d03plc integrates a system of linear or nonlinear convection-diffusion 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 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 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

#include <nag.h>

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 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 PDE solution values

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

The optional coupled ODEs are of the general form

R_{i} (t, V, \dot{V}, ξ, U^{*}, U_{x}^{*}, U_{t}^{*}) = 0, i = 1, 2, \dots, nv,

(3)

where the vector

V

is the set of ODE solution values

V (t) = {[V_{1} (t), \dots, V_{nv} (t)]}^{T},

\dot{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

\dot{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),

ξ

represents a vector of

n_{ξ}

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

R = L - M \dot{V} - N U_{t}^{*},

(4)

where

R = {[R_{1}, \dots, R_{nv}]}^{T}

L

is a vector of length nv,

M

is an nv by nv matrix,

N

is an nv by

(n_{ξ} \times npde)

matrix and the entries in

L

M

and

N

may depend on

t

ξ

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}

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)

and

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

, for

j = 2, 3, \dots, 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 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,

(5)

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

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

(6)

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} .

(7)

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 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 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 the d03pfc documentation.

The boundary conditions must be specified in bndary in the form

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

(8)

at the left-hand boundary, and

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

(9)

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 algebraic-differential equation system which is defined by the functions

R_{i}

must be specified in odedef. You must also specify the coupling points

ξ

(if any) in the array xi.

The problem is subject to the following restrictions:

(i)In (1), ${\dot{V}}_{j} (t)$ , for $j = 1, 2, \dots, nv$ , may only appear linearly in the functions $S_{i}$ , for $i = 1, 2, \dots, npde$ , 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) $t_{0} < t_{out}$ , 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 + 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 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

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: $npde$ – 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}

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

\dot{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: $npde$ – 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: $nv$ – Integer Input

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

7: $v [nv]$ – const double Input

On entry: if

nv > 0

v [i - 1]

contains the value of the component

V_{i} (t)

, for

i = 1, 2, \dots, nv

8: $vdot [nv]$ – const double Input

On entry: if

nv > 0

vdot [i - 1]

contains the value of component

{\dot{V}}_{i} (t)

, for

i = 1, 2, \dots, nv

Note:

{\dot{V}}_{i} (t)

, for

i = 1, 2, \dots, nv

, may only appear linearly in

S_{j}

, for

j = 1, 2, \dots, npde

9: $p [npde \times npde]$ – 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, V)

, for

i = 1, 2, \dots, npde

and

j = 1, 2, \dots, npde

10: $c [npde]$ – double Output

On exit:

c [i - 1]

must be set to the value of

C_{i} (x, t, U, V)

, for

i = 1, 2, \dots, npde

11: $d [npde]$ – double Output

On exit:

d [i - 1]

must be set to the value of

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

, for

i = 1, 2, \dots, npde

12: $s [npde]$ – double Output

On exit:

s [i - 1]

must be set to the value of

S_{i} (x, t, U, V, \dot{V})

, for

i = 1, 2, \dots, npde

13: $ires$ – 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 $ires = 3$ , d03plc returns to the calling function with the error indicator set to $fail . code =$ NE_FAILED_DERIV.

14: $comm$ – 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 floating-point 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: $numflx$ – 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 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: $npde$ – 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: $nv$ – Integer Input

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

5: $v [nv]$ – const double Input

On entry: if

nv > 0

v [i - 1]

contains the value of the component

V_{i} (t)

, for

i = 1, 2, \dots, nv

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

On entry:

uleft [i - 1]

contains the left value of the component

U_{i} (x)

, for

i = 1, 2, \dots, npde

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

On entry:

uright [i - 1]

contains the right value of the component

U_{i} (x)

, for

i = 1, 2, \dots, npde

8: $flux [npde]$ – double Output

On exit:

flux [i - 1]

must be set to the numerical flux

{\hat{F}}_{i}

, for

i = 1, 2, \dots, npde

9: $ires$ – 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 $ires = 3$ , d03plc returns to the calling function with the error indicator set to $fail . code =$ NE_FAILED_DERIV.

10: $comm$ – 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: $saved$ – 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 d03plc. If your code inadvertently does return any NaNs or infinities, d03plc 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 (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: $npde$ – 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: $u [npde \times npts]$ – 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:

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, \dots, npts

Note: if banded matrix algebra is to be used then the functions

G_{i}^{L}

and

G_{i}^{R}

may depend on the value of

U_{i} (x, t)

at the boundary point and the two adjacent points only.

