NAG CL Interface
d03plc (dim1_parab_convdiff_dae)
1
Purpose
d03plc integrates a system of linear or nonlinear convectiondiffusion equations in one space dimension, with optional source terms and scope for coupled ordinary differential equations (ODEs). The system must be posed in conservative form. Convection terms are discretized using a sophisticated upwind scheme involving a usersupplied numerical flux function based on the solution of a Riemann problem at each mesh point. The method of lines is employed to reduce the partial differential equations (PDEs) to a system of ODEs, and the resulting system is solved using a backward differentiation formula (BDF) method or a Theta method.
2
Specification
void 
d03plc (Integer npde,
double *ts,
double tout,
void 
(*pdedef)(Integer npde,
double t,
double x,
const double u[],
const double ux[],
Integer nv,
const double v[],
const double vdot[],
double p[],
double c[],
double d[],
double s[],
Integer *ires,
Nag_Comm *comm),


void 
(*numflx)(Integer npde,
double t,
double x,
Integer nv,
const double v[],
const double uleft[],
const double uright[],
double flux[],
Integer *ires,
Nag_Comm *comm, Nag_D03_Save *saved),


void 
(*bndary)(Integer npde,
Integer npts,
double t,
const double x[],
const double u[],
Integer nv,
const double v[],
const double vdot[],
Integer ibnd,
double g[],
Integer *ires,
Nag_Comm *comm),


double u[],
Integer npts,
const double x[],
Integer nv,
void 
(*odedef)(Integer npde,
double t,
Integer nv,
const double v[],
const double vdot[],
Integer nxi,
const double xi[],
const double ucp[],
const double ucpx[],
const double ucpt[],
double r[],
Integer *ires,
Nag_Comm *comm),


Integer nxi,
const double xi[],
Integer neqn,
const double rtol[],
const double atol[],
Integer itol,
Nag_NormType norm,
Nag_LinAlgOption laopt,
const double algopt[],
double rsave[],
Integer lrsave,
Integer isave[],
Integer lisave,
Integer itask,
Integer itrace,
const char *outfile,
Integer *ind,
Nag_Comm *comm, Nag_D03_Save *saved,
NagError *fail) 

The function may be called by the names: d03plc, nag_pde_dim1_parab_convdiff_dae or nag_pde_parab_1d_cd_ode.
3
Description
d03plc integrates the system of convectiondiffusion equations in conservative form:
or the hyperbolic convectiononly system:
for
$i=1,2,\dots ,{\mathbf{npde}}\text{, \hspace{1em}}a\le x\le b\text{, \hspace{1em}}t\ge {t}_{0}$, where the vector
$U$ is the set of PDE solution values
The optional coupled ODEs are of the general form
where the vector
$V$ is the set of ODE solution values
$\stackrel{.}{V}$ denotes its derivative with respect to time, and
${U}_{x}$ is the spatial derivative of
$U$.
In
(1),
${P}_{i,j}$,
${F}_{i}$ and
${C}_{i}$ depend on
$x$,
$t$,
$U$ and
$V$;
${D}_{i}$ depends on
$x$,
$t$,
$U$,
${U}_{x}$ and
$V$; and
${S}_{i}$ depends on
$x$,
$t$,
$U$,
$V$ and
linearly on
$\stackrel{.}{V}$. Note that
${P}_{i,j}$,
${F}_{i}$,
${C}_{i}$ and
${S}_{i}$ must not depend on any space derivatives, and
${P}_{i,j}$,
${F}_{i}$,
${C}_{i}$ and
${D}_{i}$ must not depend on any time derivatives. In terms of conservation laws,
${F}_{i}$,
$\frac{{C}_{i}\partial {D}_{i}}{\partial x}$ and
${S}_{i}$ are the convective flux, diffusion and source terms respectively.
In
(3),
$\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 PDE spatial mesh points.
${U}^{*}$,
${U}_{x}^{*}$ and
${U}_{t}^{*}$ are the functions
$U$,
${U}_{x}$ and
${U}_{t}$ evaluated at these coupling points. Each
${R}_{i}$ may depend only linearly on time derivatives. Hence
(3) may be written more precisely as
where
$R={\left[{R}_{1},\dots ,{R}_{{\mathbf{nv}}}\right]}^{\mathrm{T}}$,
$L$ is a vector of length
nv,
$M$ is an
nv by
nv matrix,
$N$ is an
nv by
$\left({n}_{\xi}\times {\mathbf{npde}}\right)$ matrix and the entries in
$L$,
$M$ and
$N$ may depend on
$t$,
$\xi $,
${U}^{*}$,
${U}_{x}^{*}$ and
$V$. In practice you only need to supply a vector of information to define the ODEs and not the matrices
$L$,
$M$ and
$N$. (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 $b={x}_{{\mathbf{npts}}}$ are the leftmost and rightmost points of a userdefined mesh ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{npts}}}$. The initial values of the functions $U\left(x,t\right)$ and $V\left(t\right)$ must be given at $t={t}_{0}$.
The PDEs are approximated by a system of ODEs in time for the values of
${U}_{i}$ at mesh points using a spatial discretization method similar to the centraldifference scheme used in
d03pcc,
d03phc and
d03ppc, but with the flux
${F}_{i}$ replaced by a
numerical flux, which is a representation of the flux taking into account the direction of the flow of information at that point (i.e., the direction of the characteristics). Simple central differencing of the numerical flux then becomes a sophisticated upwind scheme in which the correct direction of upwinding is automatically achieved.
The numerical flux vector,
${\hat{F}}_{i}$ say, must be calculated by you in terms of the
left and
right values of the solution vector
$U$ (denoted by
${U}_{L}$ and
${U}_{R}$ respectively), at each midpoint of the mesh
${x}_{\mathit{j}\frac{1}{2}}=\left({x}_{\mathit{j}1}+{x}_{\mathit{j}}\right)/2$, for
$\mathit{j}=2,3,\dots ,{\mathbf{npts}}$. The left and right values are calculated by
d03plc from two adjacent mesh points using a standard upwind technique combined with a Van Leer slopelimiter (see
LeVeque (1990)). The physically correct value for
${\hat{F}}_{i}$ is derived from the solution of the Riemann problem given by
where
$y=x{x}_{j\frac{1}{2}}$, i.e.,
$y=0$ corresponds to
$x={x}_{j\frac{1}{2}}$, with discontinuous initial values
$U={U}_{L}$ for
$y<0$ and
$U={U}_{R}$ for
$y>0$, using an
approximate Riemann solver. This applies for either of the systems
(1) or
(2); the numerical flux is independent of the functions
${P}_{i,j}$,
${C}_{i}$,
${D}_{i}$ and
${S}_{i}$. A description of several approximate Riemann solvers can be found in
LeVeque (1990) and
Berzins et al. (1989). Roe's scheme (see
Roe (1981)) is perhaps the easiest to understand and use, and a brief summary follows. Consider the system of PDEs
${U}_{t}+{F}_{x}=0$ or equivalently
${U}_{t}+A{U}_{x}=0$. Provided the system is linear in
$U$, i.e., the Jacobian matrix
$A$ does not depend on
$U$, the numerical flux
$\hat{F}$ is given by
where
${F}_{L}$ (
${F}_{R}$) is the flux
$F$ calculated at the left (right) value of
$U$, denoted by
${U}_{L}$ (
${U}_{R}$); the
${\lambda}_{k}$ are the eigenvalues of
$A$; the
${e}_{k}$ are the right eigenvectors of
$A$; and the
${\alpha}_{k}$ are defined by
An example is given in
Section 10 and in the
d03pfc documentation.
If the system is nonlinear, Roe's scheme requires that a linearized Jacobian is found (see
Roe (1981)).
The functions
${P}_{i,j}$,
${C}_{i}$,
${D}_{i}$ and
${S}_{i}$ (but
not
${F}_{i}$) must be specified in
pdedef. The numerical flux
${\hat{F}}_{i}$ must be supplied in a separate
numflx.
For problems in the form
(2)) the NAG defined null void function pointer, NULLFN, can be supplied in the call to
d03plc.
The boundary condition specification has sufficient flexibility to allow for different types of problems. For secondorder problems, i.e.,
${D}_{i}$ depending on
${U}_{x}$, a boundary condition is required for each PDE at both boundaries for the problem to be wellposed. If there are no secondorder terms present, then the continuous PDE problem generally requires exactly one boundary condition for each PDE, that is
npde boundary conditions in total. However, in common with most discretization schemes for firstorder problems, a
numerical boundary condition is required at the other boundary for each PDE. In order to be consistent with the characteristic directions of the PDE system, the numerical boundary conditions must be derived from the solution inside the domain in some manner (see below). You must supply both types of boundary condition, i.e., a total of
npde conditions at each boundary point.
The position of each boundary condition should be chosen with care. In simple terms, if information is flowing into the domain then a physical boundary condition is required at that boundary, and a numerical boundary condition is required at the other boundary. In many cases the boundary conditions are simple, e.g., for the linear advection equation. In general you should calculate the characteristics of the PDE system and specify a physical boundary condition for each of the characteristic variables associated with incoming characteristics, and a numerical boundary condition for each outgoing characteristic.
A common way of providing numerical boundary conditions is to extrapolate the characteristic variables from the inside of the domain (note that when using banded matrix algebra the fixed bandwidth means that only linear extrapolation is allowed, i.e., using information at just two interior points adjacent to the boundary). For problems in which the solution is known to be uniform (in space) towards a boundary during the period of integration then extrapolation is unnecessary; the numerical boundary condition can be supplied as the known solution at the boundary. Another method of supplying numerical boundary conditions involves the solution of the characteristic equations associated with the outgoing characteristics. Examples of both methods can be found in
Section 10 and in the
d03pfc documentation.
The boundary conditions must be specified in
bndary in the form
at the lefthand boundary, and
at the righthand boundary.
Note that spatial derivatives at the boundary are not passed explicitly to
bndary, but they can be calculated using values of
$U$ at and adjacent to the boundaries if required. However, it should be noted that instabilities may occur if such onesided differencing opposes the characteristic direction at the boundary.
The algebraicdifferential equation system which is defined by the functions
${R}_{i}$ must be specified in
odedef. You must also specify the coupling points
$\xi $ (if any) in the array
xi.
The problem is subject to the following restrictions:

