NAG Library Function Document
nag_pde_parab_1d_cd (d03pfc)
1
Purpose
nag_pde_parab_1d_cd (d03pfc) integrates a system of linear or nonlinear convectiondiffusion equations in one space dimension, with optional source terms. 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 PDEs to a system of ordinary differential equations (ODEs), and the resulting system is solved using a backward differentiation formula (BDF) method.
2
Specification
#include <nag.h> 
#include <nagd03.h> 
void 
nag_pde_parab_1d_cd (Integer npde,
double *ts,
double tout,
void 
(*pdedef)(Integer npde,
double t,
double x,
const double u[],
const double ux[],
double p[],
double c[],
double d[],
double s[],
Integer *ires,
Nag_Comm *comm),


double u[],
Integer npts,
const double x[],
const double acc[],
double tsmax,
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) 

3
Description
nag_pde_parab_1d_cd (d03pfc) 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 solution values
The functions
${P}_{i,j}$,
${F}_{i}$,
${C}_{i}$ and
${S}_{i}$ depend on
$x$,
$t$ and
$U$; and
${D}_{i}$ depends on
$x$,
$t$,
$U$ and
${U}_{x}$, where
${U}_{x}$ is the spatial derivative of
$U$. Note that
${P}_{i,j}$,
${F}_{i}$,
${C}_{i}$ and
${S}_{i}$ must not depend on any space derivatives; and none of the functions may depend on 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.
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 $U\left(x,t\right)$ 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
nag_pde_parab_1d_fd (d03pcc),
nag_pde_parab_1d_fd_ode (d03phc) and
nag_pde_parab_1d_fd_ode_remesh (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
${x}_{j1/2}=\left({x}_{j1}+{x}_{j}\right)/2$, for
$j=2,3,\dots ,{\mathbf{npts}}$. The left and right values are calculated by
nag_pde_parab_1d_cd (d03pfc) 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
$y=x{x}_{j1/2}$, i.e.,
$y=0$ corresponds to
$x={x}_{j1/2}$, 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.
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 a
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
nag_pde_parab_1d_cd (d03pfc).
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 conditions, 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 only linear extrapolation is allowed in this function (for greater flexibility the function
nag_pde_parab_1d_cd_ode (d03plc) should be used). 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. Examples can be found in
Section 10.
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 problem is subject to the following restrictions:
(i) 
${P}_{i,j}$, ${F}_{i}$, ${C}_{i}$ and ${S}_{i}$ must not depend on any space derivatives; 
(ii) 
${P}_{i,j}$, ${F}_{i}$, ${C}_{i}$, ${D}_{i}$ and ${S}_{i}$ 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 ${\mathbf{npde}}\times {\mathbf{npts}}$ ODEs in the time direction. This system is then integrated forwards in time using a BDF method.
For further details of the algorithm, 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
5
Arguments
 1:
$\mathbf{npde}$ – IntegerInput

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:
$\mathbf{tout}$ – doubleInput

On entry: the final value of $t$ to which the integration is to be carried out.
 4:
$\mathbf{pdedef}$ – function, supplied by the userExternal Function

pdedef must evaluate the functions
${P}_{i,j}$,
${C}_{i}$,
${D}_{i}$ and
${S}_{i}$ which partially define the system of PDEs.
${P}_{i,j}$,
${C}_{i}$ and
${S}_{i}$ may depend on
$x$,
$t$ and
$U$;
${D}_{i}$ may depend on
$x$,
$t$,
$U$ and
${U}_{x}$.
pdedef is called approximately midway between each pair of mesh points in turn by
nag_pde_parab_1d_cd (d03pfc). For problems in the form
(2)) the NAG defined null void function pointer, NULLFN, can be supplied in the call to
nag_pde_parab_1d_cd (d03pfc).
The specification of
pdedef is:
void 
pdedef (Integer npde,
double t,
double x,
const double u[],
const double ux[],
double p[],
double c[],
double d[],
double s[],
Integer *ires,
Nag_Comm *comm)


 1:
$\mathbf{npde}$ – IntegerInput

On entry: the number of PDEs in the system.
 2:
$\mathbf{t}$ – doubleInput

On entry: the current value of the independent variable $t$.
 3:
$\mathbf{x}$ – doubleInput

On entry: the current value of the space variable $x$.
 4:
$\mathbf{u}\left[{\mathbf{npde}}\right]$ – const doubleInput

