer *nbkpts, const double xbkpts[], const Integer *npoly, const Integer *npts, double x[], void (NAG_CALL *uinit)(const Integer *npde, const Integer *npts, const double x[], double u[]),const double *acc, double rsave[], const Integer *lrsave, Integer isave[], const Integer *lisave, const Integer *itask, const Integer *itrace, Integer *ind, Integer *ifail)

### 2.2Specification for d03pda

Fortran Interface
 Subroutine d03pda ( npde, m, ts, tout, u, npts, x, acc, ind,
 Integer, Intent (In) :: npde, m, nbkpts, npoly, npts, lrsave, lisave, itask, itrace Integer, Intent (Inout) :: isave(lisave), ind, iuser(*), iwsav(505), ifail Real (Kind=nag_wp), Intent (In) :: tout, xbkpts(nbkpts), acc Real (Kind=nag_wp), Intent (Inout) :: ts, u(npde,npts), rsave(lrsave), ruser(*), rwsav(1100) Real (Kind=nag_wp), Intent (Out) :: x(npts) Logical, Intent (Inout) :: lwsav(100) Character (80), Intent (Inout) :: cwsav(10) External :: pdedef, bndary, uinit
#include nagmk26.h
 void d03pda_ (const Integer *npde, const Integer *m, double *ts, const double *tout, void (NAG_CALL *pdedef)(const Integer *npde, const double *t, const double x[], const Integer *nptl, const double u[], const double ux[], double p[], double q[], double r[], Integer *ires, Integer iuser[], double ruser[]),void (NAG_CALL *bndary)(const Integer *npde, const double *t, const double u[], const double ux[], const Integer *ibnd, double beta[], double gamma[], Integer *ires, Integer iuser[], double ruser[]),double u[], const Integer *nbkpts, const double xbkpts[], const Integer *npoly, const Integer *npts, double x[], void (NAG_CALL *uinit)(const Integer *npde, const Integer *npts, const double x[], double u[], Integer iuser[], double ruser[]),const double *acc, double rsave[], const Integer *lrsave, Integer isave[], const Integer *lisave, const Integer *itask, const Integer *itrace, Integer *ind, Integer iuser[], double ruser[], char cwsav[], logical lwsav[], Integer iwsav[], double rwsav[], Integer *ifail, const Charlen length_cwsav)

## 3Description

d03pdf/d03pda integrates the system of parabolic equations:
 $∑j=1npdePi,j ∂Uj ∂t +Qi=x-m ∂∂x xmRi, i=1,2,…,npde, a≤x≤b,t≥t0,$ (1)
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
 $U x,t = U 1 x,t ,…, U npde x,t T ,$ (2)
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 user-defined set of break-points ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{nbkpts}}}$. 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
 $βix,tRix,t,U,Ux=γix,t,U,Ux, i=1,2,…,npde,$ (3)
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 break-points and internally selected points for each element in turn, that is ${P}_{i,j}$, ${Q}_{i}$ and ${R}_{i}$ are evaluated twice at each break-point. Any discontinuities in these functions must therefore be at one or more of the break-points ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{nbkpts}}}$; (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 break-points by a Chebyshev polynomial of degree npoly. The interval between each pair of break-points is treated by d03pdf/d03pda 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 break-points. In the case of just one element, the break-points are the boundaries. The user-defined break-points 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 break-points and the method is similar to an ordinary finite element method.
In total there are $\left({\mathbf{nbkpts}}-1\right)×{\mathbf{npoly}}+1$ mesh points in the spatial direction, and ${\mathbf{npde}}×\left(\left({\mathbf{nbkpts}}-1\right)×{\mathbf{npoly}}+1\right)$ ODEs in the time direction; one ODE at each break-point for each PDE component and (${\mathbf{npoly}}-1$) ODEs for each PDE component between each pair of break-points. The system is then integrated forwards in time using a backward differentiation formula method.
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 elliptic-parabolic 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

