# NAG Library Routine Document

## 1Purpose

d03pjf/d03pja integrates a system of linear or nonlinear parabolic partial differential equations (PDEs), in one space variable with scope for coupled ordinary differential equations (ODEs). 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 ODEs. The resulting system is solved using a backward differentiation formula (BDF) method or a Theta method (switching between Newton's method and functional iteration).
d03pja is a version of d03pjf that has additional arguments in order to make it safe for use in multithreaded applications (see Section 5).

## 2Specification

### 2.1Specification for d03pjf

Fortran Interface
 Subroutine d03pjf ( npde, m, ts, tout, u, npts, x, nv, nxi, xi, neqn, rtol, atol, itol, norm, ind,
 Integer, Intent (In) :: npde, m, nbkpts, npoly, npts, nv, nxi, neqn, itol, lrsave, lisave, itask, itrace Integer, Intent (Inout) :: isave(lisave), ind, ifail Real (Kind=nag_wp), Intent (In) :: tout, xbkpts(nbkpts), xi(nxi), rtol(*), atol(*), algopt(30) Real (Kind=nag_wp), Intent (Inout) :: ts, u(neqn), rsave(lrsave) Real (Kind=nag_wp), Intent (Out) :: x(npts) Character (1), Intent (In) :: norm, laopt External :: pdedef, bndary, odedef, uvinit
#include nagmk26.h
 void d03pjf_ (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[], const Integer *nv, const double v[], const double vdot[], 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 *nv, const double v[], const double vdot[], 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[], const Integer *nv, void (NAG_CALL *odedef)(const Integer *npde, const double *t, const Integer *nv, const double v[], const double vdot[], const Integer *nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], Integer *ires),const Integer *nxi, const double xi[], const Integer *neqn, void (NAG_CALL *uvinit)(const Integer *npde, const Integer *npts, const double x[], double u[], const Integer *nv, double v[]),const double rtol[], const double atol[], const Integer *itol, const char *norm, const char *laopt, const double algopt[], double rsave[], const Integer *lrsave, Integer isave[], const Integer *lisave, const Integer *itask, const Integer *itrace, Integer *ind, Integer *ifail, const Charlen length_norm, const Charlen length_laopt)

### 2.2Specification for d03pja

Fortran Interface
 Subroutine d03pja ( npde, m, ts, tout, u, npts, x, nv, nxi, xi, neqn, rtol, atol, itol, norm, ind,
 Integer, Intent (In) :: npde, m, nbkpts, npoly, npts, nv, nxi, neqn, itol, lrsave, lisave, itask, itrace Integer, Intent (Inout) :: isave(lisave), ind, iuser(*), iwsav(505), ifail Real (Kind=nag_wp), Intent (In) :: tout, xbkpts(nbkpts), xi(nxi), rtol(*), atol(*), algopt(30) Real (Kind=nag_wp), Intent (Inout) :: ts, u(neqn), rsave(lrsave), ruser(*), rwsav(1100) Real (Kind=nag_wp), Intent (Out) :: x(npts) Logical, Intent (Inout) :: lwsav(100) Character (1), Intent (In) :: norm, laopt Character (80), Intent (Inout) :: cwsav(10) External :: pdedef, bndary, odedef, uvinit
#include nagmk26.h
 void d03pja_ (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[], const Integer *nv, const double v[], const double vdot[], 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 *nv, const double v[], const double vdot[], 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[], const Integer *nv, void (NAG_CALL *odedef)(const Integer *npde, const double *t, const Integer *nv, const double v[], const double vdot[], const Integer *nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], Integer *ires, Integer iuser[], double ruser[]),const Integer *nxi, const double xi[], const Integer *neqn, void (NAG_CALL *uvinit)(const Integer *npde, const Integer *npts, const double x[], double u[], const Integer *nv, double v[], Integer iuser[], double ruser[]),const double rtol[], const double atol[], const Integer *itol, const char *norm, const char *laopt, const double algopt[], 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_norm, const Charlen length_laopt, const Charlen length_cwsav)