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 doubleInput

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:
$\mathbf{p}\left[{\mathbf{npde}}\times {\mathbf{npde}}\right]$ – doubleOutput

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 ${P}_{\mathit{i},\mathit{j}}\left(x,t,U\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npde}}$.
 7:
$\mathbf{c}\left[{\mathbf{npde}}\right]$ – doubleOutput

On exit: ${\mathbf{c}}\left[\mathit{i}1\right]$ must be set to the value of ${C}_{\mathit{i}}\left(x,t,U\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
 8:
$\mathbf{d}\left[{\mathbf{npde}}\right]$ – doubleOutput

On exit: ${\mathbf{d}}\left[\mathit{i}1\right]$ must be set to the value of ${D}_{\mathit{i}}\left(x,t,U,{U}_{x}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
 9:
$\mathbf{s}\left[{\mathbf{npde}}\right]$ – doubleOutput

On exit: ${\mathbf{s}}\left[\mathit{i}1\right]$ must be set to the value of ${S}_{\mathit{i}}\left(x,t,U\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
 10:
$\mathbf{ires}$ – 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 ${\mathbf{ires}}=3$, nag_pde_parab_1d_cd (d03pfc) returns to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.
 11:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
pdedef.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
nag_pde_parab_1d_cd (d03pfc) you may allocate memory and initialize these pointers with various quantities for use by
pdedef when called from
nag_pde_parab_1d_cd (d03pfc) (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: pdedef should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
nag_pde_parab_1d_cd (d03pfc). If your code inadvertently
does return any NaNs or infinities,
nag_pde_parab_1d_cd (d03pfc) is likely to produce unexpected results.
 5:
$\mathbf{numflx}$ – function, supplied by the userExternal 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
nag_pde_parab_1d_cd (d03pfc).
The specification of
numflx is:
 1:
$\mathbf{npde}$ – IntegerInput

On entry: the number of PDEs in the system.
 2:
$\mathbf{t}$ – doubleInput

On entry: the current value of the independent variable $t$.
 3:
$\mathbf{x}$ – doubleInput

On entry: the current value of the space variable $x$.
 4:
$\mathbf{uleft}\left[{\mathbf{npde}}\right]$ – const doubleInput

On entry: ${\mathbf{uleft}}\left[\mathit{i}1\right]$ contains the left value of the component ${U}_{\mathit{i}}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
 5:
$\mathbf{uright}\left[{\mathbf{npde}}\right]$ – const doubleInput

On entry: ${\mathbf{uright}}\left[\mathit{i}1\right]$ contains the right value of the component ${U}_{\mathit{i}}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
 6:
$\mathbf{flux}\left[{\mathbf{npde}}\right]$ – doubleOutput

On exit: ${\mathbf{flux}}\left[\mathit{i}1\right]$ must be set to the numerical flux ${\hat{F}}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
 7:
$\mathbf{ires}$ – 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 ${\mathbf{ires}}=3$, nag_pde_parab_1d_cd (d03pfc) returns to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.
 8:
$\mathbf{comm}$ – 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
nag_pde_parab_1d_cd (d03pfc) you may allocate memory and initialize these pointers with various quantities for use by
numflx when called from
nag_pde_parab_1d_cd (d03pfc) (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
 9:
$\mathbf{saved}$ – Nag_D03_Save *Communication Structure

If
numflx calls one of the approximate Riemann solvers
nag_pde_parab_1d_euler_roe (d03puc),
nag_pde_parab_1d_euler_osher (d03pvc),
nag_pde_parab_1d_euler_hll (d03pwc) or
nag_pde_parab_1d_euler_exact (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
nag_pde_parab_1d_cd (d03pfc). If your code inadvertently
does return any NaNs or infinities,
nag_pde_parab_1d_cd (d03pfc) is likely to produce unexpected results.
 6:
$\mathbf{bndary}$ – function, supplied by the userExternal Function

bndary must evaluate the functions
${G}_{i}^{L}$ and
${G}_{i}^{R}$ which describe the physical and numerical boundary conditions, as given by
(6) and
(7).
The specification of
bndary is:
void 
bndary (Integer npde,
Integer npts,
double t,
const double x[],
const double u[],
Integer ibnd,
double g[],
Integer *ires,
Nag_Comm *comm)


 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:
$\mathbf{t}$ – doubleInput

On entry: the current value of the independent variable $t$.
 4:
$\mathbf{x}\left[{\mathbf{npts}}\right]$ – const doubleInput