## 5Arguments

1:     $\mathbf{npde}$ – IntegerInput
On entry: the number of PDEs in the system to be solved.
Constraint: ${\mathbf{npde}}\ge 1$.
2:     $\mathbf{m}$ – IntegerInput
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:     $\mathbf{ts}$ – Real (Kind=nag_wp)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:     $\mathbf{tout}$ – Real (Kind=nag_wp)Input
On entry: the final value of $t$ to which the integration is to be carried out.
5:     $\mathbf{pdedef}$ – Subroutine, supplied by the user.External Procedure
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 for d03pdf is:
Fortran Interface
 Subroutine pdedef ( npde, t, x, nptl, u, ux, p, q, r, ires)
 Integer, Intent (In) :: npde, nptl Integer, Intent (Inout) :: ires Real (Kind=nag_wp), Intent (In) :: t, x(nptl), u(npde,nptl), ux(npde,nptl) Real (Kind=nag_wp), Intent (Out) :: p(npde,npde,nptl), q(npde,nptl), r(npde,nptl)
#include nagmk26.h
 void pdedef (const Integer *npde, const double *t, const double x[], const Integer *nptl, const double u[], const double ux[], double p[], double q[], double r[], Integer *ires)
The specification of pdedef for d03pda is:
Fortran Interface
 Subroutine pdedef ( npde, t, x, nptl, u, ux, p, q, r, ires,
 Integer, Intent (In) :: npde, nptl Integer, Intent (Inout) :: ires, iuser(*) Real (Kind=nag_wp), Intent (In) :: t, x(nptl), u(npde,nptl), ux(npde,nptl) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: p(npde,npde,nptl), q(npde,nptl), r(npde,nptl)
#include nagmk26.h
 void pdedef (const Integer *npde, const double *t, const double x[], const Integer *nptl, const double u[], const double ux[], double p[], double q[], double r[], Integer *ires, Integer iuser[], double ruser[])
1:     $\mathbf{npde}$ – IntegerInput
On entry: the number of PDEs in the system.
2:     $\mathbf{t}$ – Real (Kind=nag_wp)Input
On entry: the current value of the independent variable $t$.
3:     $\mathbf{x}\left({\mathbf{nptl}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: contains a set of mesh points at which ${P}_{i,j}$, ${Q}_{i}$ and ${R}_{i}$ are to be evaluated. ${\mathbf{x}}\left(1\right)$ and ${\mathbf{x}}\left({\mathbf{nptl}}\right)$ contain successive user-supplied break-points and the elements of the array will satisfy ${\mathbf{x}}\left(1\right)<{\mathbf{x}}\left(2\right)<\cdots <{\mathbf{x}}\left({\mathbf{nptl}}\right)$.
4:     $\mathbf{nptl}$ – IntegerInput
On entry: the number of points at which evaluations are required (the value of ${\mathbf{npoly}}+1$).
5:     $\mathbf{u}\left({\mathbf{npde}},{\mathbf{nptl}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{u}}\left(\mathit{i},\mathit{j}\right)$ contains the value of the component ${U}_{\mathit{i}}\left(x,t\right)$ where $x={\mathbf{x}}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nptl}}$.
6:     $\mathbf{ux}\left({\mathbf{npde}},{\mathbf{nptl}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{ux}}\left(\mathit{i},\mathit{j}\right)$ contains the value of the component $\frac{\partial {U}_{\mathit{i}}\left(x,t\right)}{\partial x}$ where $x={\mathbf{x}}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nptl}}$.
7:     $\mathbf{p}\left({\mathbf{npde}},{\mathbf{npde}},{\mathbf{nptl}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{p}}\left(\mathit{i},\mathit{j},\mathit{k}\right)$ must be set to the value of ${P}_{\mathit{i},\mathit{j}}\left(x,t,U,{U}_{x}\right)$ where $x={\mathbf{x}}\left(\mathit{k}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$, $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{k}=1,2,\dots ,{\mathbf{nptl}}$.
8:     $\mathbf{q}\left({\mathbf{npde}},{\mathbf{nptl}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{q}}\left(\mathit{i},\mathit{j}\right)$ must be set to the value of ${Q}_{\mathit{i}}\left(x,t,U,{U}_{x}\right)$ where $x={\mathbf{x}}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nptl}}$.
9:     $\mathbf{r}\left({\mathbf{npde}},{\mathbf{nptl}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{r}}\left(\mathit{i},\mathit{j}\right)$ must be set to the value of ${R}_{\mathit{i}}\left(x,t,U,{U}_{x}\right)$ where $x={\mathbf{x}}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nptl}}$.
10:   $\mathbf{ires}$ – IntegerInput/Output
On entry: set to $-1\text{​ or ​}1$.
On exit: should usually remain unchanged. However, you may set ires to force the integration routine to take certain actions as described below:
${\mathbf{ires}}=2$
Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to ${\mathbf{ifail}}={\mathbf{6}}$.
${\mathbf{ires}}=3$
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 ${\mathbf{ires}}=3$, d03pdf/d03pda returns to the calling subroutine with the error indicator set to ${\mathbf{ifail}}={\mathbf{4}}$.
Note: the following are additional arguments for specific use with d03pda. Users of d03pdf therefore need not read the remainder of this description.
11:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
12:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
pdedef is called with the arguments iuser and ruser as supplied to d03pdf/d03pda. You should use the arrays iuser and ruser to supply information to pdedef.
pdedef must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03pdf/d03pda is called. Arguments denoted as Input must not be changed by this procedure.
Note: pdedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pdf/d03pda. If your code inadvertently does return any NaNs or infinities, d03pdf/d03pda is likely to produce unexpected results.