6: $nv$ – Integer Input

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

7: $v [nv]$ – const double Input

On entry: if

nv > 0

v [i - 1]

contains the value of the component

V_{i} (t)

, for

i = 1, 2, \dots, nv

8: $vdot [nv]$ – const double Input

On entry: if

nv > 0

vdot [i - 1]

contains the value of component

{\dot{V}}_{i} (t)

, for

i = 1, 2, \dots, nv

Note:

{\dot{V}}_{i} (t)

, for

i = 1, 2, \dots, nv

, may only appear linearly in

G_{j}^{L}

and

G_{j}^{R}

, for

j = 1, 2, \dots, npde

9: $ibnd$ – 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 $x = b$ .

10: $g [npde]$ – double Output

On exit:

g [i - 1]

must contain the

i

th component of either

G_{i}^{L}

G_{i}^{R}

in (8) and (9), depending on the value of ibnd, for

i = 1, 2, \dots, npde

11: $ires$ – 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 $ires = 3$ , d03plc returns to the calling function with the error indicator set to $fail . code =$ NE_FAILED_DERIV.

12: $comm$ – 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 floating-point 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: $u [neqn]$ – double Input/Output

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

$u [npde \times (j - 1) + i - 1]$ contain $U_{i} (x_{j}, t_{0})$ , for $i = 1, 2, \dots, npde$ and $j = 1, 2, \dots, npts$ , and
$u [npts \times npde + i - 1]$ contain $V_{i} (t_{0})$ , for $i = 1, 2, \dots, nv$ .

On exit: the computed solution

U_{i} (x_{j}, t)

, for

i = 1, 2, \dots, npde

and

j = 1, 2, \dots, npts

, and

V_{k} (t)

, for

k = 1, 2, \dots, nv

, evaluated at

t = ts

, as follows:

$u [npde \times (j - 1) + i - 1]$ contain $U_{i} (x_{j}, t)$ , for $i = 1, 2, \dots, npde$ and $j = 1, 2, \dots, npts$ , and
$u [npts \times npde + i - 1]$ contain $V_{i} (t)$ , for $i = 1, 2, \dots, nv$ .

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: $nv$ – Integer Input

On entry: the number of coupled ODE components.

Constraint:

nv \geq 0

11: $odedef$ – function, supplied by the user External Function

odedef must evaluate the functions

R

, which define the system of ODEs, as given in (4).

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: $npde$ – 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: $nv$ – Integer Input

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

4: $v [nv]$ – const double Input

On entry: if

nv > 0

v [i - 1]

contains the value of the component

V_{i} (t)

, for

i = 1, 2, \dots, nv

5: $vdot [nv]$ – const double Input

On entry: if

nv > 0

vdot [i - 1]

contains the value of component

{\dot{V}}_{i} (t)

, for

i = 1, 2, \dots, nv

6: $nxi$ – Integer Input

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

7: $xi [nxi]$ – const double Input

On entry: if

nxi > 0

xi [i - 1]

contains the ODE/PDE coupling point,

ξ_{i}

, for

i = 1, 2, \dots, nxi

8: $ucp [npde \times nxi]$ – const double Input

Note: the

(i, j)

th element of the matrix is stored in

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

On entry: if

nxi > 0

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

contains the value of

U_{i} (x, t)

at the coupling point

x = ξ_{j}

, for

i = 1, 2, \dots, npde

and

j = 1, 2, \dots, nxi

9: $ucpx [npde \times nxi]$ – const double Input

Note: the

(i, j)

th element of the matrix is stored in

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

On entry: if

nxi > 0

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

contains the value of

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

at the coupling point

x = ξ_{j}

, for

i = 1, 2, \dots, npde

and

j = 1, 2, \dots, nxi

10: $ucpt [npde \times nxi]$ – const double Input

Note: the

(i, j)

th element of the matrix is stored in

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

On entry: if

nxi > 0

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

contains the value of

\frac{\partial U_{i}}{\partial t}

at the coupling point

x = ξ_{j}

, for

i = 1, 2, \dots, npde

and

j = 1, 2, \dots, nxi

11: $r [nv]$ – double Output

On exit:

r [i - 1]

must contain the

i

th component of

R

, for

i = 1, 2, \dots, nv

, where

R

is defined as

R = L - M \dot{V} - N U_{t}^{*},

(10)

R = - M \dot{V} - N U_{t}^{*} .

(11)

The definition of

R

is determined by the input value of ires.

12: $ires$ – Integer * Input/Output

On entry: the form of

R

that must be returned in the array r.