(i)In (1), ${\stackrel{.}{V}}_{j}\left(t\right)$, for $j=1,2,\dots ,{\mathbf{nv}}$, may only appear linearly in the functions ${S}_{i}$, for $i=1,2,\dots ,{\mathbf{npde}}$, with a similar restriction for ${G}_{i}^{L}$ and ${G}_{i}^{R}$;

(ii)${P}_{i,j}$, ${F}_{i}$, ${C}_{i}$ and ${S}_{i}$ must not depend on any space derivatives; and ${P}_{i,j}$, ${F}_{i}$, ${C}_{i}$ and ${D}_{i}$ 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 terms ${P}_{i,j}$, ${C}_{i}$, ${D}_{i}$ and ${S}_{i}$ is done by calling the pdedef at a point approximately midway between each pair of mesh points in turn. Any discontinuities in these functions must therefore be at one or more of the mesh points ${x}_{1},{x}_{2},\dots ,{x}_{{\mathbf{npts}}}$;

(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.
In total there are
${\mathbf{npde}}\times {\mathbf{npts}}+{\mathbf{nv}}$ ODEs in the time direction. This system is then integrated forwards in time using a BDF or Theta method, optionally switching between Newton's method and functional iteration (see
Berzins et al. (1989)).
For further details of the scheme, see
Pennington and Berzins (1994) and the references therein.
4
References
Berzins M, Dew P M and Furzeland R M (1989) Developing software for timedependent problems using the method of lines and differentialalgebraic integrators Appl. Numer. Math. 5 375–397
Hirsch C (1990) Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows John Wiley
LeVeque R J (1990) Numerical Methods for Conservation Laws Birkhäuser Verlag
Pennington S V and Berzins M (1994) New NAG Library software for firstorder partial differential equations ACM Trans. Math. Softw. 20 63–99
Roe P L (1981) Approximate Riemann solvers, parameter vectors, and difference schemes J. Comput. Phys. 43 357–372
Sod G A (1978) A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws J. Comput. Phys. 27 1–31
5
Arguments

1:
$\mathbf{npde}$ – Integer
Input

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

2:
$\mathbf{ts}$ – double *
Input/Output

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

3:
$\mathbf{tout}$ – double
Input

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

4:
$\mathbf{pdedef}$ – function, supplied by the user
External Function

pdedef must evaluate the functions
${P}_{i,j}$,
${C}_{i}$,
${D}_{i}$ and
${S}_{i}$ which partially define the system of PDEs.
${P}_{i,j}$ and
${C}_{i}$ may depend on
$x$,
$t$,
$U$ and
$V$;
${D}_{i}$ may depend on
$x$,
$t$,
$U$,
${U}_{x}$ and
$V$; and
${S}_{i}$ may depend on
$x$,
$t$,
$U$,
$V$ and linearly on
$\stackrel{.}{V}$.
pdedef is called approximately midway between each pair of mesh points in turn by
d03plc. For problems in the form
(2)) the NAG defined null void function pointer, NULLFN, can be supplied in the call to
d03plc.
The specification of
pdedef is:
void 
pdedef (Integer npde,
double t,
double x,
const double u[],
const double ux[],
Integer nv,
const double v[],
const double vdot[],
double p[],
double c[],
double d[],
double s[],
Integer *ires,
Nag_Comm *comm)



1:
$\mathbf{npde}$ – Integer
Input

On entry: the number of PDEs in the system.

2:
$\mathbf{t}$ – double
Input

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

3:
$\mathbf{x}$ – double
Input

On entry: the current value of the space variable $x$.