## 3Description

d03pjf/d03pja integrates the system of parabolic-elliptic equations and coupled ODEs
 $∑j=1npdePi,j ∂Uj ∂t +Qi=x-m ∂∂x xmRi, i=1,2,…,npde, a≤x≤b,t≥t0,$ (1)
 $Fit,V,V.,ξ,U*,Ux*,R*,Ut*,Uxt*=0, i=1,2,…,nv,$ (2)
where (1) defines the PDE part and (2) generalizes the coupled ODE part of the problem.
In (1), ${P}_{i,j}$ and ${R}_{i}$ depend on $x$, $t$, $U$, ${U}_{x}$, and $V$; ${Q}_{i}$ depends on $x$, $t$, $U$, ${U}_{x}$, $V$ and linearly on $\stackrel{.}{V}$. The vector $U$ is the set of PDE solution values
 $U x,t = U 1 x,t ,…, U npde x,t T ,$
and the vector ${U}_{x}$ is the 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 vector $V$ is the set of ODE solution values
 $Vt=V1t,…,VnvtT,$
and $\stackrel{.}{V}$ denotes its derivative with respect to time.
In (2), $\xi$ represents a vector of ${n}_{\xi }$ spatial coupling points at which the ODEs are coupled to the PDEs. These points may or may not be equal to some of the PDE spatial mesh points. ${U}^{*}$, ${U}_{x}^{*}$, ${R}^{*}$, ${U}_{t}^{*}$ and ${U}_{xt}^{*}$ are the functions $U$, ${U}_{x}$, $R$, ${U}_{t}$ and ${U}_{xt}$ evaluated at these coupling points. Each ${F}_{i}$ may only depend linearly on time derivatives. Hence the equation (2) may be written more precisely as
 $F=G-AV.-B Ut* Uxt* ,$ (3)
where $F={\left[{F}_{1},\dots ,{F}_{{\mathbf{nv}}}\right]}^{\mathrm{T}}$, $G$ is a vector of length nv, $A$ is an nv by nv matrix, $B$ is an nv by $\left({n}_{\xi }×{\mathbf{npde}}\right)$ matrix and the entries in $G$, $A$ and $B$ may depend on $t$, $\xi$, ${U}^{*}$, ${U}_{x}^{*}$ and $V$. In practice you need only supply a vector of information to define the ODEs and not the matrices $A$ and $B$. (See Section 5 for the specification of odedef.)
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 PDE system which is defined by the functions ${P}_{i,j}$, ${Q}_{i}$ and ${R}_{i}$ must be specified in pdedef.
The initial values of the functions $U\left(x,t\right)$ and $V\left(t\right)$ must be given at $t={t}_{0}$. These values are calculated in uvinit.
The functions ${R}_{i}$ which may be thought of as fluxes, are also used in the definition of the boundary conditions. The boundary conditions must have the form
 $βix,tRix,t,U,Ux,V=γix,t,U,Ux,V,V., i=1,2,…,npde,$ (4)
where $x=a$ or $x=b$. The functions ${\gamma }_{i}$ may only depend linearly on $\stackrel{.}{V}$.
The boundary conditions must be specified in bndary.
The algebraic-differential equation system which is defined by the functions ${F}_{i}$ must be specified in odedef. You must also specify the coupling points $\xi$ in the array xi. Thus, the problem is subject to the following restrictions:
 (i) in (1), ${\stackrel{.}{V}}_{\mathit{j}}\left(t\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{nv}}$, may only appear linearly in the functions ${Q}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$, with a similar restriction for $\gamma$; (ii) ${P}_{\mathit{i},j}$ and the flux ${R}_{\mathit{i}}$ must not depend on any time derivatives; (iii) ${t}_{0}<{t}_{\mathrm{out}}$, so that integration is in the forward direction; (iv) 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 mesh points; (v) at least one of the functions ${P}_{i,j}$ must be nonzero so that there is a time derivative present in the PDE problem; (vi) 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 either by specifying the solution at $x=0.0$ or by specifying a zero flux there, that is ${\beta }_{i}=1.0$ and ${\gamma }_{i}=0.0$.
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 d03pjf/d03pja 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. 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 ${\mathbf{npde}}×\left(\left({\mathbf{nbkpts}}-1\right)×{\mathbf{npoly}}+1\right)+{\mathbf{nv}}$ ODEs in the time direction; one ODE at each break-point for each PDE component, ${\mathbf{npoly}}-1$ ODEs for each PDE component between each pair of break-points, and nv coupled ODEs. The system is then integrated forwards in time using a Backward Differentiation Formula (BDF) method or a Theta method.