6:     $\mathbf{bndary}$ – Subroutine, supplied by the user.External Procedure
bndary must compute the functions ${\beta }_{i}$ and ${\gamma }_{i}$ which define the boundary conditions as in equation (3).
The specification of bndary for d03pdf is:
Fortran Interface
 Subroutine bndary ( npde, t, u, ux, ibnd, beta, ires)
 Integer, Intent (In) :: npde, ibnd Integer, Intent (Inout) :: ires Real (Kind=nag_wp), Intent (In) :: t, u(npde), ux(npde) Real (Kind=nag_wp), Intent (Out) :: beta(npde), gamma(npde)
#include nagmk26.h
 void bndary (const Integer *npde, const double *t, const double u[], const double ux[], const Integer *ibnd, double beta[], double gamma[], Integer *ires)
The specification of bndary for d03pda is:
Fortran Interface
 Subroutine bndary ( npde, t, u, ux, ibnd, beta, ires,
 Integer, Intent (In) :: npde, ibnd Integer, Intent (Inout) :: ires, iuser(*) Real (Kind=nag_wp), Intent (In) :: t, u(npde), ux(npde) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: beta(npde), gamma(npde)
#include nagmk26.h
 void bndary (const Integer *npde, const double *t, const double u[], const double ux[], const Integer *ibnd, double beta[], double gamma[], Integer *ires, Integer iuser[], double ruser[])
1:     $\mathbf{npde}$ – IntegerInput
On entry: the number of PDEs in the system.
2:     $\mathbf{t}$ – Real (Kind=nag_wp)Input
On entry: the current value of the independent variable $t$.
3:     $\mathbf{u}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{u}}\left(\mathit{i}\right)$ 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:     $\mathbf{ux}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{ux}}\left(\mathit{i}\right)$ 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:     $\mathbf{ibnd}$ – IntegerInput
On entry: specifies which boundary conditions are to be evaluated.
${\mathbf{ibnd}}=0$
bndary must set up the coefficients of the left-hand boundary, $x=a$.
${\mathbf{ibnd}}\ne 0$
bndary must set up the coefficients of the right-hand boundary, $x=b$.
6:     $\mathbf{beta}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{beta}}\left(\mathit{i}\right)$ 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:     $\mathbf{gamma}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{gamma}}\left(\mathit{i}\right)$ 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:     $\mathbf{ires}$ – IntegerInput/Output
On entry: set to $-1\text{​ or ​}1$.
On exit: should usually remain unchanged. However, you may set ires to force the integration routine to take certain actions as described below:
${\mathbf{ires}}=2$
Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to ${\mathbf{ifail}}={\mathbf{6}}$.
${\mathbf{ires}}=3$
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 ${\mathbf{ires}}=3$, d03pdf/d03pda returns to the calling subroutine with the error indicator set to ${\mathbf{ifail}}={\mathbf{4}}$.
Note: the following are additional arguments for specific use with d03pda. Users of d03pdf therefore need not read the remainder of this description.
9:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
10:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
bndary is called with the arguments iuser and ruser as supplied to d03pdf/d03pda. You should use the arrays iuser and ruser to supply information to bndary.
bndary must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03pdf/d03pda is called. Arguments denoted as Input must not be changed by this procedure.
Note: bndary should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pdf/d03pda. If your code inadvertently does return any NaNs or infinities, d03pdf/d03pda is likely to produce unexpected results.
7:     $\mathbf{u}\left({\mathbf{npde}},{\mathbf{npts}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{ind}}=1$ the value of u must be unchanged from the previous call.
On exit: ${\mathbf{u}}\left(i,j\right)$ will contain the computed solution at $t={\mathbf{ts}}$.
8:     $\mathbf{nbkpts}$ – IntegerInput
On entry: the number of break-points in the interval $\left[a,b\right]$.
Constraint: ${\mathbf{nbkpts}}\ge 2$.
9:     $\mathbf{xbkpts}\left({\mathbf{nbkpts}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the values of the break-points in the space direction. ${\mathbf{xbkpts}}\left(1\right)$ must specify the left-hand boundary, $a$, and ${\mathbf{xbkpts}}\left({\mathbf{nbkpts}}\right)$ must specify the right-hand boundary, $b$.
Constraint: ${\mathbf{xbkpts}}\left(1\right)<{\mathbf{xbkpts}}\left(2\right)<\cdots <{\mathbf{xbkpts}}\left({\mathbf{nbkpts}}\right)$.
10:   $\mathbf{npoly}$ – IntegerInput
On entry: the degree of the Chebyshev polynomial to be used in approximating the PDE solution between each pair of break-points.
Constraint: $1\le {\mathbf{npoly}}\le 49$.
11:   $\mathbf{npts}$ – IntegerInput
On entry: the number of mesh points in the interval $\left[a,b\right]$.
Constraint: ${\mathbf{npts}}=\left({\mathbf{nbkpts}}-1\right)×{\mathbf{npoly}}+1$.
12:   $\mathbf{x}\left({\mathbf{npts}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the mesh points chosen by d03pdf/d03pda in the spatial direction. The values of x will satisfy ${\mathbf{x}}\left(1\right)<{\mathbf{x}}\left(2\right)<\cdots <{\mathbf{x}}\left({\mathbf{npts}}\right)$.
13:   $\mathbf{uinit}$ – Subroutine, supplied by the user.External Procedure
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 for d03pdf is:
Fortran Interface
 Subroutine uinit ( npde, npts, x, u)
 Integer, Intent (In) :: npde, npts Real (Kind=nag_wp), Intent (In) :: x(npts) Real (Kind=nag_wp), Intent (Out) :: u(npde,npts)