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

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

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

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

6:
$\mathbf{nv}$ – Integer
Input

On entry: the number of coupled ODEs in the system.

7:
$\mathbf{v}\left[{\mathbf{nv}}\right]$ – const double
Input

On entry: if ${\mathbf{nv}}>0$, ${\mathbf{v}}\left[\mathit{i}1\right]$ contains the value of the component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.

8:
$\mathbf{vdot}\left[{\mathbf{nv}}\right]$ – const double
Input

On entry: if
${\mathbf{nv}}>0$,
${\mathbf{vdot}}\left[\mathit{i}1\right]$ 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
${S}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.

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

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

10:
$\mathbf{c}\left[{\mathbf{npde}}\right]$ – double
Output

On exit: ${\mathbf{c}}\left[\mathit{i}1\right]$ must be set to the value of ${C}_{\mathit{i}}\left(x,t,U,V\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.

11:
$\mathbf{d}\left[{\mathbf{npde}}\right]$ – double
Output

On exit: ${\mathbf{d}}\left[\mathit{i}1\right]$ must be set to the value of ${D}_{\mathit{i}}\left(x,t,U,{U}_{x},V\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.

12:
$\mathbf{s}\left[{\mathbf{npde}}\right]$ – double
Output

On exit: ${\mathbf{s}}\left[\mathit{i}1\right]$ must be set to the value of ${S}_{\mathit{i}}\left(x,t,U,V,\stackrel{.}{V}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.

13:
$\mathbf{ires}$ – Integer *
Input/Output

On entry: set to $1\text{or}1$.
On exit: should usually remain unchanged. However, you may set
ires to force the integration function to take certain actions as described below:
 ${\mathbf{ires}}=2$
 Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
 ${\mathbf{ires}}=3$
 Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{ires}}=3$ when a physically meaningless input or output value has been generated. If you consecutively set ${\mathbf{ires}}=3$, d03plc returns to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.

14:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
pdedef.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
d03plc you may allocate memory and initialize these pointers with various quantities for use by
pdedef when called from
d03plc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: pdedef should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
d03plc. If your code inadvertently
does return any NaNs or infinities,
d03plc is likely to produce unexpected results.

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

numflx must supply the numerical flux for each PDE given the
left and
right values of the solution vector
${\mathbf{u}}$.
numflx is called approximately midway between each pair of mesh points in turn by
d03plc.
The specification of
numflx is:
void 
numflx (Integer npde,
double t,
double x,
Integer nv,
const double v[],
const double uleft[],
const double uright[],
double flux[],
Integer *ires,
Nag_Comm *comm, Nag_D03_Save *saved)



1:
$\mathbf{npde}$ – Integer
Input

On entry: the number of PDEs in the system.

2:
$\mathbf{t}$ – double
Input

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

3:
$\mathbf{x}$ – double
Input

On entry: the current value of the space variable $x$.

4:
$\mathbf{nv}$ – Integer
Input

On entry: the number of coupled ODEs in the system.

5:
$\mathbf{v}\left[{\mathbf{nv}}\right]$ – const double
Input

On entry: if ${\mathbf{nv}}>0$, ${\mathbf{v}}\left[\mathit{i}1\right]$ contains the value of the component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.

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

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

7:
$\mathbf{uright}\left[{\mathbf{npde}}\right]$ – const double
Input

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

8:
$\mathbf{flux}\left[{\mathbf{npde}}\right]$ – double
Output

On exit: ${\mathbf{flux}}\left[\mathit{i}1\right]$ must be set to the numerical flux ${\hat{F}}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.

9:
$\mathbf{ires}$ – Integer *
Input/Output

On entry: set to $1\text{or}1$.
On exit: should usually remain unchanged. However, you may set
ires to force the integration function to take certain actions as described below:
 ${\mathbf{ires}}=2$
 Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
 ${\mathbf{ires}}=3$
 Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{ires}}=3$ when a physically meaningless input or output value has been generated. If you consecutively set ${\mathbf{ires}}=3$, d03plc returns to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.

10:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
numflx.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
d03plc you may allocate memory and initialize these pointers with various quantities for use by
numflx when called from
d03plc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).

11:
$\mathbf{saved}$ – Nag_D03_Save *
Communication Structure

If
numflx calls one of the approximate Riemann solvers
d03puc,
d03pvc,
d03pwc or
d03pxc then
saved is used to pass data concerning the computation to the solver. You should not change the components of
saved.
Note: numflx should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
d03plc. If your code inadvertently
does return any NaNs or infinities,
d03plc is likely to produce unexpected results.

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

bndary must evaluate the functions
${G}_{i}^{L}$ and
${G}_{i}^{R}$ which describe the physical and numerical boundary conditions, as given by
(8) and
(9).
The specification of
bndary is:
void 
bndary (Integer npde,
Integer npts,
double t,
const double x[],
const double u[],
Integer nv,
const double v[],
const double vdot[],
Integer ibnd,
double g[],
Integer *ires,
Nag_Comm *comm)



1:
$\mathbf{npde}$ – Integer
Input

On entry: the number of PDEs in the system.

2:
$\mathbf{npts}$ – Integer
Input

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

3:
$\mathbf{t}$ – double
Input

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

4:
$\mathbf{x}\left[{\mathbf{npts}}\right]$ – const double
Input

On entry: the mesh points in the spatial direction. ${\mathbf{x}}\left[0\right]$ corresponds to the lefthand boundary, $a$, and ${\mathbf{x}}\left[{\mathbf{npts}}1\right]$ corresponds to the righthand boundary, $b$.

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

Note: the $\left(i,j\right)$th element of the matrix $U$ is stored in ${\mathbf{u}}\left[\left(j1\right)\times {\mathbf{npde}}+i1\right]$.
On entry:
${\mathbf{u}}\left[\left(\mathit{j}1\right)\times {\mathbf{npde}}+\mathit{i}1\right]$ contains the value of the component
${U}_{\mathit{i}}\left(x,t\right)$ at
$x={\mathbf{x}}\left[\mathit{j}1\right]$, for
$\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and
$\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
Note: if banded matrix algebra is to be used then the functions ${G}_{\mathit{i}}^{L}$ and ${G}_{\mathit{i}}^{R}$ may depend on the value of ${U}_{\mathit{i}}\left(x,t\right)$ at the boundary point and the two adjacent points only.

6:
$\mathbf{nv}$ – Integer
Input

On entry: the number of coupled ODEs in the system.

7:
$\mathbf{v}\left[{\mathbf{nv}}\right]$ – const double
Input

On entry: if ${\mathbf{nv}}>0$, ${\mathbf{v}}\left[\mathit{i}1\right]$ contains the value of the component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.

8:
$\mathbf{vdot}\left[{\mathbf{nv}}\right]$ – const double
Input

On entry: if
${\mathbf{nv}}>0$,
${\mathbf{vdot}}\left[\mathit{i}1\right]$ 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
${G}_{\mathit{j}}^{L}$ and ${G}_{\mathit{j}}^{R}$, for $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.

