d03pfc integrates a system of linear or nonlinear convection-diffusion 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 user-supplied 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>

void

d03pfc (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),

void	(numflx)(Integer npde, double t, double x, 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 ibnd, double g[], 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)

The function may be called by the names: d03pfc, nag_pde_dim1_parab_convdiff or nag_pde_parab_1d_cd.

3 Description

d03pfc integrates the system of convection-diffusion equations in conservative form:

\sum_{j = 1}^{npde} P_{i, j} \frac{\partial U_{j}}{\partial t} + \frac{\partial F_{i}}{\partial x} = C_{i} \frac{\partial D_{i}}{\partial x} + S_{i},

(1)

or the hyperbolic convection-only system:

\frac{\partial U_{i}}{\partial t} + \frac{\partial F_{i}}{\partial x} = 0,

(2)

for

i = 1, 2, \dots, npde, a \leq x \leq b, t \geq t_{0}

, where the vector

U

is the set of solution values

U (x, t) = {[U_{1} (x, t), \dots, U_{npde} (x, t)]}^{T} .

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}

t_{out}

, over the space interval

a \leq x \leq b

, where

a = x_{1}

and

b = x_{npts}

are the leftmost and rightmost points of a user-defined mesh

x_{1}, x_{2}, \dots, x_{npts}

. The initial values of the functions

U (x, t)

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 central-difference 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 mid-point of the mesh

x_{j - 1 / 2} = (x_{j - 1} + x_{j}) / 2

, for

j = 2, 3, \dots, npts

. The left and right values are calculated by d03pfc from two adjacent mesh points using a standard upwind technique combined with a Van Leer slope-limiter (see LeVeque (1990)). The physically correct value for

{\hat{F}}_{i}

is derived from the solution of the Riemann problem given by

\frac{\partial U_{i}}{\partial t} + \frac{\partial F_{i}}{\partial y} = 0,

(3)

where

y = x - x_{j - 1 / 2}

, i.e.,

y = 0

corresponds to

x = x_{j - 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

\hat{F} = \frac{1}{2} (F_{L} + F_{R}) - \frac{1}{2} \sum_{k = 1}^{npde} α_{k} | λ_{k} | e_{k},

(4)

where

F_{L}

(

F_{R}

) is the flux

F

calculated at the left (right) value of

U

, denoted by

U_{L}

(

U_{R}

); the

λ_{k}

are the eigenvalues of

A

; the

e_{k}

are the right eigenvectors of

A

; and the

α_{k}

are defined by

U_{R} - U_{L} = \sum_{k = 1}^{npde} α_{k} e_{k} .

(5)

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 d03pfc.

The boundary condition specification has sufficient flexibility to allow for different types of problems. For second-order 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 well-posed. If there are no second-order 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 first-order 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 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.

The boundary conditions must be specified in bndary in the form

G_{i}^{L} (x, t, U) = 0 at ​ x = a, i = 1, 2, \dots, npde,

(6)

at the left-hand boundary, and

G_{i}^{R} (x, t, U) = 0 at ​ x = b, i = 1, 2, \dots, npde,

(7)

at the right-hand 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 one-sided 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) $t_{0} < t_{out}$ , 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 $x_{1}, x_{2}, \dots, x_{npts}$ ;
(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

npde \times 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 time-dependent problems using the method of lines and differential-algebraic 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 first-order 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: $npde$ – Integer Input

On entry: the number of PDEs to be solved.

Constraint:

npde \geq 1

2: $ts$ – 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

ts = tout

Constraint:

ts < tout

3: $tout$ – double Input

On entry: the final value of

t

to which the integration is to be carried out.

4: $pdedef$ – 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}

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 d03pfc. For problems in the form (2)) the NAG defined null void function pointer, NULLFN, can be supplied in the call to 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: $npde$ – Integer Input

On entry: the number of PDEs in the system.