#include nagmk26.h
 void uinit (const Integer *npde, const Integer *npts, const double x[], double u[])
The specification of uinit for d03pda is:
Fortran Interface
 Subroutine uinit ( npde, npts, x, u,
 Integer, Intent (In) :: npde, npts Integer, Intent (Inout) :: iuser(*) Real (Kind=nag_wp), Intent (In) :: x(npts) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: u(npde,npts)
#include nagmk26.h
 void uinit (const Integer *npde, const Integer *npts, const double x[], double u[], Integer iuser[], double ruser[])
1:     $\mathbf{npde}$ – IntegerInput
On entry: the number of PDEs in the system.
2:     $\mathbf{npts}$ – IntegerInput
On entry: the number of mesh points in the interval $\left[a,b\right]$.
3:     $\mathbf{x}\left({\mathbf{npts}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{x}}\left(\mathit{j}\right)$, contains the values of the $\mathit{j}$th mesh point, for $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
4:     $\mathbf{u}\left({\mathbf{npde}},{\mathbf{npts}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{u}}\left(\mathit{i},\mathit{j}\right)$ 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 $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
Note: the following are additional arguments for specific use with d03pda. Users of d03pdf therefore need not read the remainder of this description.
5:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
6:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
uinit is called with the arguments iuser and ruser as supplied to d03pdf/d03pda. You should use the arrays iuser and ruser to supply information to uinit.
uinit must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03pdf/d03pda is called. Arguments denoted as Input must not be changed by this procedure.
Note: uinit should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pdf/d03pda. If your code inadvertently does return any NaNs or infinities, d03pdf/d03pda is likely to produce unexpected results.
14:   $\mathbf{acc}$ – Real (Kind=nag_wp)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:
 $Ei,j=acc×1.0+uij.$
Constraint: ${\mathbf{acc}}>0.0$.
15:   $\mathbf{rsave}\left({\mathbf{lrsave}}\right)$ – Real (Kind=nag_wp) arrayCommunication 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 routine because it contains required information about the iteration.
16:   $\mathbf{lrsave}$ – IntegerInput
On entry: the dimension of the array rsave as declared in the (sub)program from which d03pdf/d03pda is called.
Constraint: ${\mathbf{lrsave}}\ge 11×{\mathbf{npde}}×{\mathbf{npts}}+50+\mathit{nwkres}+\mathit{lenode}$.
17:   $\mathbf{isave}\left({\mathbf{lisave}}\right)$ – Integer arrayCommunication 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 routine because it contains required information about the iteration. In particular:
${\mathbf{isave}}\left(1\right)$
Contains the number of steps taken in time.
${\mathbf{isave}}\left(2\right)$
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.
${\mathbf{isave}}\left(3\right)$
Contains the number of Jacobian evaluations performed by the time integrator.
${\mathbf{isave}}\left(4\right)$
Contains the order of the last backward differentiation formula method used.
${\mathbf{isave}}\left(5\right)$
Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the $LU$ decomposition of the Jacobian matrix.
18:   $\mathbf{lisave}$ – IntegerInput
On entry: the dimension of the array isave as declared in the (sub)program from which d03pdf/d03pda is called.
Constraint: ${\mathbf{lisave}}\ge {\mathbf{npde}}×{\mathbf{npts}}+24$.
19:   $\mathbf{itask}$ – IntegerInput
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:   $\mathbf{itrace}$ – IntegerInput
On entry: the level of trace information required from d03pdf/d03pda 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 on the current error message unit (see x04aaf).
${\mathbf{itrace}}>0$
Output from the underlying ODE solver is printed on the current advisory message unit (see x04abf). 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. You are advised to set ${\mathbf{itrace}}=0$, unless you are experienced with Sub-chapter D02M–N.
21:   $\mathbf{ind}$ – IntegerInput/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 routine. In this case, only the arguments tout and ifail should be reset between calls to d03pdf/d03pda.
Constraint: ${\mathbf{ind}}=0$ or $1$.
On exit: ${\mathbf{ind}}=1$.
22:   $\mathbf{ifail}$ – IntegerInput/Output
Note: for d03pda, ifail does not occur in this position in the argument list. See the additional arguments described below.
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
Note: the following are additional arguments for specific use with d03pda. Users of d03pdf therefore need not read the remainder of this description.
22:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
iuser is not used by d03pdf/d03pda, but is passed directly to pdedef, bndary and uinit and may be used to pass information to these routines.
23:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
24:   $\mathbf{cwsav}\left(10\right)$ – Character(80) arrayCommunication Array
25:   $\mathbf{lwsav}\left(100\right)$ – Logical arrayCommunication Array
26:   $\mathbf{iwsav}\left(505\right)$ – Integer arrayCommunication Array
27:   $\mathbf{rwsav}\left(1100\right)$ – Real (Kind=nag_wp) arrayCommunication Array
28:   $\mathbf{ifail}$ – IntegerInput/Output
Note: see the argument description for ifail above.