9:
$\mathbf{ibnd}$ – Integer
Input

On entry: specifies which boundary conditions are to be evaluated.
 ${\mathbf{ibnd}}=0$
 bndary must evaluate the lefthand boundary condition at $x=a$.
 ${\mathbf{ibnd}}\ne 0$
 bndary must evaluate the righthand boundary condition at $x=b$.

10:
$\mathbf{g}\left[{\mathbf{npde}}\right]$ – double
Output

On exit:
${\mathbf{g}}\left[\mathit{i}1\right]$ must contain the
$\mathit{i}$th component of either
${G}_{\mathit{i}}^{L}$ or
${G}_{\mathit{i}}^{R}$ in
(8) and
(9), depending on the value of
ibnd, for
$\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.

11:
$\mathbf{ires}$ – Integer *
Input/Output

On entry: set to $1\text{or}1$.
On exit: should usually remain unchanged. However, you may set
ires to force the integration function to take certain actions as described below:
 ${\mathbf{ires}}=2$
 Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
 ${\mathbf{ires}}=3$
 Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{ires}}=3$ when a physically meaningless input or output value has been generated. If you consecutively set ${\mathbf{ires}}=3$, d03plc returns to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.

12:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
bndary.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
d03plc you may allocate memory and initialize these pointers with various quantities for use by
bndary when called from
d03plc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: bndary should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
d03plc. If your code inadvertently
does return any NaNs or infinities,
d03plc is likely to produce unexpected results.

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

On entry: the initial values of the dependent variables defined as follows:

${\mathbf{u}}\left[{\mathbf{npde}}\times \left(\mathit{j}1\right)+\mathit{i}1\right]$ contain ${U}_{\mathit{i}}\left({x}_{\mathit{j}},{t}_{0}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$, and

${\mathbf{u}}\left[{\mathbf{npts}}\times {\mathbf{npde}}+\mathit{i}1\right]$ contain ${V}_{\mathit{i}}\left({t}_{0}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.
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:

${\mathbf{u}}\left[{\mathbf{npde}}\times \left(\mathit{j}1\right)+\mathit{i}1\right]$ 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

${\mathbf{u}}\left[{\mathbf{npts}}\times {\mathbf{npde}}+\mathit{i}1\right]$ contain ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.

8:
$\mathbf{npts}$ – Integer
Input

On entry: the number of mesh points in the interval $\left[a,b\right]$.
Constraint:
${\mathbf{npts}}\ge 3$.

9:
$\mathbf{x}\left[{\mathbf{npts}}\right]$ – const double
Input

On entry: the mesh points in the space direction. ${\mathbf{x}}\left[0\right]$ must specify the lefthand boundary, $a$, and ${\mathbf{x}}\left[{\mathbf{npts}}1\right]$ must specify the righthand boundary, $b$.
Constraint:
${\mathbf{x}}\left[0\right]<{\mathbf{x}}\left[1\right]<\cdots <{\mathbf{x}}\left[{\mathbf{npts}}1\right]$.

10:
$\mathbf{nv}$ – Integer
Input

On entry: the number of coupled ODE components.
Constraint:
${\mathbf{nv}}\ge 0$.

11:
$\mathbf{odedef}$ – function, supplied by the user
External Function

odedef must evaluate the functions
$R$, which define the system of ODEs, as given in
(4).
If
${\mathbf{nv}}=0$,
odedef will never be called and the NAG defined null void function pointer, NULLFN, can be supplied in the call to
d03plc.
The specification of
odedef is:
void 
odedef (Integer npde,
double t,
Integer nv,
const double v[],
const double vdot[],
Integer nxi,
const double xi[],
const double ucp[],
const double ucpx[],
const double ucpt[],
double r[],
Integer *ires,
Nag_Comm *comm)



1:
$\mathbf{npde}$ – Integer
Input

On entry: the number of PDEs in the system.

2:
$\mathbf{t}$ – double
Input

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

3:
$\mathbf{nv}$ – Integer
Input

On entry: the number of coupled ODEs in the system.

4:
$\mathbf{v}\left[{\mathbf{nv}}\right]$ – const double
Input

On entry: if ${\mathbf{nv}}>0$, ${\mathbf{v}}\left[\mathit{i}1\right]$ contains the value of the component ${V}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.

5:
$\mathbf{vdot}\left[{\mathbf{nv}}\right]$ – const double
Input

On entry: if ${\mathbf{nv}}>0$, ${\mathbf{vdot}}\left[\mathit{i}1\right]$ contains the value of component ${\stackrel{.}{V}}_{\mathit{i}}\left(t\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$.

6:
$\mathbf{nxi}$ – Integer
Input

On entry: the number of ODE/PDE coupling points.

7:
$\mathbf{xi}\left[{\mathbf{nxi}}\right]$ – const double
Input

On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{xi}}\left[\mathit{i}1\right]$ contains the ODE/PDE coupling point, ${\xi}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxi}}$.

8:
$\mathbf{ucp}\left[{\mathbf{npde}}\times {\mathbf{nxi}}\right]$ – const double
Input

Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{ucp}}\left[\left(j1\right)\times {\mathbf{npde}}+i1\right]$.
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{ucp}}\left[\left(\mathit{j}1\right)\times {\mathbf{npde}}+\mathit{i}1\right]$ 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:
$\mathbf{ucpx}\left[{\mathbf{npde}}\times {\mathbf{nxi}}\right]$ – const double
Input

Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{ucpx}}\left[\left(j1\right)\times {\mathbf{npde}}+i1\right]$.
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{ucpx}}\left[\left(\mathit{j}1\right)\times {\mathbf{npde}}+\mathit{i}1\right]$ 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:
$\mathbf{ucpt}\left[{\mathbf{npde}}\times {\mathbf{nxi}}\right]$ – const double
Input

Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{ucpt}}\left[\left(j1\right)\times {\mathbf{npde}}+i1\right]$.
On entry: if ${\mathbf{nxi}}>0$, ${\mathbf{ucpt}}\left[\left(\mathit{j}1\right)\times {\mathbf{npde}}+\mathit{i}1\right]$ 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 $\mathit{j}=1,2,\dots ,{\mathbf{nxi}}$.

11:
$\mathbf{r}\left[{\mathbf{nv}}\right]$ – double
Output

On exit:
${\mathbf{r}}\left[\mathit{i}1\right]$ must contain the
$\mathit{i}$th component of
$R$, for
$\mathit{i}=1,2,\dots ,{\mathbf{nv}}$, where
$R$ is defined as
or
The definition of
$R$ is determined by the input value of
ires.

12:
$\mathbf{ires}$ – Integer *
Input/Output

On entry: the form of
$R$ that must be returned in the array
r.
 ${\mathbf{ires}}=1$
 Equation (10) must be used.
 ${\mathbf{ires}}=1$
 Equation (11) must be used.
On exit: should usually remain unchanged. However, you may reset
ires to force the integration function to take certain actions, as described below:
 ${\mathbf{ires}}=2$
 Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP.
 ${\mathbf{ires}}=3$
 Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set ${\mathbf{ires}}=3$ when a physically meaningless input or output value has been generated. If you consecutively set ${\mathbf{ires}}=3$, d03plc returns to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.

13:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
odedef.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
d03plc you may allocate memory and initialize these pointers with various quantities for use by
odedef when called from
d03plc (see
Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: odedef should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
d03plc. If your code inadvertently
does return any NaNs or infinities,
d03plc is likely to produce unexpected results.

12:
$\mathbf{nxi}$ – Integer
Input

On entry: the number of ODE/PDE coupling points.
Constraints:
 if ${\mathbf{nv}}=0$, ${\mathbf{nxi}}=0$;
 if ${\mathbf{nv}}>0$, ${\mathbf{nxi}}\ge 0$.

13:
$\mathbf{xi}\left[{\mathbf{nxi}}\right]$ – const double
Input

On entry: ${\mathbf{xi}}\left[\mathit{i}1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{nxi}}$, must be set to the ODE/PDE coupling points.
Constraint:
${\mathbf{x}}\left[0\right]\le {\mathbf{xi}}\left[0\right]<{\mathbf{xi}}\left[1\right]<\cdots <{\mathbf{xi}}\left[{\mathbf{nxi}}1\right]\le {\mathbf{x}}\left[{\mathbf{npts}}1\right]$.

14:
$\mathbf{neqn}$ – Integer
Input

On entry: the number of ODEs in the time direction.
Constraint:
${\mathbf{neqn}}={\mathbf{npde}}\times {\mathbf{npts}}+{\mathbf{nv}}$.

15:
$\mathbf{rtol}\left[\mathit{dim}\right]$ – const double
Input

Note: the dimension,
dim, of the array
rtol
must be at least
 $1$ when ${\mathbf{itol}}=1$ or $2$;
 ${\mathbf{neqn}}$ when ${\mathbf{itol}}=3$ or $4$.
On entry: the relative local error tolerance.
Constraint:
${\mathbf{rtol}}\left[i1\right]\ge 0.0$ for all relevant $i$.

16:
$\mathbf{atol}\left[\mathit{dim}\right]$ – const double
Input

Note: the dimension,
dim, of the array
atol
must be at least
 $1$ when ${\mathbf{itol}}=1$ or $3$;
 ${\mathbf{neqn}}$ when ${\mathbf{itol}}=2$ or $4$.
On entry: the absolute local error tolerance.
Constraint:
${\mathbf{atol}}\left[i1\right]\ge 0.0$ for all relevant
$i$.
Note: corresponding elements of
rtol and
atol cannot both be
$0.0$.

17:
$\mathbf{itol}$ – Integer
Input

On entry: a value to indicate the form of the local error test.
If
${e}_{\mathit{i}}$ is the estimated local error for
${\mathbf{u}}\left[\mathit{i}1\right]$, for
$\mathit{i}=1,2,\dots ,{\mathbf{neqn}}$, and
$\Vert \text{\hspace{1em}}\Vert $ denotes the norm, the error test to be satisfied is
$\Vert {e}_{\mathit{i}}\Vert <1.0$.
itol indicates to
d03plc whether to interpret either or both of
rtol and
atol as a vector or scalar in the formation of the weights
${w}_{\mathit{i}}$ used in the calculation of the norm (see the description of
norm):
itol  rtol  atol  ${w}_{\mathit{i}}$ 
1  scalar  scalar  ${\mathbf{rtol}}\left[0\right]\times \left{\mathbf{u}}\left[\mathit{i}1\right]\right+{\mathbf{atol}}\left[0\right]$ 
2  scalar  vector  ${\mathbf{rtol}}\left[0\right]\times \left{\mathbf{u}}\left[\mathit{i}1\right]\right+{\mathbf{atol}}\left[\mathit{i}1\right]$ 
3  vector  scalar  ${\mathbf{rtol}}\left[\mathit{i}1\right]\times \left{\mathbf{u}}\left[\mathit{i}1\right]\right+{\mathbf{atol}}\left[0\right]$ 
4  vector  vector  ${\mathbf{rtol}}\left[\mathit{i}1\right]\times \left{\mathbf{u}}\left[\mathit{i}1\right]\right+{\mathbf{atol}}\left[\mathit{i}1\right]$ 
Constraint:
$1\le {\mathbf{itol}}\le 4$.

18:
$\mathbf{norm}$ – Nag_NormType
Input

On entry: the type of norm to be used.
 ${\mathbf{norm}}=\mathrm{Nag\_OneNorm}$
 Averaged ${L}_{1}$ norm.
 ${\mathbf{norm}}=\mathrm{Nag\_TwoNorm}$
 Averaged ${L}_{2}$ norm.
If
${U}_{\mathrm{norm}}$ denotes the norm of the vector
u of length
neqn, then for the averaged
${L}_{1}$ norm
and for the averaged
${L}_{2}$ norm
See the description of
itol for the formulation of the weight vector
$w$.
Constraint:
${\mathbf{norm}}=\mathrm{Nag\_OneNorm}$ or $\mathrm{Nag\_TwoNorm}$.

19:
$\mathbf{laopt}$ – Nag_LinAlgOption
Input

On entry: the type of matrix algebra required.
 ${\mathbf{laopt}}=\mathrm{Nag\_LinAlgFull}$
 Full matrix methods to be used.
 ${\mathbf{laopt}}=\mathrm{Nag\_LinAlgBand}$
 Banded matrix methods to be used.
 ${\mathbf{laopt}}=\mathrm{Nag\_LinAlgSparse}$
 Sparse matrix methods to be used.
Constraint:
${\mathbf{laopt}}=\mathrm{Nag\_LinAlgFull}$, $\mathrm{Nag\_LinAlgBand}$ or $\mathrm{Nag\_LinAlgSparse}$.
Note: you are recommended to use the banded option when no coupled ODEs are present (${\mathbf{nv}}=0$). Also, the banded option should not be used if the boundary conditions involve solution components at points other than the boundary and the immediately adjacent two points.