## 4References

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
Berzins M, Dew P M and Furzeland R M (1988) Software tools for time-dependent equations in simulation and optimization of large systems Proc. IMA Conf. Simulation and Optimization (ed A J Osiadcz) 35–50 Clarendon Press, Oxford
Berzins M and Furzeland R M (1992) An adaptive theta method for the solution of stiff and nonstiff differential equations Appl. Numer. Math. 9 1–19
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 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 functions ${P}_{i,j}$, ${Q}_{i}$ and ${R}_{i}$ which define the system of PDEs. The functions may depend on $x$, $t$, $U$, ${U}_{x}$ and $V$; ${Q}_{i}$ may depend linearly on $\stackrel{.}{V}$. The functions must be evaluated at a set of points.
The specification of pdedef for d03pjf is:
Fortran Interface
 Subroutine pdedef ( npde, t, x, nptl, u, ux, nv, v, vdot, p, q, r, ires)
 Integer, Intent (In) :: npde, nptl, nv Integer, Intent (Inout) :: ires Real (Kind=nag_wp), Intent (In) :: t, x(nptl), u(npde,nptl), ux(npde,nptl), v(nv), vdot(nv) 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[], const Integer *nv, const double v[], const double vdot[], double p[], double q[], double r[], Integer *ires)
The specification of pdedef for d03pja is:
Fortran Interface
 Subroutine pdedef ( npde, t, x, nptl, u, ux, nv, v, vdot, p, q, r, ires,
 Integer, Intent (In) :: npde, nptl, nv Integer, Intent (Inout) :: ires, iuser(*) Real (Kind=nag_wp), Intent (In) :: t, x(nptl), u(npde,nptl), ux(npde,nptl), v(nv), vdot(nv) 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[], const Integer *nv, const double v[], const double vdot[], 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:     $\mathbf{nv}$ – IntegerInput
On entry: the number of coupled ODEs in the system.
8:     $\mathbf{v}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nv}}>0$, ${\mathbf{v}}\left(\mathit{i}\right)$ contains the value of the component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
9:     $\mathbf{vdot}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nv}}>0$, ${\mathbf{vdot}}\left(\mathit{i}\right)$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
Note: ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$, may only appear linearly in ${Q}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
10:   $\mathbf{p}\left({\mathbf{npde}},{\mathbf{npde}},{\mathbf{nptl}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{p}}\left(\mathit{i},\mathit{j},\mathit{k}\right)$ must be set to the value of ${P}_{\mathit{i},\mathit{j}}\left(x,t,U,{U}_{x},V\right)$ where $x={\mathbf{x}}\left(\mathit{k}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$, $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{k}=1,2,\dots ,{\mathbf{nptl}}$.
11:   $\mathbf{q}\left({\mathbf{npde}},{\mathbf{nptl}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{q}}\left(\mathit{i},\mathit{j}\right)$ must be set to the value of ${Q}_{\mathit{i}}\left(x,t,U,{U}_{x},V,\stackrel{.}{V}\right)$ where $x={\mathbf{x}}\left(\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nptl}}$.
12:   $\mathbf{r}\left({\mathbf{npde}},{\mathbf{nptl}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{r}}\left(\mathit{i},\mathit{j}\right)$ must be set to the value of ${R}_{\mathit{i}}\left(x,t,U,{U}_{x},V\right)$ where $x={\mathbf{x}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nptl}}$.
13:   $\mathbf{ires}$ – 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 ${\mathbf{ires}}=3$, d03pjf/d03pja 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 d03pja. Users of d03pjf therefore need not read the remainder of this description.
14:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
15:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
pdedef is called with the arguments iuser and ruser as supplied to d03pjf/d03pja. 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 d03pjf/d03pja 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 d03pjf/d03pja. If your code inadvertently does return any NaNs or infinities, d03pjf/d03pja 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 (4).
The specification of bndary for d03pjf is:
Fortran Interface
 Subroutine bndary ( npde, t, u, ux, nv, v, vdot, ibnd, beta, ires)
 Integer, Intent (In) :: npde, nv, ibnd Integer, Intent (Inout) :: ires Real (Kind=nag_wp), Intent (In) :: t, u(npde), ux(npde), v(nv), vdot(nv) 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 *nv, const double v[], const double vdot[], const Integer *ibnd, double beta[], double gamma[], Integer *ires)
The specification of bndary for d03pja is:
Fortran Interface
 Subroutine bndary ( npde, t, u, ux, nv, v, vdot, ibnd, beta, ires,
 Integer, Intent (In) :: npde, nv, ibnd Integer, Intent (Inout) :: ires, iuser(*) Real (Kind=nag_wp), Intent (In) :: t, u(npde), ux(npde), v(nv), vdot(nv) 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 *nv, const double v[], const double vdot[], 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:     $\mathbf{nv}$ – IntegerInput
On entry: the number of coupled ODEs in the system.
6:     $\mathbf{v}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nv}}>0$, ${\mathbf{v}}\left(\mathit{i}\right)$ contains the value of the component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
7:     $\mathbf{vdot}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nv}}>0$, ${\mathbf{vdot}}\left(\mathit{i}\right)$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
Note: ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$, may only appear linearly in ${Q}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
8:     $\mathbf{ibnd}$ – 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, $x=b$.
9:     $\mathbf{beta}\left({\mathbf{npde}}\right)$ – 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 $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
10:   $\mathbf{gamma}\left({\mathbf{npde}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{gamma}}\left(\mathit{i}\right)$ must be set to the value of ${\gamma }_{\mathit{i}}\left(x,t,U,{U}_{x},V,\stackrel{.}{V}\right)$ at the boundary specified by ibnd, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
11:   $\mathbf{ires}$ – 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 ${\mathbf{ires}}=3$, d03pjf/d03pja 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 d03pja. Users of d03pjf therefore need not read the remainder of this description.
12:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
13:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
bndary is called with the arguments iuser and ruser as supplied to d03pjf/d03pja. 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 d03pjf/d03pja 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 d03pjf/d03pja. If your code inadvertently does return any NaNs or infinities, d03pjf/d03pja is likely to produce unexpected results.