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{tout}}\le {\mathbf{ts}}$, or ${\mathbf{tout}}-{\mathbf{ts}}$ is too small, or ${\mathbf{itask}}\ne 1$, $2$ or $3$, or ${\mathbf{m}}\ne 0$, $1$ or $2$, or ${\mathbf{m}}>0$ and ${\mathbf{xbkpts}}\left(1\right)<0.0$, or ${\mathbf{npde}}<1$, or ${\mathbf{nbkpts}}<2$, or ${\mathbf{npoly}}<1$ or ${\mathbf{npoly}}>49$, or ${\mathbf{npts}}\ne \left({\mathbf{nbkpts}}-1\right)×{\mathbf{npoly}}+1$, or ${\mathbf{acc}}\le 0.0$, or ${\mathbf{ind}}\ne 0$ or $1$, or break-points ${\mathbf{xbkpts}}\left(i\right)$ are not ordered, or lrsave is too small, or lisave is too small.
${\mathbf{ifail}}=2$
The underlying ODE solver cannot make any further progress across the integration range from the current point $t={\mathbf{ts}}$ with the supplied value of acc. The components of u contain the computed values at the current point $t={\mathbf{ts}}$.
${\mathbf{ifail}}=3$
In the underlying ODE solver, there were repeated errors or corrector convergence test failures on an attempted step, before completing the requested task. The problem may have a singularity or acc is too small for the integration to continue. Integration was successful as far as $t={\mathbf{ts}}$.
${\mathbf{ifail}}=4$
In setting up the ODE system, the internal initialization routine was unable to initialize the derivative of the ODE system. This could be due to the fact that ires was repeatedly set to $3$ in at least pdedef or bndary, when the residual in the underlying ODE solver was being evaluated.
${\mathbf{ifail}}=5$
In solving the ODE system, a singular Jacobian has been encountered. You should check your problem formulation.
${\mathbf{ifail}}=6$
When evaluating the residual in solving the ODE system, ires was set to $2$ in at least pdedef or bndary. Integration was successful as far as $t={\mathbf{ts}}$.
${\mathbf{ifail}}=7$
The value of acc is so small that the routine is unable to start the integration in time.
${\mathbf{ifail}}=8$
In one of pdedef or bndary, ires was set to an invalid value.
${\mathbf{ifail}}=9$ (d02nnf)
A serious error has occurred in an internal call to the specified routine. Check the problem specification and all arguments and array dimensions. Setting ${\mathbf{itrace}}=1$ may provide more information. If the problem persists, contact NAG.
${\mathbf{ifail}}=10$
The required task has been completed, but it is estimated that a small change in acc is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when ${\mathbf{itask}}\ne 2$.)
${\mathbf{ifail}}=11$
An error occurred during Jacobian formulation of the ODE system (a more detailed error description may be directed to the current error message unit).
${\mathbf{ifail}}=12$
Not applicable.
${\mathbf{ifail}}=13$
Not applicable.
${\mathbf{ifail}}=14$
The flux function ${R}_{i}$ was detected as depending on time derivatives, which is not permissible.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