2: $t$ – double Input

On entry: the current value of the independent variable

t

3: $x$ – double Input

On entry: the current value of the space variable

x

4: $u [npde]$ – const double Input

On entry:

u [i - 1]

contains the value of the component

U_{i} (x, t)

, for

i = 1, 2, \dots, npde

5: $ux [npde]$ – const double Input

On entry:

ux [i - 1]

contains the value of the component

\frac{\partial U_{i} (x, t)}{\partial x}

, for

i = 1, 2, \dots, npde

6: $p [npde \times npde]$ – double Output

Note: the

(i, j)

th element of the matrix

P

is stored in

p [(j - 1) \times npde + i - 1]

On exit:

p [(j - 1) \times npde + i - 1]

must be set to the value of

P_{i, j} (x, t, U)

, for

i = 1, 2, \dots, npde

and

j = 1, 2, \dots, npde

7: $c [npde]$ – double Output

On exit:

c [i - 1]

must be set to the value of

C_{i} (x, t, U)

, for

i = 1, 2, \dots, npde

8: $d [npde]$ – double Output

On exit:

d [i - 1]

must be set to the value of

D_{i} (x, t, U, U_{x})

, for

i = 1, 2, \dots, npde

9: $s [npde]$ – double Output

On exit:

s [i - 1]

must be set to the value of

S_{i} (x, t, U)

, for

i = 1, 2, \dots, npde

10: $ires$ – Integer * Input/Output

On entry: set to

−1

1

On exit: should usually remain unchanged. However, you may set ires to force the integration function to take certain actions as described below:

$ires = 2$: Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to $fail . code =$ NE_USER_STOP.
$ires = 3$: Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $ires = 3$ when a physically meaningless input or output value has been generated. If you consecutively set $ires = 3$ , d03pfc returns to the calling function with the error indicator set to $fail . code =$ NE_FAILED_DERIV.

11: $comm$ – 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 d03pfc you may allocate memory and initialize these pointers with various quantities for use by pdedef when called from d03pfc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

Note: pdedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pfc. If your code inadvertently does return any NaNs or infinities, d03pfc is likely to produce unexpected results.

5: $numflx$ – 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

u

. numflx is called approximately midway between each pair of mesh points in turn by d03pfc.

The specification of numflx is:

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

1: $npde$ – Integer Input

On entry: the number of PDEs in the system.

2: $t$ – double Input

On entry: the current value of the independent variable

t

3: $x$ – double Input

On entry: the current value of the space variable

x

4: $uleft [npde]$ – const double Input

On entry:

uleft [i - 1]

contains the left value of the component

U_{i} (x)

, for

i = 1, 2, \dots, npde

5: $uright [npde]$ – const double Input

On entry:

uright [i - 1]

contains the right value of the component

U_{i} (x)

, for

i = 1, 2, \dots, npde

6: $flux [npde]$ – double Output

On exit:

flux [i - 1]

must be set to the numerical flux

{\hat{F}}_{i}

, for

i = 1, 2, \dots, npde

7: $ires$ – Integer * Input/Output

On entry: set to

−1

1

On exit: should usually remain unchanged. However, you may set ires to force the integration function to take certain actions as described below:

$ires = 2$: Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to $fail . code =$ NE_USER_STOP.
$ires = 3$: Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $ires = 3$ when a physically meaningless input or output value has been generated. If you consecutively set $ires = 3$ , d03pfc returns to the calling function with the error indicator set to $fail . code =$ NE_FAILED_DERIV.

8: $comm$ – 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 d03pfc you may allocate memory and initialize these pointers with various quantities for use by numflx when called from d03pfc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

9: $saved$ – 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 floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pfc. If your code inadvertently does return any NaNs or infinities, d03pfc is likely to produce unexpected results.

6: $bndary$ – 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 (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: $npde$ – Integer Input

On entry: the number of PDEs in the system.

2: $npts$ – Integer Input

On entry: the number of mesh points in the interval

[a, b]

3: $t$ – double Input

On entry: the current value of the independent variable

t

4: $x [npts]$ – const double Input

On entry: the mesh points in the spatial direction.

x [0]

corresponds to the left-hand boundary,

a

, and

x [npts - 1]

corresponds to the right-hand boundary,

b

5: $u [npde \times 3]$ – const double Input

Note: the

(i, j)

th element of the matrix

U

is stored in

u [(j - 1) \times npde + i - 1]

On entry: contains the value of solution components in the boundary region.

ibnd = 0

u [(j - 1) \times npde + i - 1]

contains the value of the component

U_{i} (x, t)

x = x [j - 1]

, for

i = 1, 2, \dots, npde

and

j = 1, 2, 3

ibnd \neq 0

u [(j - 1) \times npde + i - 1]

contains the value of the component

U_{i} (x, t)

x = x [npts - j]

, for

i = 1, 2, \dots, npde

and

j = 1, 2, 3

6: $ibnd$ – Integer Input

On entry: specifies which boundary conditions are to be evaluated.