$ires = 1$: Equation (10) must be used.
$ires = −1$: 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:

$ires = 2$: 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 $ires = 3$ , d03plc returns to the calling function with the error indicator set to $fail . code =$ NE_FAILED_DERIV.

13: $comm$ – 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 floating-point 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: $nxi$ – Integer Input

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

Constraints:

if $nv = 0$ , $nxi = 0$ ;
if $nv > 0$ , $nxi \geq 0$ .

13: $xi [nxi]$ – const double Input

On entry:

xi [i - 1]

, for

i = 1, 2, \dots, nxi

, must be set to the ODE/PDE coupling points.

Constraint:

x [0] \leq xi [0] < xi [1] < \dots < xi [nxi - 1] \leq x [npts - 1]

14: $neqn$ – Integer Input

On entry: the number of ODEs in the time direction.

Constraint:

neqn = npde \times npts + nv

15: $rtol [\dim]$ – const double Input

Note: the dimension, dim, of the array rtol must be at least

$1$ when $itol = 1$ or $2$ ;
$neqn$ when $itol = 3$ or $4$ .

On entry: the relative local error tolerance.

Constraint:

rtol [i - 1] \geq 0.0

for all relevant

i

16: $atol [\dim]$ – const double Input

Note: the dimension, dim, of the array atol must be at least

$1$ when $itol = 1$ or $3$ ;
$neqn$ when $itol = 2$ or $4$ .

On entry: the absolute local error tolerance.

Constraint:

atol [i - 1] \geq 0.0

for all relevant

i

Note: corresponding elements of rtol and atol cannot both be

0.0

17: $itol$ – Integer Input

On entry: a value to indicate the form of the local error test. If

e_{i}

is the estimated local error for

u [i - 1]

, for

i = 1, 2, \dots, neqn

, and

‖ ‖

denotes the norm, the error test to be satisfied is

‖ e_{i} ‖ < 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_{i}

used in the calculation of the norm (see the description of norm):

itol	rtol	atol	$w_{i}$
1	scalar	scalar	$rtol [0] \times \| u [i - 1] \| + atol [0]$
2	scalar	vector	$rtol [0] \times \| u [i - 1] \| + atol [i - 1]$
3	vector	scalar	$rtol [i - 1] \times \| u [i - 1] \| + atol [0]$
4	vector	vector	$rtol [i - 1] \times \| u [i - 1] \| + atol [i - 1]$

Constraint:

1 \leq itol \leq 4

18: $norm$ – Nag_NormType Input

On entry: the type of norm to be used.

$norm = Nag_OneNorm$: Averaged $L_{1}$ norm.
$norm = Nag_TwoNorm$: Averaged $L_{2}$ norm.

U_{norm}

denotes the norm of the vector u of length neqn, then for the averaged

L_{1}

norm

U_{norm} = \frac{1}{neqn} \sum_{i = 1}^{neqn} u [i - 1] / w_{i},

and for the averaged

L_{2}

norm

U_{norm} = \sqrt{\frac{1}{neqn} \sum_{i = 1}^{neqn} {(u [i - 1] / w_{i})}^{2}} .

See the description of itol for the formulation of the weight vector

w

Constraint:

norm = Nag_OneNorm

Nag_TwoNorm

19: $laopt$ – Nag_LinAlgOption Input

On entry: the type of matrix algebra required.

$laopt = Nag_LinAlgFull$: Full matrix methods to be used.
$laopt = Nag_LinAlgBand$: Banded matrix methods to be used.
$laopt = Nag_LinAlgSparse$: Sparse matrix methods to be used.

Constraint:

laopt = Nag_LinAlgFull

Nag_LinAlgBand

Nag_LinAlgSparse

Note: you are recommended to use the banded option when no coupled ODEs are present (

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: $algopt [30]$ – 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,

algopt [0]

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:

$algopt [0]$: Selects the ODE integration method to be used. If $algopt [0] = 1.0$ , a BDF method is used and if $algopt [0] = 2.0$ , a Theta method is used. The default is $algopt [0] = 1.0$ .

algopt [0] = 2.0

, then

algopt [i - 1]

, for

i = 2, 3, 4

, are not used.