d03pdf/d03pda 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 break-points 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.

## 8Parallelism and Performance

d03pdf/d03pda is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d03pdf/d03pda 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

d03pdf/d03pda is designed to solve parabolic systems (possibly including elliptic equations) with second-order derivatives in space. The argument specification allows you to include equations with only first-order 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.

## 10Example

The problem consists of a fourth-order PDE which can be written as a pair of second-order elliptic-parabolic PDEs for ${U}_{1}\left(x,t\right)$ and ${U}_{2}\left(x,t\right)$,
 $0= ∂2U1 ∂x2 -U2$ (4)
 $∂U2 ∂t = ∂2U2 ∂x2 +U2 ∂U1 ∂x -U1 ∂U2 ∂x$ (5)
where $-1\le x\le 1$ and $t\ge 0$. The boundary conditions are given by
 $∂U1 ∂x =0 and U1=1 at ​x=-1, and ∂U1 ∂x =0 and U1=-1 at ​x=1.$
The initial conditions at $t=0$ are given by
 $U1=-sin⁡πx2 and U2=π24sin⁡πx2.$
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 ${\gamma }_{2}$.

### 10.1Program Text

Note: the following programs illustrate the use of d03pdf and d03pda.

Program Text (d03pdfe.f90)

Program Text (d03pdae.f90)

### 10.2Program Data

Program Data (d03pdfe.d)

Program Data (d03pdae.d)

### 10.3Program Results

Program Results (d03pdfe.r)

Program Results (d03pdae.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017