$ibnd = 0$: bndary must evaluate the left-hand boundary condition at $x = a$ .
$ibnd \neq 0$: bndary must evaluate the right-hand boundary condition at $x = b$ .

7: $g [npde]$ – double Output

On exit:

g [i - 1]

must contain the

i

th component of either

g^{L}

g^{R}

in (6) and (7), depending on the value of ibnd, for

i = 1, 2, \dots, npde

8: $ires$ – Integer * Input/Output

On entry: set to

−1

1

On exit: should usually remain unchanged. However, you may set ires to force the integration function to take certain actions as described below:

$ires = 2$: Indicates to the integrator that control should be passed back immediately to the calling function with the error indicator set to $fail . code =$ NE_USER_STOP.
$ires = 3$: Indicates to the integrator that the current time step should be abandoned and a smaller time step used instead. You may wish to set $ires = 3$ when a physically meaningless input or output value has been generated. If you consecutively set $ires = 3$ , d03pfc returns to the calling function with the error indicator set to $fail . code =$ NE_FAILED_DERIV.

9: $comm$ – 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 d03pfc you may allocate memory and initialize these pointers with various quantities for use by bndary when called from d03pfc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

Note: bndary should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03pfc. If your code inadvertently does return any NaNs or infinities, d03pfc is likely to produce unexpected results.

7: $u [npde \times npts]$ – double Input/Output

Note: the

(i, j)

th element of the matrix

U

is stored in

u [(j - 1) \times npde + i - 1]

On entry:

u [(j - 1) \times npde + i - 1]

must contain the initial value of

U_{i} (x, t)

x = x [j - 1]

and

t = ts

, for

i = 1, 2, \dots, npde

and

j = 1, 2, \dots, npts

On exit:

u [(j - 1) \times npde + i - 1]

will contain the computed solution

U_{i} (x, t)

x = x [j - 1]

and

t = ts

, for

i = 1, 2, \dots, npde

and

j = 1, 2, \dots, npts

8: $npts$ – Integer Input

On entry: the number of mesh points in the interval

[a, b]

Constraint:

npts \geq 3

9: $x [npts]$ – const double Input

On entry: the mesh points in the space direction.

x [0]

must specify the left-hand boundary,

a

, and

x [npts - 1]

must specify the right-hand boundary,

b

Constraint:

x [0] < x [1] < \dots < x [npts - 1]

10: $acc [2]$ – const double Input

On entry: the components of acc contain the relative and absolute error tolerances used in the local error test in the time integration.

E (i, j)

is the estimated error for

U_{i}

at the

j

th mesh point, the error test is

E (i, j) = acc [0] \times u [(j - 1) \times npde + i - 1] + acc [1] .

Constraint:

acc [0]

and

acc [1] \geq 0.0

(but not both zero).

11: $tsmax$ – double Input

On entry: the maximum absolute step size to be allowed in the time integration. If

tsmax = 0.0

then no maximum is imposed.

Constraint:

tsmax \geq 0.0

12: $rsave [lrsave]$ – double Communication Array

ind = 0

, rsave need not be set on entry.

ind = 1

, rsave must be unchanged from the previous call to the function because it contains required information about the iteration.

13: $lrsave$ – Integer Input

On entry: the dimension of the array rsave.

Constraint:

lrsave \geq (11 + 9 \times npde) \times npde \times npts + (32 + 3 \times npde) \times npde + 7 \times npts + 54

14: $isave [lisave]$ – Integer Communication Array

ind = 0

, isave need not be set on entry.

ind = 1

, isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular:

$isave [0]$: Contains the number of steps taken in time.
$isave [1]$: 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.
$isave [2]$: Contains the number of Jacobian evaluations performed by the time integrator.
$isave [3]$: Contains the order of the last backward differentiation formula method used.
$isave [4]$: Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the $L U$ decomposition of the Jacobian matrix.

15: $lisave$ – Integer Input

On entry: the dimension of the array isave.

Constraint:

lisave \geq npde \times npts + 24

16: $itask$ – Integer Input

On entry: the task to be performed by the ODE integrator.