$algopt [1]$: Specifies the maximum order of the BDF integration formula to be used. $algopt [1]$ may be $1.0$ , $2.0$ , $3.0$ , $4.0$ or $5.0$ . The default value is $algopt [1] = 5.0$ .
$algopt [2]$: Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the BDF method. If $algopt [2] = 1.0$ a modified Newton iteration is used and if $algopt [2] = 2.0$ 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 $algopt [2] = 1.0$ .
$algopt [3]$: 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 $P_{i, j} = 0.0$ , for $j = 1, 2, \dots, npde$ , for some $i$ or when there is no ${\dot{V}}_{i} (t)$ dependence in the coupled ODE system. If $algopt [3] = 1.0$ , the Petzold test is used. If $algopt [3] = 2.0$ , the Petzold test is not used. The default value is $algopt [3] = 1.0$ .

algopt [0] = 1.0

algopt [i - 1]

, for

i = 5, 6, 7

, are not used.

$algopt [4]$: Specifies the value of Theta to be used in the Theta integration method. $0.51 \leq algopt [4] \leq 0.99$ . The default value is $algopt [4] = 0.55$ .
$algopt [5]$: Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the Theta method. If $algopt [5] = 1.0$ , a modified Newton iteration is used and if $algopt [5] = 2.0$ , a functional iteration method is used. The default value is $algopt [5] = 1.0$ .
$algopt [6]$: 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 $algopt [6] = 1.0$ , switching is allowed and if $algopt [6] = 2.0$ , switching is not allowed. The default value is $algopt [6] = 1.0$ .
$algopt [10]$: Specifies a point in the time direction, $t_{crit}$ , beyond which integration must not be attempted. The use of $t_{crit}$ is described under the argument itask. If $algopt [0] \neq 0.0$ , a value of $0.0$ for $algopt [10]$ , say, should be specified even if itask subsequently specifies that $t_{crit}$ will not be used.
$algopt [11]$: Specifies the minimum absolute step size to be allowed in the time integration. If this option is not required, $algopt [11]$ should be set to $0.0$ .
$algopt [12]$: Specifies the maximum absolute step size to be allowed in the time integration. If this option is not required, $algopt [12]$ should be set to $0.0$ .
$algopt [13]$: Specifies the initial step size to be attempted by the integrator. If $algopt [13] = 0.0$ , the initial step size is calculated internally.
$algopt [14]$: Specifies the maximum number of steps to be attempted by the integrator in any one call. If $algopt [14] = 0.0$ , no limit is imposed.
$algopt [22]$: 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 $\dot{V}$ . If $algopt [22] = 1.0$ , a modified Newton iteration is used and if $algopt [22] = 2.0$ , functional iteration is used. The default value is $algopt [22] = 1.0$ .

algopt [28]

and

algopt [29]

are used only for the sparse matrix algebra option, i.e.,

laopt = Nag_LinAlgSparse

$algopt [28]$: Governs the choice of pivots during the decomposition of the first Jacobian matrix. It should lie in the range $0.0 < algopt [28] < 1.0$ , with smaller values biasing the algorithm towards maintaining sparsity at the expense of numerical stability. If $algopt [28]$ lies outside the range then the default value is used. If the functions regard the Jacobian matrix as numerically singular, increasing $algopt [28]$ towards $1.0$ may help, but at the cost of increased fill-in. The default value is $algopt [28] = 0.1$ .
$algopt [29]$: Used as the relative pivot threshold during subsequent Jacobian decompositions (see $algopt [28]$ ) below which an internal error is invoked. $algopt [29]$ must be greater than zero, otherwise the default value is used. If $algopt [29]$ 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 $algopt [28]$ ). The default value is $algopt [29] = 0.0001$ .

21: $rsave [lrsave]$ – 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.

22: $lrsave$ – Integer Input

On entry: the dimension of the array rsave. Its size depends on the type of matrix algebra selected.

laopt = Nag_LinAlgFull

lrsave \geq neqn \times neqn + neqn + nwkres + lenode

laopt = Nag_LinAlgBand

lrsave \geq (3 mlu + 1) \times neqn + nwkres + lenode

laopt = Nag_LinAlgSparse

lrsave \geq 4 neqn + 11 neqn / 2 + 1 + nwkres + lenode

Where

mlu

is the lower or upper half bandwidths such that

for PDE problems only (no coupled ODEs),: $mlu = 3 npde - 1;$
for coupled PDE/ODE problems,: $mlu = neqn - 1 .$

Where

nwkres

is defined by

if $nv > 0 and nxi > 0$ ,: $nwkres = npde (2 npts + 6 nxi + 3 npde + 26) + nxi + nv + 7 npts + 2;$
if $nv > 0 and nxi = 0$ ,: $nwkres = npde (2 npts + 3 npde + 32) + nv + 7 npts + 3;$
if $nv = 0$ ,: $nwkres = npde (2 npts + 3 npde + 32) + 7 npts + 4 .$

Where

lenode

is defined by

if the BDF method is used,: $lenode = (6 + int (algopt [1])) \times neqn + 50;$
if the Theta method is used,: $lenode = 9 neqn + 50,$

Note: when

laopt = 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

itrace > 0

and the function returns with

fail . code =

NE_INT_2.