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:
$\mathbf{u}\left[{\mathbf{npde}}\times 3\right]$ – const doubleInput

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: contains the value of solution components in the boundary region.
If ${\mathbf{ibnd}}=0$, ${\mathbf{u}}\left[\left(\mathit{j}1\right)\times {\mathbf{npde}}+\mathit{i}1\right]$ contains the value of the component ${U}_{\mathit{i}}\left(\mathrm{x},t\right)$ at $x={\mathbf{x}}\left[\mathit{j}1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,3$.
If ${\mathbf{ibnd}}\ne 0$, ${\mathbf{u}}\left[\left(\mathit{j}1\right)\times {\mathbf{npde}}+\mathit{i}1\right]$ contains the value of the component ${U}_{\mathit{i}}\left(x,t\right)$ at $x={\mathbf{x}}\left[{\mathbf{npts}}\mathit{j}\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,3$.
 6:
$\mathbf{ibnd}$ – IntegerInput

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 $x=b$.
 7:
$\mathbf{g}\left[{\mathbf{npde}}\right]$ – doubleOutput

On exit:
${\mathbf{g}}\left[\mathit{i}1\right]$ must contain the
$\mathit{i}$th component of either
${{\mathbf{g}}}^{L}$ or
${{\mathbf{g}}}^{R}$ in
(6) and
(7), depending on the value of
ibnd, for
$\mathit{i}=1,2,\dots ,{\mathbf{npde}}$.
 8:
$\mathbf{ires}$ – 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 ${\mathbf{ires}}=3$, nag_pde_parab_1d_cd (d03pfc) returns to the calling function with the error indicator set to ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_FAILED_DERIV.
 9:
$\mathbf{comm}$ – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
bndary.
 user – double *
 iuser – Integer *
 p – Pointer
The type Pointer will be
void *. Before calling
nag_pde_parab_1d_cd (d03pfc) you may allocate memory and initialize these pointers with various quantities for use by
bndary when called from
nag_pde_parab_1d_cd (d03pfc) (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: bndary should not return floatingpoint NaN (Not a Number) or infinity values, since these are not handled by
nag_pde_parab_1d_cd (d03pfc). If your code inadvertently
does return any NaNs or infinities,
nag_pde_parab_1d_cd (d03pfc) is likely to produce unexpected results.
 7:
$\mathbf{u}\left[{\mathbf{npde}}\times {\mathbf{npts}}\right]$ – doubleInput/Output

Note: the $\left(i,j\right)$th element of the matrix $U$ is stored in ${\mathbf{u}}\left[\left(j1\right)\times {\mathbf{npde}}+i1\right]$.
On entry: ${\mathbf{u}}\left[\left(\mathit{j}1\right)\times {\mathbf{npde}}+\mathit{i}1\right]$ must contain the initial value of ${U}_{\mathit{i}}\left(x,t\right)$ at $x={\mathbf{x}}\left[\mathit{j}1\right]$ and $t={\mathbf{ts}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
On exit: ${\mathbf{u}}\left[\left(\mathit{j}1\right)\times {\mathbf{npde}}+\mathit{i}1\right]$ will contain the computed solution ${U}_{\mathit{i}}\left(x,t\right)$ at $x={\mathbf{x}}\left[\mathit{j}1\right]$ and $t={\mathbf{ts}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{npde}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{npts}}$.
 8:
$\mathbf{npts}$ – IntegerInput

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 doubleInput

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:
$\mathbf{acc}\left[2\right]$ – const doubleInput

On entry: the components of
acc contain the relative and absolute error tolerances used in the local error test in the time integration.
If
$\mathrm{E}\left(i,j\right)$ is the estimated error for
${U}_{i}$ at the
$j$th mesh point, the error test is
Constraint:
${\mathbf{acc}}\left[0\right]$ and ${\mathbf{acc}}\left[1\right]\ge 0.0$ (but not both zero).
 11:
$\mathbf{tsmax}$ – doubleInput

On entry: the maximum absolute step size to be allowed in the time integration. If ${\mathbf{tsmax}}=0.0$ then no maximum is imposed.
Constraint:
${\mathbf{tsmax}}\ge 0.0$.
 12:
$\mathbf{rsave}\left[{\mathbf{lrsave}}\right]$ – doubleCommunication 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.
 13:
$\mathbf{lrsave}$ – IntegerInput