7:     $\mathbf{u}\left({\mathbf{neqn}}\right)$ – 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: the computed solution ${U}_{\mathit{i}}\left({x}_{\mathit{j}},t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$, and ${V}_{\mathit{k}}\left(t\right)$, for $\mathit{k}=1,2,\dots ,{\mathbf{nv}}$, evaluated at $t={\mathbf{ts}}$, as follows:
• ${\mathbf{u}}\left({\mathbf{npde}}×\left(\mathit{j}-1\right)+\mathit{i}\right)$ contain ${U}_{\mathit{i}}\left({x}_{\mathit{j}},t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$, and
• ${\mathbf{u}}\left({\mathbf{npts}}×{\mathbf{npde}}+\mathit{i}\right)$ contain ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
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 d03pjf/d03pja 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:   $\mathbf{nv}$ – IntegerInput
On entry: the number of coupled ODE components.
Constraint: ${\mathbf{nv}}\ge 0$.
14:   $\mathbf{odedef}$ – Subroutine, supplied by the NAG Library or the user.External Procedure
odedef must evaluate the functions $F$, which define the system of ODEs, as given in (3).
If you wish to compute the solution of a system of PDEs only (${\mathbf{nv}}=0$), odedef must be the dummy routine d03pck for d03pjf (or d53pck for d03pja). d03pck and d53pck are included in the NAG Library.
The specification of odedef for d03pjf is:
Fortran Interface
 Subroutine odedef ( npde, t, nv, v, vdot, nxi, xi, ucp, ucpx, rcp, ucpt, f, ires)
 Integer, Intent (In) :: npde, nv, nxi Integer, Intent (Inout) :: ires Real (Kind=nag_wp), Intent (In) :: t, v(nv), vdot(nv), xi(nxi), ucp(npde,nxi), ucpx(npde,nxi), rcp(npde,nxi), ucpt(npde,nxi), ucptx(npde,nxi) Real (Kind=nag_wp), Intent (Out) :: f(nv)
#include nagmk26.h
 void odedef (const Integer *npde, const double *t, const Integer *nv, const double v[], const double vdot[], const Integer *nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], Integer *ires)
The specification of odedef for d03pja is:
Fortran Interface
 Subroutine odedef ( npde, t, nv, v, vdot, nxi, xi, ucp, ucpx, rcp, ucpt, f, ires,
 Integer, Intent (In) :: npde, nv, nxi Integer, Intent (Inout) :: ires, iuser(*) Real (Kind=nag_wp), Intent (In) :: t, v(nv), vdot(nv), xi(nxi), ucp(npde,nxi), ucpx(npde,nxi), rcp(npde,nxi), ucpt(npde,nxi), ucptx(npde,nxi) Real (Kind=nag_wp), Intent (Inout) :: ruser(*) Real (Kind=nag_wp), Intent (Out) :: f(nv)
#include nagmk26.h
 void odedef (const Integer *npde, const double *t, const Integer *nv, const double v[], const double vdot[], const Integer *nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], 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:     $\mathbf{nv}$ – IntegerInput
On entry: the number of coupled ODEs in the system.
4:     $\mathbf{v}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nv}}>0$, ${\mathbf{v}}\left(\mathit{i}\right)$ contains the value of the component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
5:     $\mathbf{vdot}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nv}}>0$, ${\mathbf{vdot}}\left(\mathit{i}\right)$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
6:     $\mathbf{nxi}$ – IntegerInput
On entry: the number of ODE/PDE coupling points.
7:     $\mathbf{xi}\left({\mathbf{nxi}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{xi}}\left(\mathit{i}\right)$ contains the ODE/PDE coupling points, ${\xi }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxi}}$.
8:     $\mathbf{ucp}\left({\mathbf{npde}},{\mathbf{nxi}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{ucp}}\left(\mathit{i},\mathit{j}\right)$ contains the value of ${U}_{\mathit{i}}\left(x,t\right)$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
9:     $\mathbf{ucpx}\left({\mathbf{npde}},{\mathbf{nxi}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{ucpx}}\left(\mathit{i},\mathit{j}\right)$ contains the value of $\frac{\partial {U}_{\mathit{i}}\left(x,t\right)}{\partial x}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
10:   $\mathbf{rcp}\left({\mathbf{npde}},{\mathbf{nxi}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{rcp}}\left(\mathit{i},\mathit{j}\right)$ contains the value of the flux ${R}_{\mathit{i}}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
11:   $\mathbf{ucpt}\left({\mathbf{npde}},{\mathbf{nxi}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{ucpt}}\left(\mathit{i},\mathit{j}\right)$ contains the value of $\frac{\partial {U}_{\mathit{i}}}{\partial t}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
12:   $\mathbf{ucptx}\left({\mathbf{npde}},{\mathbf{nxi}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{ucptx}}\left(\mathit{i},\mathit{j}\right)$ contains the value of $\frac{{\partial }^{2}{U}_{\mathit{i}}}{\partial x\partial t}$ at the coupling point $x={\xi }_{\mathit{j}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.
13:   $\mathbf{f}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{f}}\left(\mathit{i}\right)$ must contain the $\mathit{i}$th component of $F$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$, where $F$ is defined as
 $F=G-AV.-B Ut* Uxt* ,$ (5)
or
 $F=-AV.-B Ut* Uxt* .$ (6)
The definition of $F$ is determined by the input value of ires.
14:   $\mathbf{ires}$ – IntegerInput/Output
On entry: the form of $F$ that must be returned in the array f.
${\mathbf{ires}}=1$
Equation (5) must be used.
${\mathbf{ires}}=-1$
Equation (6) must be used.
On exit: should usually remain unchanged. However, you may reset 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 ${\mathbf{ires}}=3$, d03pjf/d03pja 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 d03pja. Users of d03pjf therefore need not read the remainder of this description.
15:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
16:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
odedef is called with the arguments iuser and ruser as supplied to d03pjf/d03pja. You should use the arrays iuser and ruser to supply information to odedef.
odedef must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03pjf/d03pja is called. Arguments denoted as Input must not be changed by this procedure.
Note: odedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pjf/d03pja. If your code inadvertently does return any NaNs or infinities, d03pjf/d03pja is likely to produce unexpected results.
15:   $\mathbf{nxi}$ – IntegerInput
On entry: the number of ODE/PDE coupling points.
Constraints:
• if ${\mathbf{nv}}=0$, ${\mathbf{nxi}}=0$;
• if ${\mathbf{nv}}>0$, ${\mathbf{nxi}}\ge 0$.
16:   $\mathbf{xi}\left({\mathbf{nxi}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{xi}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxi}}$, must be set to the ODE/PDE coupling points.