20:
$\mathbf{algopt}\left[30\right]$ – const double
Input

On entry: may be set to control various options available in the integrator. If you wish to employ all the default options,
${\mathbf{algopt}}\left[0\right]$ 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:
 ${\mathbf{algopt}}\left[0\right]$
 Selects the ODE integration method to be used. If ${\mathbf{algopt}}\left[0\right]=1.0$, a BDF method is used and if ${\mathbf{algopt}}\left[0\right]=2.0$, a Theta method is used. The default is ${\mathbf{algopt}}\left[0\right]=1.0$.
If ${\mathbf{algopt}}\left[0\right]=2.0$, then
${\mathbf{algopt}}\left[\mathit{i}1\right]$, for $\mathit{i}=2,3,4$, are not used.
 ${\mathbf{algopt}}\left[1\right]$
 Specifies the maximum order of the BDF integration formula to be used. ${\mathbf{algopt}}\left[1\right]$ may be $1.0$, $2.0$, $3.0$, $4.0$ or $5.0$. The default value is ${\mathbf{algopt}}\left[1\right]=5.0$.
 ${\mathbf{algopt}}\left[2\right]$
 Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the BDF method. If ${\mathbf{algopt}}\left[2\right]=1.0$ a modified Newton iteration is used and if ${\mathbf{algopt}}\left[2\right]=2.0$ 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 ${\mathbf{algopt}}\left[2\right]=1.0$.
 ${\mathbf{algopt}}\left[3\right]$
 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 ${\mathbf{algopt}}\left[3\right]=1.0$, the Petzold test is used. If ${\mathbf{algopt}}\left[3\right]=2.0$, the Petzold test is not used. The default value is ${\mathbf{algopt}}\left[3\right]=1.0$.
If ${\mathbf{algopt}}\left[0\right]=1.0$,
${\mathbf{algopt}}\left[\mathit{i}1\right]$, for $\mathit{i}=5,6,7$, are not used.
 ${\mathbf{algopt}}\left[4\right]$
 Specifies the value of Theta to be used in the Theta integration method. $0.51\le {\mathbf{algopt}}\left[4\right]\le 0.99$. The default value is ${\mathbf{algopt}}\left[4\right]=0.55$.
 ${\mathbf{algopt}}\left[5\right]$
 Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the Theta method. If ${\mathbf{algopt}}\left[5\right]=1.0$, a modified Newton iteration is used and if ${\mathbf{algopt}}\left[5\right]=2.0$, a functional iteration method is used. The default value is ${\mathbf{algopt}}\left[5\right]=1.0$.
 ${\mathbf{algopt}}\left[6\right]$
 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 ${\mathbf{algopt}}\left[6\right]=1.0$, switching is allowed and if ${\mathbf{algopt}}\left[6\right]=2.0$, switching is not allowed. The default value is ${\mathbf{algopt}}\left[6\right]=1.0$.
 ${\mathbf{algopt}}\left[10\right]$
 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 ${\mathbf{algopt}}\left[0\right]\ne 0.0$, a value of $0.0$ for ${\mathbf{algopt}}\left[10\right]$, say, should be specified even if itask subsequently specifies that ${t}_{\mathrm{crit}}$ will not be used.
 ${\mathbf{algopt}}\left[11\right]$
 Specifies the minimum absolute step size to be allowed in the time integration. If this option is not required, ${\mathbf{algopt}}\left[11\right]$ should be set to $0.0$.
 ${\mathbf{algopt}}\left[12\right]$
 Specifies the maximum 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 initial step size to be attempted by the integrator. If ${\mathbf{algopt}}\left[13\right]=0.0$, the initial step size is calculated internally.
 ${\mathbf{algopt}}\left[14\right]$
 Specifies the maximum number of steps to be attempted by the integrator in any one call. If ${\mathbf{algopt}}\left[14\right]=0.0$, no limit is imposed.
 ${\mathbf{algopt}}\left[22\right]$
 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 ${\mathbf{algopt}}\left[22\right]=1.0$, a modified Newton iteration is used and if ${\mathbf{algopt}}\left[22\right]=2.0$, functional iteration is used. The default value is ${\mathbf{algopt}}\left[22\right]=1.0$.
${\mathbf{algopt}}\left[28\right]$ and ${\mathbf{algopt}}\left[29\right]$ are used only for the sparse matrix algebra option, i.e., ${\mathbf{laopt}}=\mathrm{Nag\_LinAlgSparse}$.
 ${\mathbf{algopt}}\left[28\right]$
 Governs the choice of pivots during the decomposition of the first Jacobian matrix. It should lie in the range $0.0<{\mathbf{algopt}}\left[28\right]<1.0$, with smaller values biasing the algorithm towards maintaining sparsity at the expense of numerical stability. If ${\mathbf{algopt}}\left[28\right]$ lies outside the range then the default value is used. If the functions regard the Jacobian matrix as numerically singular, increasing ${\mathbf{algopt}}\left[28\right]$ towards $1.0$ may help, but at the cost of increased fillin. The default value is ${\mathbf{algopt}}\left[28\right]=0.1$.
 ${\mathbf{algopt}}\left[29\right]$
 Used as the relative pivot threshold during subsequent Jacobian decompositions (see ${\mathbf{algopt}}\left[28\right]$) below which an internal error is invoked. ${\mathbf{algopt}}\left[29\right]$ must be greater than zero, otherwise the default value is used. If ${\mathbf{algopt}}\left[29\right]$ is greater than $1.0$ no check is made on the pivot size, and this may be a necessary option if the Jacobian matrix is found to be numerically singular (see ${\mathbf{algopt}}\left[28\right]$). The default value is ${\mathbf{algopt}}\left[29\right]=0.0001$.

21:
$\mathbf{rsave}\left[{\mathbf{lrsave}}\right]$ – double
Communication Array

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

22:
$\mathbf{lrsave}$ – Integer
Input

On entry: the dimension of the array
rsave.
Its size depends on the type of matrix algebra selected.
If ${\mathbf{laopt}}=\mathrm{Nag\_LinAlgFull}$, ${\mathbf{lrsave}}\ge {\mathbf{neqn}}\times {\mathbf{neqn}}+{\mathbf{neqn}}+\mathit{nwkres}+\mathit{lenode}$.
If ${\mathbf{laopt}}=\mathrm{Nag\_LinAlgBand}$, ${\mathbf{lrsave}}\ge \left(3\mathit{mlu}+1\right)\times {\mathbf{neqn}}+\mathit{nwkres}+\mathit{lenode}$.
If ${\mathbf{laopt}}=\mathrm{Nag\_LinAlgSparse}$, ${\mathbf{lrsave}}\ge 4{\mathbf{neqn}}+11{\mathbf{neqn}}/2+1+\mathit{nwkres}+\mathit{lenode}$.
Where
$\mathit{mlu}$ is the lower or upper half bandwidths such that
 for PDE problems only (no coupled ODEs),
 $\mathit{mlu}=3{\mathbf{npde}}1\text{;}$
 for coupled PDE/ODE problems,
 $\mathit{mlu}={\mathbf{neqn}}1\text{.}$
Where
$\mathit{nwkres}$ is defined by
 if ${\mathbf{nv}}>0\text{ and}{\mathbf{nxi}}>0$,
 $\mathit{nwkres}={\mathbf{npde}}\left(2{\mathbf{npts}}+6{\mathbf{nxi}}+3{\mathbf{npde}}+26\right)+{\mathbf{nxi}}+{\mathbf{nv}}+7{\mathbf{npts}}+2\text{;}$
 if ${\mathbf{nv}}>0\text{ and}{\mathbf{nxi}}=0$,
 $\mathit{nwkres}={\mathbf{npde}}\left(2{\mathbf{npts}}+3{\mathbf{npde}}+32\right)+{\mathbf{nv}}+7{\mathbf{npts}}+3\text{;}$
 if ${\mathbf{nv}}=0$,
 $\mathit{nwkres}={\mathbf{npde}}\left(2{\mathbf{npts}}+3{\mathbf{npde}}+32\right)+7{\mathbf{npts}}+4\text{.}$
Where
$\mathit{lenode}$ is defined by
 if the BDF method is used,
 $\mathit{lenode}=\left(6+\mathrm{int}\left({\mathbf{algopt}}\left[1\right]\right)\right)\times {\mathbf{neqn}}+50\text{;}$
 if the Theta method is used,
 $\mathit{lenode}=9{\mathbf{neqn}}+50\text{,}$
Note: when
${\mathbf{laopt}}=\mathrm{Nag\_LinAlgSparse}$, 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 function returns with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT_2.