$itask = 1$: Normal computation of output values $u$ at $t = tout$ (by overshooting and interpolating).
$itask = 2$: Take one step in the time direction and return.
$itask = 3$: Stop at first internal integration point at or beyond $t = tout$ .

Constraint:

itask = 1

2

3

17: $itrace$ – Integer Input

On entry: the level of trace information required from d03pfc and the underlying ODE solver. itrace may take the value

−1

0

1

2

3

$itrace = −1$: No output is generated.
$itrace = 0$: Only warning messages from the PDE solver are printed.
$itrace > 0$: 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.

itrace < −1

−1

is assumed and similarly if

itrace > 3

3

is assumed.

The advisory messages are given in greater detail as itrace increases.

18: $outfile$ – 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: $ind$ – Integer * Input/Output

On entry: indicates whether this is a continuation call or a new integration.

$ind = 0$: Starts or restarts the integration in time.
$ind = 1$: Continues the integration after an earlier exit from the function. In this case, only the argument tout should be reset between calls to d03pfc.

Constraint:

ind = 0

1

On exit:

ind = 1

20: $comm$ – Nag_Comm *

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

21: $saved$ – 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: $fail$ – 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 acc are unlikely to result in a changed solution. $acc [0] = ⟨ value ⟩$ , $acc [1] = ⟨ value ⟩$ .
The required task has been completed, but it is estimated that a small change in the values of acc is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when $itask \neq 2$ .)
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 $⟨ value ⟩$ 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 $ires = 3$ in pdedef, numflx, or bndary.
NE_FAILED_START: Values in acc are too small to start integration: $acc [0] = ⟨ value ⟩$ , $acc [1] = ⟨ value ⟩$ .
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: $ts = ⟨ value ⟩$ .
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 $t = ts$ . 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 acc. $ts = ⟨ value ⟩$ , $acc [0] = ⟨ value ⟩$ , $acc [1] = ⟨ value ⟩$ .
NE_INCOMPAT_PARAM: On entry, $acc [0]$ and $acc [1]$ are both zero.
NE_INT: ires set to an invalid value in call to pdedef, numflx, or bndary.

On entry, $ind = ⟨ value ⟩$ .
Constraint: $ind = 0$ or $1$ .

On entry, $itask = ⟨ value ⟩$ .
Constraint: $itask = 1$ , $2$ or $3$ .

On entry, $npde = ⟨ value ⟩$ .
Constraint: $npde \geq 1$ .

On entry, $npts = ⟨ value ⟩$ .
Constraint: $npts \geq 3$ .

On entry, on initial entry $ind = 1$ .
Constraint: on initial entry $ind = 0$ .
NE_INT_2: On entry, $lisave = ⟨ value ⟩$ .
Constraint: $lisave \geq ⟨ value ⟩$ .

On entry, $lrsave = ⟨ value ⟩$ .
Constraint: $lrsave \geq ⟨ value ⟩$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.

Serious error in internal call to an auxiliary. Increase itrace for further details.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_CLOSE_FILE: Cannot close file $⟨ value ⟩$ .
NE_NOT_STRICTLY_INCREASING: On entry, $i = ⟨ value ⟩$ , $x [i - 1] = ⟨ value ⟩$ , $j = ⟨ value ⟩$ and $x [j - 1] = ⟨ value ⟩$ .
Constraint: $x [0] < x [1] < \dots < x [npts - 1]$ .
NE_NOT_WRITE_FILE: Cannot open file $⟨ value ⟩$ for writing.
NE_REAL: On entry, $acc [0] = ⟨ value ⟩$ .
Constraint: $acc [0] \geq 0.0$ .

On entry, $acc [1] = ⟨ value ⟩$ .
Constraint: $acc [1] \geq 0.0$ .

On entry, $tsmax = ⟨ value ⟩$ .
Constraint: $tsmax \geq 0.0$ .
NE_REAL_2: On entry, $tout = ⟨ value ⟩$ and $ts = ⟨ value ⟩$ .
Constraint: $tout > ts$ .

On entry, $tout - ts$ is too small: $tout = ⟨ value ⟩$ and $ts = ⟨ value ⟩$ .
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, $ires = 2$ has been set in pdedef, numflx, or bndary. Integration is successful as far as ts: $ts = ⟨ value ⟩$ .

7 Accuracy

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

d03pfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

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 implementation-specific information.