Constraint: ${\mathbf{xbkpts}}\left(1\right)\le {\mathbf{xi}}\left(1\right)<{\mathbf{xi}}\left(2\right)<\cdots <{\mathbf{xi}}\left({\mathbf{nxi}}\right)\le {\mathbf{xbkpts}}\left({\mathbf{nbkpts}}\right)$.
17:   $\mathbf{neqn}$ – IntegerInput
On entry: the number of ODEs in the time direction.
Constraint: ${\mathbf{neqn}}={\mathbf{npde}}×{\mathbf{npts}}+{\mathbf{nv}}$.
18:   $\mathbf{uvinit}$ – Subroutine, supplied by the user.External Procedure
uvinit must compute the initial values of the PDE and the ODE components ${U}_{\mathit{i}}\left({x}_{\mathit{j}},{t}_{0}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$, and ${V}_{\mathit{k}}\left({t}_{0}\right)$, for $\mathit{k}=1,2,\dots ,{\mathbf{nv}}$.
The specification of uvinit for d03pjf is:
Fortran Interface
 Subroutine uvinit ( npde, npts, x, u, nv, v)
 Integer, Intent (In) :: npde, npts, nv Real (Kind=nag_wp), Intent (In) :: x(npts) Real (Kind=nag_wp), Intent (Out) :: u(npde,npts), v(nv)
#include nagmk26.h
 void uvinit (const Integer *npde, const Integer *npts, const double x[], double u[], const Integer *nv, double v[])
The specification of uvinit for d03pja is:
Fortran Interface
 Subroutine uvinit ( npde, npts, x, u, nv, v,
 Integer, Intent (In) :: npde, npts, nv 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), v(nv)
#include nagmk26.h
 void uvinit (const Integer *npde, const Integer *npts, const double x[], double u[], const Integer *nv, double v[], 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: ${\mathbf{x}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npts}}$, contains the current values of the space variable ${x}_{\mathit{i}}$.
4:     $\mathbf{u}\left({\mathbf{npde}},{\mathbf{npts}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{u}}\left(\mathit{i},\mathit{j}\right)$ contains the value of the component ${U}_{\mathit{i}}\left({x}_{\mathit{j}},{t}_{0}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
5:     $\mathbf{nv}$ – IntegerInput
On entry: the number of coupled ODEs in the system.
6:     $\mathbf{v}\left({\mathbf{nv}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{v}}\left(\mathit{i}\right)$ contains the value of component ${V}_{\mathit{i}}\left({t}_{0}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
Note: the following are additional arguments for specific use with d03pja. Users of d03pjf therefore need not read the remainder of this description.
7:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
8:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
uvinit is called with the arguments iuser and ruser as supplied to d03pjf/d03pja. You should use the arrays iuser and ruser to supply information to uvinit.
uvinit must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03pjf/d03pja is called. Arguments denoted as Input must not be changed by this procedure.
Note: uvinit should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pjf/d03pja. If your code inadvertently does return any NaNs or infinities, d03pjf/d03pja is likely to produce unexpected results.
19:   $\mathbf{rtol}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array rtol must be at least $1$ if ${\mathbf{itol}}=1$ or $2$ and at least ${\mathbf{neqn}}$ if ${\mathbf{itol}}=3$ or $4$.
On entry: the relative local error tolerance.
Constraint: ${\mathbf{rtol}}\left(i\right)\ge 0.0$ for all relevant $i$.
20:   $\mathbf{atol}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array atol must be at least $1$ if ${\mathbf{itol}}=1$ or $3$ and at least ${\mathbf{neqn}}$ if ${\mathbf{itol}}=2$ or $4$.
On entry: the absolute local error tolerance.
Constraint: ${\mathbf{atol}}\left(i\right)\ge 0.0$ for all relevant $i$.
Note: corresponding elements of rtol and atol cannot both be $0.0$.
21:   $\mathbf{itol}$ – IntegerInput
On entry: a value to indicate the form of the local error test. itol indicates to d03pjf/d03pja whether to interpret either or both of rtol or atol as a vector or scalar. The error test to be satisfied is $‖{e}_{i}/{w}_{i}‖<1.0$, where ${w}_{i}$ is defined as follows:
 itol rtol atol ${w}_{i}$ 1 scalar scalar ${\mathbf{rtol}}\left(1\right)×\left|{U}_{i}\right|+{\mathbf{atol}}\left(1\right)$ 2 scalar vector ${\mathbf{rtol}}\left(1\right)×\left|{U}_{i}\right|+{\mathbf{atol}}\left(i\right)$ 3 vector scalar ${\mathbf{rtol}}\left(i\right)×\left|{U}_{i}\right|+{\mathbf{atol}}\left(1\right)$ 4 vector vector ${\mathbf{rtol}}\left(i\right)×\left|{U}_{i}\right|+{\mathbf{atol}}\left(i\right)$
In the above, ${e}_{\mathit{i}}$ denotes the estimated local error for the $\mathit{i}$th component of the coupled PDE/ODE system in time, ${\mathbf{u}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{neqn}}$.
The choice of norm used is defined by the argument norm.
Constraint: $1\le {\mathbf{itol}}\le 4$.
22:   $\mathbf{norm}$ – Character(1)Input
On entry: the type of norm to be used.
${\mathbf{norm}}=\text{'M'}$
Maximum norm.
${\mathbf{norm}}=\text{'A'}$
Averaged ${L}_{2}$ norm.
If ${{\mathbf{u}}}_{\mathrm{norm}}$ denotes the norm of the vector u of length neqn, then for the averaged ${L}_{2}$ norm
 $unorm=1neqn∑i=1neqnui/wi2,$
while for the maximum norm
 $u norm = maxi ui / wi .$
See the description of itol for the formulation of the weight vector $w$.
Constraint: ${\mathbf{norm}}=\text{'M'}$ or $\text{'A'}$.
23:   $\mathbf{laopt}$ – Character(1)Input
On entry: the type of matrix algebra required.
${\mathbf{laopt}}=\text{'F'}$
Full matrix methods to be used.
${\mathbf{laopt}}=\text{'B'}$
Banded matrix methods to be used.
${\mathbf{laopt}}=\text{'S'}$
Sparse matrix methods to be used.
Constraint: ${\mathbf{laopt}}=\text{'F'}$, $\text{'B'}$ or $\text{'S'}$.
Note: you are recommended to use the banded option when no coupled ODEs are present (i.e., ${\mathbf{nv}}=0$).
24:   $\mathbf{algopt}\left(30\right)$ – Real (Kind=nag_wp) arrayInput
On entry: may be set to control various options available in the integrator. If you wish to employ all the default options, ${\mathbf{algopt}}\left(1\right)$ should be set to $0.0$. Default values will also be used for any other elements of algopt set to zero. The permissible values, default values, and meanings are as follows:
${\mathbf{algopt}}\left(1\right)$
Selects the ODE integration method to be used. If ${\mathbf{algopt}}\left(1\right)=1.0$, a BDF method is used and if ${\mathbf{algopt}}\left(1\right)=2.0$, a Theta method is used. The default value is ${\mathbf{algopt}}\left(1\right)=1.0$.
If ${\mathbf{algopt}}\left(1\right)=2.0$, ${\mathbf{algopt}}\left(\mathit{i}\right)$, for $\mathit{i}=2,3,4$ are not used.
${\mathbf{algopt}}\left(2\right)$
Specifies the maximum order of the BDF integration formula to be used. ${\mathbf{algopt}}\left(2\right)$ may be $1.0$, $2.0$, $3.0$, $4.0$ or $5.0$. The default value is ${\mathbf{algopt}}\left(2\right)=5.0$.