23:
$\mathbf{isave}\left[{\mathbf{lisave}}\right]$ – Integer
Communication Array

If
${\mathbf{ind}}=0$,
isave need not be set.
If
${\mathbf{ind}}=1$,
isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular the following components of the array
isave concern the efficiency of the integration:
 ${\mathbf{isave}}\left[0\right]$
 Contains the number of steps taken in time.
 ${\mathbf{isave}}\left[1\right]$
 Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves evaluating the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.
 ${\mathbf{isave}}\left[2\right]$
 Contains the number of Jacobian evaluations performed by the time integrator.
 ${\mathbf{isave}}\left[3\right]$
 Contains the order of the BDF method last used in the time integration, if applicable. When the Theta method is used, ${\mathbf{isave}}\left[3\right]$ contains no useful information.
 ${\mathbf{isave}}\left[4\right]$
 Contains the number of Newton iterations performed by the time integrator. Each iteration involves residual evaluation of the resulting ODE system followed by a backsubstitution using the $LU$ decomposition of the Jacobian matrix.

24:
$\mathbf{lisave}$ – Integer
Input

On entry: the dimension of the array
isave. Its size depends on the type of matrix algebra selected:
 if ${\mathbf{laopt}}=\mathrm{Nag\_LinAlgFull}$, ${\mathbf{lisave}}\ge 24$;
 if ${\mathbf{laopt}}=\mathrm{Nag\_LinAlgBand}$, ${\mathbf{lisave}}\ge {\mathbf{neqn}}+24$;
 if ${\mathbf{laopt}}=\mathrm{Nag\_LinAlgSparse}$, ${\mathbf{lisave}}\ge 25\times {\mathbf{neqn}}+24$.
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 if
${\mathbf{itrace}}>0$ and the function returns with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT_2.

25:
$\mathbf{itask}$ – Integer
Input

On entry: the task to be performed by the ODE integrator.
 ${\mathbf{itask}}=1$
 Normal computation of output values u at $t={\mathbf{tout}}$ (by overshooting and interpolating).
 ${\mathbf{itask}}=2$
 Take one step in the time direction 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 $5$.

26:
$\mathbf{itrace}$ – Integer
Input

On entry: the level of trace information required from
d03plc and the underlying ODE solver.
itrace may take the value
$1$,
$0$,
$1$,
$2$ or
$3$.
 ${\mathbf{itrace}}=1$
 No output is generated.
 ${\mathbf{itrace}}=0$
 Only warning messages from the PDE solver are printed.
 ${\mathbf{itrace}}>0$
 Output from the underlying ODE solver is printed. This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.
If ${\mathbf{itrace}}<1$, $1$ is assumed and similarly if ${\mathbf{itrace}}>3$, $3$ is assumed.
The advisory messages are given in greater detail as
itrace increases.

27:
$\mathbf{outfile}$ – const char *
Input

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

28:
$\mathbf{ind}$ – Integer *
Input/Output

On entry: indicates whether this is a continuation call or a new integration.
 ${\mathbf{ind}}=0$
 Starts or restarts the integration in time.
 ${\mathbf{ind}}=1$
 Continues the integration after an earlier exit from the function. In this case, only the
argument tout
should be reset between calls to d03plc.
Constraint:
${\mathbf{ind}}=0$ or $1$.
On exit: ${\mathbf{ind}}=1$.

29:
$\mathbf{comm}$ – Nag_Comm *

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

30:
$\mathbf{saved}$ – Nag_D03_Save *
Communication Structure

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

31:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ACC_IN_DOUBT

Integration completed, but small changes in
atol or
rtol are unlikely to result in a changed solution.
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
$5$.)
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_FAILED_DERIV

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

atol and
rtol were too small to start integration.
 NE_FAILED_STEP

Error during Jacobian formulation for ODE system. Increase
itrace for further details.
Repeated errors in an attempted step of underlying ODE solver. Integration was successful as far as
ts:
${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
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. Incorrect specification of boundary conditions may also result in this error.
Underlying ODE solver cannot make further progress from the point
ts with the supplied values of
atol and
rtol.
${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
 NE_INT

ires set to an invalid value in a call to usersupplied functions
pdedef,
numflx,
bndary or
odedef.
On entry, ${\mathbf{ind}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ind}}=0$ or $1$.
On entry, ${\mathbf{itask}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{itask}}=1$, $2$, $3$, $4$ or $5$.
On entry, ${\mathbf{itol}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{itol}}=1$, $2$, $3$ or $4$.
On entry, ${\mathbf{npde}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{npde}}\ge 1$.
On entry, ${\mathbf{npts}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{npts}}\ge 3$.
On entry, ${\mathbf{nv}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nv}}\ge 0$.
On entry, on initial entry ${\mathbf{ind}}=1$.
Constraint: on initial entry ${\mathbf{ind}}=0$.
 NE_INT_2

On entry, $\mathit{i}=\u2329\mathit{\text{value}}\u232a$ and $\mathit{j}=\u2329\mathit{\text{value}}\u232a$.
Constraint: corresponding elements ${\mathbf{atol}}\left[\mathit{i}1\right]$ and ${\mathbf{rtol}}\left[\mathit{j}1\right]$ cannot both be $0.0$.
On entry, ${\mathbf{lisave}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{lisave}}\ge \u2329\mathit{\text{value}}\u232a$.
On entry, ${\mathbf{lrsave}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{lrsave}}\ge \u2329\mathit{\text{value}}\u232a$.
On entry, ${\mathbf{nv}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{nxi}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nxi}}=0$ when ${\mathbf{nv}}=0$.
On entry, ${\mathbf{nv}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{nxi}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nxi}}\ge 0$ when ${\mathbf{nv}}>0$.
When using the sparse option
lisave or
lrsave is too small:
${\mathbf{lisave}}=\u2329\mathit{\text{value}}\u232a$,
${\mathbf{lrsave}}=\u2329\mathit{\text{value}}\u232a$.
 NE_INT_4

On entry, ${\mathbf{neqn}}=\u2329\mathit{\text{value}}\u232a$, ${\mathbf{npde}}=\u2329\mathit{\text{value}}\u232a$, ${\mathbf{npts}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{nv}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{neqn}}={\mathbf{npde}}\times {\mathbf{npts}}+{\mathbf{nv}}$.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Serious error in internal call to an auxiliary. Increase
itrace for further details.
 NE_ITER_FAIL