On entry: the dimension of the array
rsave.
Constraint:
${\mathbf{lrsave}}\ge \left(11+9\times {\mathbf{npde}}\right)\times {\mathbf{npde}}\times {\mathbf{npts}}+\left(32+3\times {\mathbf{npde}}\right)\times {\mathbf{npde}}+7\times \phantom{\rule{0ex}{0ex}}{\mathbf{npts}}+54$.
 14:
$\mathbf{isave}\left[{\mathbf{lisave}}\right]$ – IntegerCommunication Array

If
${\mathbf{ind}}=0$,
isave need not be set on entry.
If
${\mathbf{ind}}=1$,
isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular:
 ${\mathbf{isave}}\left[0\right]$
 Contains the number of steps taken in time.
 ${\mathbf{isave}}\left[1\right]$
 Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.
 ${\mathbf{isave}}\left[2\right]$
 Contains the number of Jacobian evaluations performed by the time integrator.
 ${\mathbf{isave}}\left[3\right]$
 Contains the order of the last backward differentiation formula method used.
 ${\mathbf{isave}}\left[4\right]$
 Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a backsubstitution using the $LU$ decomposition of the Jacobian matrix.
 15:
$\mathbf{lisave}$ – IntegerInput

On entry: the dimension of the array
isave.
Constraint:
${\mathbf{lisave}}\ge {\mathbf{npde}}\times {\mathbf{npts}}+24$.
 16:
$\mathbf{itask}$ – IntegerInput

On entry: the task to be performed by the ODE integrator.
 ${\mathbf{itask}}=1$
 Normal computation of output values ${\mathbf{u}}$ 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 $t={\mathbf{tout}}$.
Constraint:
${\mathbf{itask}}=1$, $2$ or $3$.
 17:
$\mathbf{itrace}$ – IntegerInput

On entry: the level of trace information required from
nag_pde_parab_1d_cd (d03pfc) 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.
 18:
$\mathbf{outfile}$ – 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.
 19:
$\mathbf{ind}$ – 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 arguments tout and fail should be reset between calls to nag_pde_parab_1d_cd (d03pfc).
Constraint:
${\mathbf{ind}}=0$ or $1$.
On exit: ${\mathbf{ind}}=1$.
 20:
$\mathbf{comm}$ – Nag_Comm *

The NAG communication argument (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
 21:
$\mathbf{saved}$ – 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.
 22:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
 NE_ACC_IN_DOUBT

Integration completed, but small changes in
acc are unlikely to result in a changed solution.
${\mathbf{acc}}\left[0\right]=\u2329\mathit{\text{value}}\u232a$,
${\mathbf{acc}}\left[1\right]=\u2329\mathit{\text{value}}\u232a$.
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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, or
bndary.
 NE_FAILED_START

Values in
acc are too small to start integration:
${\mathbf{acc}}\left[0\right]=\u2329\mathit{\text{value}}\u232a$,
${\mathbf{acc}}\left[1\right]=\u2329\mathit{\text{value}}\u232a$.
 NE_FAILED_STEP

Error during Jacobian formulation for ODE system. Increase
itrace for further details.
Repeated errors in an attempted step of underlying ODE solver. Integration was successful as far as
ts:
${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
Underlying ODE solver cannot make further progress from the point
ts with the supplied values of
acc.
${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$,
${\mathbf{acc}}\left[0\right]=\u2329\mathit{\text{value}}\u232a$,
${\mathbf{acc}}\left[1\right]=\u2329\mathit{\text{value}}\u232a$.
 NE_INCOMPAT_PARAM

On entry, ${\mathbf{acc}}\left[0\right]$ and ${\mathbf{acc}}\left[1\right]$ are both zero.
 NE_INT

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

On entry,
lisave is too small:
${\mathbf{lisave}}=\u2329\mathit{\text{value}}\u232a$. Minimum possible dimension:
$\u2329\mathit{\text{value}}\u232a$.
On entry,
lrsave is too small:
${\mathbf{lrsave}}=\u2329\mathit{\text{value}}\u232a$. Minimum possible dimension:
$\u2329\mathit{\text{value}}\u232a$.
 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Serious error in internal call to an auxiliary. Increase
itrace for further details.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
 NE_NOT_CLOSE_FILE

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

On entry, mesh points
x appear to be badly ordered:
$\mathit{I}=\u2329\mathit{\text{value}}\u232a$,
${\mathbf{x}}\left[\mathit{I}1\right]=\u2329\mathit{\text{value}}\u232a$,
$\mathit{J}=\u2329\mathit{\text{value}}\u232a$ and
${\mathbf{x}}\left[\mathit{J}1\right]=\u2329\mathit{\text{value}}\u232a$.
 NE_NOT_WRITE_FILE