${\mathbf{algopt}}\left(3\right)$
Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the BDF method. If ${\mathbf{algopt}}\left(3\right)=1.0$ a modified Newton iteration is used and if ${\mathbf{algopt}}\left(3\right)=2.0$ a functional iteration method is used. If functional iteration is selected and the integrator encounters difficulty, there is an automatic switch to the modified Newton iteration. The default value is ${\mathbf{algopt}}\left(3\right)=1.0$.
${\mathbf{algopt}}\left(4\right)$
Specifies whether or not the Petzold error test is to be employed. The Petzold error test results in extra overhead but is more suitable when algebraic equations are present, such as ${P}_{i,\mathit{j}}=0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$, for some $i$ or when there is no ${\stackrel{.}{V}}_{i}\left(t\right)$ dependence in the coupled ODE system. If ${\mathbf{algopt}}\left(4\right)=1.0$, the Petzold test is used. If ${\mathbf{algopt}}\left(4\right)=2.0$, the Petzold test is not used. The default value is ${\mathbf{algopt}}\left(4\right)=1.0$.
If ${\mathbf{algopt}}\left(1\right)=1.0$, ${\mathbf{algopt}}\left(\mathit{i}\right)$, for $\mathit{i}=5,6,7$, are not used.
${\mathbf{algopt}}\left(5\right)$
Specifies the value of Theta to be used in the Theta integration method. $0.51\le {\mathbf{algopt}}\left(5\right)\le 0.99$. The default value is ${\mathbf{algopt}}\left(5\right)=0.55$.
${\mathbf{algopt}}\left(6\right)$
Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the Theta method. If ${\mathbf{algopt}}\left(6\right)=1.0$, a modified Newton iteration is used and if ${\mathbf{algopt}}\left(6\right)=2.0$, a functional iteration method is used. The default value is ${\mathbf{algopt}}\left(6\right)=1.0$.
${\mathbf{algopt}}\left(7\right)$
Specifies whether or not the integrator is allowed to switch automatically between modified Newton and functional iteration methods in order to be more efficient. If ${\mathbf{algopt}}\left(7\right)=1.0$, switching is allowed and if ${\mathbf{algopt}}\left(7\right)=2.0$, switching is not allowed. The default value is ${\mathbf{algopt}}\left(7\right)=1.0$.
${\mathbf{algopt}}\left(11\right)$
Specifies a point in the time direction, ${t}_{\mathrm{crit}}$, beyond which integration must not be attempted. The use of ${t}_{\mathrm{crit}}$ is described under the argument itask. If ${\mathbf{algopt}}\left(1\right)\ne 0.0$, a value of $0.0$ for ${\mathbf{algopt}}\left(11\right)$, say, should be specified even if itask subsequently specifies that ${t}_{\mathrm{crit}}$ will not be used.
${\mathbf{algopt}}\left(12\right)$
Specifies the minimum absolute step size to be allowed in the time integration. If this option is not required, ${\mathbf{algopt}}\left(12\right)$ should be set to $0.0$.
${\mathbf{algopt}}\left(13\right)$
Specifies the maximum absolute step size to be allowed in the time integration. If this option is not required, ${\mathbf{algopt}}\left(13\right)$ should be set to $0.0$.
${\mathbf{algopt}}\left(14\right)$
Specifies the initial step size to be attempted by the integrator. If ${\mathbf{algopt}}\left(14\right)=0.0$, the initial step size is calculated internally.
${\mathbf{algopt}}\left(15\right)$
Specifies the maximum number of steps to be attempted by the integrator in any one call. If ${\mathbf{algopt}}\left(15\right)=0.0$, no limit is imposed.
${\mathbf{algopt}}\left(23\right)$
Specifies what method is to be used to solve the nonlinear equations at the initial point to initialize the values of $U$, ${U}_{t}$, $V$ and $\stackrel{.}{V}$. If ${\mathbf{algopt}}\left(23\right)=1.0$, a modified Newton iteration is used and if ${\mathbf{algopt}}\left(23\right)=2.0$, functional iteration is used. The default value is ${\mathbf{algopt}}\left(23\right)=1.0$.
${\mathbf{algopt}}\left(29\right)$ and ${\mathbf{algopt}}\left(30\right)$ are used only for the sparse matrix algebra option, ${\mathbf{laopt}}=\text{'S'}$.
${\mathbf{algopt}}\left(29\right)$
Governs the choice of pivots during the decomposition of the first Jacobian matrix. It should lie in the range $0.0<{\mathbf{algopt}}\left(29\right)<1.0$, with smaller values biasing the algorithm towards maintaining sparsity at the expense of numerical stability. If ${\mathbf{algopt}}\left(29\right)$ lies outside this range then the default value is used. If the routines regard the Jacobian matrix as numerically singular then increasing ${\mathbf{algopt}}\left(29\right)$ towards $1.0$ may help, but at the cost of increased fill-in. The default value is ${\mathbf{algopt}}\left(29\right)=0.1$.
${\mathbf{algopt}}\left(30\right)$
Is used as a relative pivot threshold during subsequent Jacobian decompositions (see ${\mathbf{algopt}}\left(29\right)$) below which an internal error is invoked. If ${\mathbf{algopt}}\left(30\right)$ is greater than $1.0$ no check is made on the pivot size, and this may be a necessary option if the Jacobian is found to be numerically singular (see ${\mathbf{algopt}}\left(29\right)$). The default value is ${\mathbf{algopt}}\left(30\right)=0.0001$.
25:   $\mathbf{rsave}\left({\mathbf{lrsave}}\right)$ – 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.
26:   $\mathbf{lrsave}$ – IntegerInput
On entry: the dimension of the array rsave as declared in the (sub)program from which d03pjf/d03pja is called. Its size depends on the type of matrix algebra selected.
If ${\mathbf{laopt}}=\text{'F'}$, ${\mathbf{lrsave}}\ge {\mathbf{neqn}}×{\mathbf{neqn}}+{\mathbf{neqn}}+\mathit{nwkres}+\mathit{lenode}$.
If ${\mathbf{laopt}}=\text{'B'}$, ${\mathbf{lrsave}}\ge \left(3×\mathit{mlu}+1\right)×{\mathbf{neqn}}+\mathit{nwkres}+\mathit{lenode}$.
If ${\mathbf{laopt}}=\text{'S'}$, ${\mathbf{lrsave}}\ge 4×{\mathbf{neqn}}+11×{\mathbf{neqn}}/2+1+\mathit{nwkres}+\mathit{lenode}$.
Where
 $\mathit{mlu}$ is the lower or upper half bandwidths such that $\mathit{mlu}=3×{\mathbf{npde}}-1$, for PDE problems only (no coupled ODEs); or $\mathit{mlu}={\mathbf{neqn}}-1$, for coupled PDE/ODE problems. $\mathit{nwkres}=\left\{\begin{array}{ll}3×{\left({\mathbf{npoly}}+1\right)}^{2}+\left({\mathbf{npoly}}+1\right)×\left[{{\mathbf{npde}}}^{2}+6×{\mathbf{npde}}+{\mathbf{nbkpts}}+1\right]+8×{\mathbf{npde}}+{\mathbf{nxi}}×\left(5×{\mathbf{npde}}+1\right)+{\mathbf{nv}}+3\text{,}& \text{when ​}{\mathbf{nv}}>0\text{​ and ​}{\mathbf{nxi}}>0\text{; or}\\ 3×{\left({\mathbf{npoly}}+1\right)}^{2}+\left({\mathbf{npoly}}+1\right)×\left[{{\mathbf{npde}}}^{2}+6×{\mathbf{npde}}+{\mathbf{nbkpts}}+1\right]+13×{\mathbf{npde}}+{\mathbf{nv}}+4\text{,}& \text{when ​}{\mathbf{nv}}>0\text{​ and ​}{\mathbf{nxi}}=0\text{; or}\\ 3×{\left({\mathbf{npoly}}+1\right)}^{2}+\left({\mathbf{npoly}}+1\right)×\left[{{\mathbf{npde}}}^{2}+6×{\mathbf{npde}}+{\mathbf{nbkpts}}+1\right]+13×{\mathbf{npde}}+5\text{,}& \text{when ​}{\mathbf{nv}}=0\text{.}\end{array}\right\$ $\mathit{lenode}=\left\{\begin{array}{ll}\left(6+\mathrm{int}\left({\mathbf{algopt}}\left(2\right)\right)\right)×{\mathbf{neqn}}+50\text{,}& \text{when the BDF method is used; or}\\ 9×{\mathbf{neqn}}+50\text{,}& \text{when the Theta method is used.}\end{array}\right\$