In solving ODE system, the maximum number of steps ${\mathbf{algopt}}\left[14\right]$ has been exceeded. ${\mathbf{algopt}}\left[14\right]=\u2329\mathit{\text{value}}\u232a$.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
 NE_NOT_CLOSE_FILE

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

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

Cannot open file $\u2329\mathit{\text{value}}\u232a$ for writing.
 NE_REAL_2

On entry, at least one point in
xi lies outside
$\left[{\mathbf{x}}\left[0\right],{\mathbf{x}}\left[{\mathbf{npts}}1\right]\right]$:
${\mathbf{x}}\left[0\right]=\u2329\mathit{\text{value}}\u232a$ and
${\mathbf{x}}\left[{\mathbf{npts}}1\right]=\u2329\mathit{\text{value}}\u232a$.
On entry, ${\mathbf{tout}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{tout}}>{\mathbf{ts}}$.
On entry, ${\mathbf{tout}}{\mathbf{ts}}$ is too small:
${\mathbf{tout}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
 NE_REAL_ARRAY

On entry, $\mathit{i}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{atol}}\left[\mathit{i}1\right]=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{atol}}\left[\mathit{i}1\right]\ge 0.0$.
On entry, $\mathit{i}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{rtol}}\left[\mathit{i}1\right]=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{rtol}}\left[\mathit{i}1\right]\ge 0.0$.
 NE_SING_JAC

Singular Jacobian of ODE system. Check problem formulation.
 NE_TIME_DERIV_DEP

The functions $P$, $D$, or $C$ appear to depend on time derivatives.
 NE_USER_STOP

In evaluating residual of ODE system,
${\mathbf{ires}}=2$ has been set in usersupplied functions
pdedef,
numflx,
bndary or
odedef. Integration is successful as far as
ts:
${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
 NE_ZERO_WTS

Zero error weights encountered during time integration.
Some error weights
${w}_{i}$ became zero during the time integration (see the description of
itol). Pure relative error control
(${\mathbf{atol}}\left[i1\right]=0.0$) was requested on a variable (the
$i$th) which has become zero. The integration was successful as far as
$t={\mathbf{ts}}$.
7
Accuracy
d03plc 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 arguments,
atol and
rtol.
8
Parallelism and Performance
d03plc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d03plc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
d03plc is designed to solve systems of PDEs in conservative form, with optional source terms which are independent of space derivatives, and optional secondorder diffusion terms. The use of the function to solve systems which are not naturally in this form is discouraged, and you are advised to use one of the centraldifference schemes for such problems.
You should be aware of the stability limitations for hyperbolic PDEs. For most problems with small error tolerances the ODE integrator does not attempt unstable time steps, but in some cases a maximum time step should be imposed using ${\mathbf{algopt}}\left[12\right]$. It is worth experimenting with this argument, particularly if the integration appears to progress unrealistically fast (with large time steps). Setting the maximum time step to the minimum mesh size is a safe measure, although in some cases this may be too restrictive.
Problems with source terms should be treated with caution, as it is known that for large source terms stable and reasonable looking solutions can be obtained which are in fact incorrect, exhibiting nonphysical speeds of propagation of discontinuities (typically one spatial mesh point per time step). It is essential to employ a very fine mesh for problems with source terms and discontinuities, and to check for nonphysical propagation speeds by comparing results for different mesh sizes. Further details and an example can be found in
Pennington and Berzins (1994).
The time taken depends on the complexity of the system and on the accuracy requested. For a given system and a fixed accuracy it is approximately proportional to
neqn.
10
Example
For this function two examples are presented, with a main program and two example problems given in Example 1 (ex1) and Example 2 (ex2).
Example 1 (ex1)
This example is a simple firstorder system with coupled ODEs arising from the use of the characteristic equations for the numerical boundary conditions.
The PDEs are
for
$x\in \left[0,1\right]$ and
$t\ge 0$.
The PDEs have an exact solution given by
where
$f\left(z\right)=\mathrm{exp}\left(\pi z\right)\mathrm{sin}\left(2\pi z\right)$,
$g\left(z\right)=\mathrm{exp}\left(2\pi z\right)\mathrm{cos}\left(2\pi z\right)$.
The initial conditions are given by the exact solution.
The characteristic variables are ${W}_{1}={U}_{1}{U}_{2}$ and ${W}_{2}={U}_{1}+{U}_{2}$, corresponding to the characteristics given by $dx/dt=1$ and $dx/dt=3$ respectively. Hence we require a physical boundary condition for ${W}_{2}$ at the lefthand boundary and for ${W}_{1}$ at the righthand boundary (corresponding to the incoming characteristics), and a numerical boundary condition for ${W}_{1}$ at the lefthand boundary and for ${W}_{2}$ at the righthand boundary (outgoing characteristics).
The physical boundary conditions are obtained from the exact solution, and the numerical boundary conditions are supplied in the form of the characteristic equations for the outgoing characteristics, that is
at the lefthand boundary, and
at the righthand boundary.
In order to specify these boundary conditions, two ODE variables
${V}_{1}$ and
${V}_{2}$ are introduced, defined by
The coupling points are therefore at
$x=0$ and
$x=1$.
The numerical boundary conditions are now
at the lefthand boundary, and
at the righthand boundary.
The spatial derivatives are evaluated at the appropriate boundary points in
bndary using onesided differences (into the domain and therefore consistent with the characteristic directions).
The numerical flux is calculated using Roe's approximate Riemann solver (see
Section 3 for details), giving
Example 2 (ex2)
This example is the standard shocktube test problem proposed by
Sod (1978) for the Euler equations of gas dynamics. The problem models the flow of a gas in a long tube following the sudden breakdown of a diaphragm separating two initial gas states at different pressures and densities. There is an exact solution to this problem which is not included explicitly as the calculation is quite lengthy. The PDEs are
where
$\rho $ is the density;
$m$ is the momentum, such that
$m=\rho u$, where
$u$ is the velocity;
$e$ is the specific energy; and
$\gamma $ is the (constant) ratio of specific heats. The pressure
$p$ is given by
The solution domain is
$0\le x\le 1$ for
$0<t\le 0.2$, with the initial discontinuity at
$x=0.5$, and initial conditions
The solution is uniform and constant at both boundaries for the spatial domain and time of integration stated, and hence the physical and numerical boundary conditions are indistinguishable and are both given by the initial conditions above. The evaluation of the numerical flux for the Euler equations is not trivial; the Roe algorithm given in
Section 3 cannot be used directly as the Jacobian is nonlinear. However, an algorithm is available using the argumentvector method (see
Roe (1981)), and this is provided in the utility function
d03puc. An alternative Approxiate Riemann Solver using Osher's scheme is provided in
d03pvc. Either
d03puc or
d03pvc can be called from
numflx.
10.1
Program Text
10.2
Program Data
None.
10.3
Program Results