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

On entry, ${\mathbf{acc}}\left[0\right]<0.0$:
${\mathbf{acc}}\left[0\right]=\u2329\mathit{\text{value}}\u232a$.
On entry, ${\mathbf{acc}}\left[1\right]<0.0$:
${\mathbf{acc}}\left[1\right]=\u2329\mathit{\text{value}}\u232a$.
On entry, ${\mathbf{tsmax}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{tsmax}}\ge 0.0$.
 NE_REAL_2

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

Singular Jacobian of ODE system. Check problem formulation.
 NE_TIME_DERIV_DEP

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
pdedef,
numflx, or
bndary. Integration is successful as far as
ts:
${\mathbf{ts}}=\u2329\mathit{\text{value}}\u232a$.
7
Accuracy
nag_pde_parab_1d_cd (d03pfc) 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 components of the accuracy argument,
acc.
8
Parallelism and Performance
nag_pde_parab_1d_cd (d03pfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_pde_parab_1d_cd (d03pfc) 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.
nag_pde_parab_1d_cd (d03pfc) 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
tsmax. 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.
10
Example
For this function two examples are presented. There is a single example program for nag_pde_parab_1d_cd (d03pfc), with a main program and the code to solve the two example problems given in Example 1 (ex1) and Example 2 (ex2).
Example 1 (ex1)
This example is a simple firstorder system which illustrates the calculation of the numerical flux using Roe's approximate Riemann solver, and the specification of numerical boundary conditions using extrapolated characteristic variables. The PDEs are
for
$x\in \left[0,1\right]$ and
$t\ge 0$. The PDEs have an exact solution given by
The initial conditions are given by the exact solution. The characteristic variables are
$2{U}_{1}+{U}_{2}$ and
$2{U}_{1}{U}_{2}$ corresponding to the characteristics given by
$dx/dt=3$ and
$dx/dt=1$ respectively. Hence a physical boundary condition is required for
$2{U}_{1}+{U}_{2}$ at the lefthand boundary, and for
$2{U}_{1}{U}_{2}$ at the righthand boundary (corresponding to the incoming characteristics); and a numerical boundary condition is required for
$2{U}_{1}{U}_{2}$ at the lefthand boundary, and for
$2{U}_{1}+{U}_{2}$ at the righthand boundary (outgoing characteristics). The physical boundary conditions are obtained from the exact solution, and the numerical boundary conditions are calculated by linear extrapolation of the appropriate characteristic variable. The numerical flux is calculated using Roe's approximate Riemann solver: Using the notation in
Section 3, the flux vector
$F$ and the Jacobian matrix
$A$ are
and the eigenvalues of
$A$ are
$3$ and
$1$ with right eigenvectors
${\left[\begin{array}{cc}1& 2\end{array}\right]}^{\mathrm{T}}$ and
${\left[\begin{array}{cc}1& 2\end{array}\right]}^{\mathrm{T}}$ respectively. Using equation
(4) the
${\alpha}_{k}$ are given by
that is
${F}_{L}$ is given by
and similarly for
${F}_{R}$. From equation
(4), the numerical flux vector is
that is
Example 2 (ex2)
This example is an advectiondiffusion equation in which the flux term depends explicitly on
$x$:
for
$x\in \left[1,1\right]$ and
$0\le t\le 10$. The argument
$\epsilon $ is taken to be
$0.01$. The two physical boundary conditions are
$U\left(1,t\right)=3.0$ and
$U\left(1,t\right)=5.0$ and the initial condition is
$U\left(x,0\right)=x+4$. The integration is run to steady state at which the solution is known to be
$U=4$ across the domain with a narrow boundary layer at both boundaries. In order to write the PDE in conservative form, a source term must be introduced, i.e.,
As in Example 1, the numerical flux is calculated using the Roe approximate Riemann solver. The Riemann problem to solve locally is
The
$x$ in the flux term is assumed to be constant at a local level, and so using the notation in
Section 3,
$F=xU$ and
$A=x$. The eigenvalue is
$x$ and the eigenvector (a scalar in this case) is
$1$. The numerical flux is therefore
10.1
Program Text
Program Text (d03pfce.c)
10.2
Program Data
None.
10.3
Program Results
Program Results (d03pfce.r)