Note: when ${\mathbf{laopt}}=\text{'S'}$, the value of lrsave may be too small when supplied to the integrator. An estimate of the minimum size of lrsave is printed on the current error message unit if ${\mathbf{itrace}}>0$ and the routine returns with ${\mathbf{ifail}}={\mathbf{15}}$.
27:   $\mathbf{isave}\left({\mathbf{lisave}}\right)$ – 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 required for subsequent calls. 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 ODE method last used in the time integration.
${\mathbf{isave}}\left(5\right)$
Contains the number of Newton iterations performed by the time integrator. Each iteration involves residual evaluation of the resulting ODE system followed by a back-substitution using the $LU$ decomposition of the Jacobian matrix.
28:   $\mathbf{lisave}$ – IntegerInput
On entry: the dimension of the array isave as declared in the (sub)program from which d03pjf/d03pja is called. Its size depends on the type of matrix algebra selected:
• if ${\mathbf{laopt}}=\text{'F'}$, ${\mathbf{lisave}}\ge 24$;
• if ${\mathbf{laopt}}=\text{'B'}$, ${\mathbf{lisave}}\ge {\mathbf{neqn}}+24$;
• if ${\mathbf{laopt}}=\text{'S'}$, ${\mathbf{lisave}}\ge 25×{\mathbf{neqn}}+24$.
Note: when using the sparse option, the value of lisave may be too small when supplied to the integrator. An estimate of the minimum size of lisave is printed on the current error message unit if ${\mathbf{itrace}}>0$ and the routine returns with ${\mathbf{ifail}}={\mathbf{15}}$.
29:   $\mathbf{itask}$ – 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 $t={\mathbf{tout}}$.
${\mathbf{itask}}=4$
Normal computation of output values u at $t={\mathbf{tout}}$ but without overshooting $t={t}_{\mathrm{crit}}$ where ${t}_{\mathrm{crit}}$ is described under the argument algopt.
${\mathbf{itask}}=5$
Take one step in the time direction and return, without passing ${t}_{\mathrm{crit}}$, where ${t}_{\mathrm{crit}}$ is described under the argument algopt.
Constraint: ${\mathbf{itask}}=1$, $2$, $3$, $4$ or $5$.
30:   $\mathbf{itrace}$ – IntegerInput
On entry: the level of trace information required from d03pjf/d03pja 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.
31:   $\mathbf{ind}$ – 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 d03pjf/d03pja.
Constraint: ${\mathbf{ind}}=0$ or $1$.
On exit: ${\mathbf{ind}}=1$.
32:   $\mathbf{ifail}$ – IntegerInput/Output
Note: for d03pja, 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 d03pja. Users of d03pjf therefore need not read the remainder of this description.
32:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace
33:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace
iuser and ruser are not used by d03pjf/d03pja, but are passed directly to pdedef, bndary, odedef and uvinit and may be used to pass information to these routines.
34:   $\mathbf{cwsav}\left(10\right)$ – Character(80) arrayCommunication Array
35:   $\mathbf{lwsav}\left(100\right)$ – Logical arrayCommunication Array
36:   $\mathbf{iwsav}\left(505\right)$ – Integer arrayCommunication Array
37:   $\mathbf{rwsav}\left(1100\right)$ – Real (Kind=nag_wp) arrayCommunication Array
38:   $\mathbf{ifail}$ – 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, ${\mathbf{tout}}-{\mathbf{ts}}$ is too small, or ${\mathbf{itask}}\ne 1$, $2$, $3$, $4$ or $5$, or ${\mathbf{m}}\ne 0$, $1$ or $2$, or at least one of the coupling point in array xi is outside the interval [${\mathbf{xbkpts}}\left(1\right),{\mathbf{xbkpts}}\left({\mathbf{nbkpts}}\right)$], or ${\mathbf{npts}}\ne \left({\mathbf{nbkpts}}-1\right)×{\mathbf{npoly}}+1$, or ${\mathbf{nbkpts}}<2$, or ${\mathbf{npde}}\le 0$, or ${\mathbf{norm}}\ne \text{'A'}$ or $\text{'M'}$, or ${\mathbf{itol}}\ne 1$, $2$, $3$ or $4$, or ${\mathbf{npoly}}<1$ or ${\mathbf{npoly}}>49$, or nv and nxi are incorrectly defined, or ${\mathbf{neqn}}\ne {\mathbf{npde}}×{\mathbf{npts}}+{\mathbf{nv}}$, or ${\mathbf{laopt}}\ne \text{'F'}$, $\text{'B'}$ or $\text{'S'}$, or ${\mathbf{ind}}\ne 0$ or $1$, or break-points ${\mathbf{xbkpts}}\left(i\right)$ are badly ordered, or lrsave is too small, or lisave is too small, or the ODE integrator has not been correctly defined; check algopt argument, or either an element of rtol or ${\mathbf{atol}}<0.0$, or all the elements of rtol and atol are zero.
${\mathbf{ifail}}=2$
The underlying ODE solver cannot make any further progress, with the values of atol and rtol, across the integration range from the current point $t={\mathbf{ts}}$. 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 error test failures on an attempted step, before completing the requested task, but the integration was successful as far as $t={\mathbf{ts}}$. The problem may have a singularity, or the error requirement may be inappropriate.
${\mathbf{ifail}}=4$
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, bndary or odedef, 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, bndary or odedef. Integration was successful as far as $t={\mathbf{ts}}$.
${\mathbf{ifail}}=7$
The values of atol and rtol are so small that the routine is unable to start the integration in time.
${\mathbf{ifail}}=8$
In one of pdedef, bndary or odedef, 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 atol and rtol is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when ${\mathbf{itask}}\ne 2$ or $5$.)
${\mathbf{ifail}}=11$
An error occurred during Jacobian formulation of the ODE system (a more detailed error description may be directed to the current error message unit).
${\mathbf{ifail}}=12$
In solving the ODE system, the maximum number of steps specified in ${\mathbf{algopt}}\left(15\right)$ have been taken.
${\mathbf{ifail}}=13$
Some error weights ${w}_{i}$ became zero during the time integration (see the description of itol). Pure relative error control (${\mathbf{atol}}\left(i\right)=0.0$) was requested on a variable (the $i$th) which has become zero. The integration was successful as far as $t={\mathbf{ts}}$.
${\mathbf{ifail}}=14$
The flux function ${R}_{i}$ was detected as depending on time derivatives, which is not permissible.
${\mathbf{ifail}}=15$
When using the sparse option, the value of lisave or lrsave was not sufficient (more detailed information may be directed to the current error message unit).
${\mathbf{ifail}}=-99$
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