9 Further Comments

d03pfc is designed to solve systems of PDEs in conservative form, with optional source terms which are independent of space derivatives, and optional second-order 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 central-difference 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 non-physical 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 non-physical 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 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 first-order 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

\begin{array}{lcl} \frac{\partial U_{1}}{\partial t} + \frac{\partial U_{1}}{\partial x} + \frac{\partial U_{2}}{\partial x} & = & 0, \\ \frac{\partial U_{2}}{\partial t} + 4 \frac{\partial U_{1}}{\partial x} + \frac{\partial U_{2}}{\partial x} & = & 0, \end{array}

for

x \in [0, 1]

and

t \geq 0

. The PDEs have an exact solution given by

\begin{array}{lcl} U_{1} (x, t) & = & \frac{1}{2} {\exp (x + t) + \exp (x - 3 t)} + \frac{1}{4} {\sin (2 π {(x - 3 t)}^{2}) - \sin (2 π {(x + t)}^{2})} + 2 t^{2} - 2 x t, \\ U_{2} (x, t) & = & \exp (x - 3 t) - \exp (x + t) + \frac{1}{2} {\sin (2 π {(x - 3 t)}^{2}) + \sin (2 π {(x - 3 t)}^{2})} + x^{2} + 5 t^{2} - 2 x t . \end{array}

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

d x / d t = 3

and

d x / d t = −1

respectively. Hence a physical boundary condition is required for

2 U_{1} + U_{2}

at the left-hand boundary, and for

2 U_{1} - U_{2}

at the right-hand boundary (corresponding to the incoming characteristics); and a numerical boundary condition is required for

2 U_{1} - U_{2}

at the left-hand boundary, and for

2 U_{1} + U_{2}

at the right-hand 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

F = [\begin{matrix} U_{1} + U_{2} \\ 4 U_{1} + U_{2} \end{matrix}] and A = [\begin{matrix} 1 & 1 \\ 4 & 1 \end{matrix}],

and the eigenvalues of

A

are

3

and

−1

with right eigenvectors

{[\begin{matrix} 1 & 2 \end{matrix}]}^{T}

and

{[\begin{matrix} −1 & 2 \end{matrix}]}^{T}

respectively. Using equation (4) the

α_{k}

are given by

[\begin{matrix} U_{1 R} - U_{1 L} \\ U_{2 R} - U_{2 L} \end{matrix}] = α_{1} [\begin{matrix} 1 \\ 2 \end{matrix}] + α_{2} [\begin{array}{r} −1 \\ 2 \end{array}],

that is

α_{1} = \frac{1}{4} (2 U_{1 R} - 2 U_{1 L} + U_{2 R} - U_{2 L}) and α_{2} = \frac{1}{4} (−2 U_{1 R} + 2 U_{1 L} + U_{2 R} - U_{2 L}) .

F_{L}

is given by

F_{L} = [\begin{array}{r} U_{1 L} + U_{2 L} \\ 4 U_{1 L} + U_{2 L} \end{array}],

and similarly for

F_{R}

. From equation (4), the numerical flux vector is

\hat{F} = \frac{1}{2} [\begin{array}{r} U_{1 L} + U_{2 L} + U_{1 R} + U_{2 R} \\ 4 U_{1 L} + U_{2 L} + 4 U_{1 R} + U_{2 R} \end{array}] - \frac{1}{2} α_{1} | 3 | [\begin{matrix} 1 \\ 2 \end{matrix}] - \frac{1}{2} α_{2} | - 1 | [\begin{array}{r} −1 \\ 2 \end{array}],

that is

\hat{F} = \frac{1}{2} [\begin{array}{r} 3 U_{1 L} - U_{1 R} + \frac{3}{2} U_{2 L} + \frac{1}{2} U_{2 R} \\ 6 U_{1 L} + 2 U_{1 R} + 3 U_{2 L} - U_{2 R} \end{array}] .

Example 2 (ex2)

This example is an advection-diffusion equation in which the flux term depends explicitly on

x

\frac{\partial U}{\partial t} + x \frac{\partial U}{\partial x} = ε \frac{\partial^{2} U}{\partial x^{2}},

for

x \in [−1, 1]

and

0 \leq t \leq 10

. The argument

ε

is taken to be

0.01

. The two physical boundary conditions are

U (−1, t) = 3.0

and

U (1, t) = 5.0

and the initial condition is

U (x, 0) = 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.,