d03pjf/d03pja controls the accuracy of the integration in the time direction but not the accuracy of the approximation in space. The spatial accuracy depends on both the number of mesh 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 atol and rtol.

## 8Parallelism and Performance

d03pjf/d03pja is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d03pjf/d03pja 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.

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

This example provides a simple coupled system of one PDE and one ODE.
 $V 1 2 ∂ U 1 ∂ t -x V 1 V . 1 ∂ U 1 ∂ x = ∂ 2 U 1 ∂ x 2 V . 1 = V 1 U 1 + ∂ U 1 ∂ x +1 +t ,$
for $t\in \left[{10}^{-4},0.1×{2}^{i}\right]\text{, }i=1,2,\dots ,5,x\in \left[0,1\right]$.
The left boundary condition at $x=0$ is
 $∂U1 ∂x =-V1exp⁡t.$
The right boundary condition at $x=1$ is
 $U1=-V1V.1.$
The initial conditions at $t={10}^{-4}$ are defined by the exact solution:
 $V1=t, and U1x,t=expt1-x-1.0, x∈0,1,$
and the coupling point is at ${\xi }_{1}=1.0$.

### 10.1Program Text

Note: the following programs illustrate the use of d03pjf and d03pja.

Program Text (d03pjfe.f90)

Program Text (d03pjae.f90)

### 10.2Program Data

Program Data (d03pjfe.d)

Program Data (d03pjae.d)

### 10.3Program Results

Program Results (d03pjfe.r)

Program Results (d03pjae.r)

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