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

ind = 0

, isave need not be set.

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:

$isave [0]$: 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 evaluating 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 BDF method last used in the time integration, if applicable. When the Theta method is used, $isave [3]$ contains no useful information.
$isave [4]$: Contains the number of Newton iterations performed by the time integrator. Each iteration involves residual evaluation of the resulting ODE system followed by a back-substitution using the $L U$ decomposition of the Jacobian matrix.

24: $lisave$ – Integer Input

On entry: the dimension of the array isave. Its size depends on the type of matrix algebra selected:

if $laopt = Nag_LinAlgFull$ , $lisave \geq 24$ ;
if $laopt = Nag_LinAlgBand$ , $lisave \geq neqn + 24$ ;
if $laopt = Nag_LinAlgSparse$ , $lisave \geq 25 \times neqn + 24$ .

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

itrace > 0

and the function returns with

fail . code =

NE_INT_2.

25: $itask$ – 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 $t = tout$ .
$itask = 4$: Normal computation of output values u at $t = tout$ but without overshooting $t = t_{crit}$ where $t_{crit}$ is described under the argument algopt.
$itask = 5$: Take one step in the time direction and return, without passing $t_{crit}$ , where $t_{crit}$ is described under the argument algopt.

Constraint:

itask = 1

2

3

4

5

26: $itrace$ – 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

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.

27: $outfile$ – 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: $ind$ – 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 d03plc.

Constraint:

ind = 0

1

On exit:

ind = 1

29: $comm$ – Nag_Comm *

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

30: $saved$ – 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: $fail$ – 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 $itask \neq 2$ 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 $⟨ value ⟩$ 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, 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: $ts = ⟨ value ⟩$ .
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 atol and rtol. $ts = ⟨ value ⟩$ .
NE_INT: ires set to an invalid value in a call to user-supplied functions pdedef, numflx, bndary or odedef.

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

On entry, $itask = ⟨ value ⟩$ .
Constraint: $itask = 1$ , $2$ , $3$ , $4$ or $5$ .

On entry, $itol = ⟨ value ⟩$ .
Constraint: $itol = 1$ , $2$ , $3$ or $4$ .

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

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

On entry, $nv = ⟨ value ⟩$ .
Constraint: $nv \geq 0$ .

On entry, on initial entry $ind = 1$ .
Constraint: on initial entry $ind = 0$ .
NE_INT_2: On entry, $i = ⟨ value ⟩$ and $j = ⟨ value ⟩$ .
Constraint: corresponding elements $atol [i - 1]$ and $rtol [j - 1]$ cannot both be $0.0$ .

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

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

On entry, $nv = ⟨ value ⟩$ and $nxi = ⟨ value ⟩$ .
Constraint: $nxi = 0$ when $nv = 0$ .

On entry, $nv = ⟨ value ⟩$ and $nxi = ⟨ value ⟩$ .
Constraint: $nxi \geq 0$ when $nv > 0$ .

When using the sparse option lisave or lrsave is too small: $lisave = ⟨ value ⟩$ , $lrsave = ⟨ value ⟩$ .
NE_INT_4: On entry, $neqn = ⟨ value ⟩$ , $npde = ⟨ value ⟩$ , $npts = ⟨ value ⟩$ and $nv = ⟨ value ⟩$ .
Constraint: $neqn = npde \times npts + nv$ .
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 $algopt [14]$ has been exceeded. $algopt [14] = ⟨ value ⟩$ .
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: $x [0] < x [1] < \dots < x [npts - 1]$ .

On entry, $i = ⟨ value ⟩$ , $xi [i] = ⟨ value ⟩$ and $xi [i - 1] = ⟨ value ⟩$ .
Constraint: $xi [i] > xi [i - 1]$ .
NE_NOT_WRITE_FILE: Cannot open file $⟨ value ⟩$ for writing.
NE_REAL_2: On entry, at least one point in xi lies outside $[x [0], x [npts - 1]]$ : $x [0] = ⟨ value ⟩$ and $x [npts - 1] = ⟨ value ⟩$ .

On entry, $tout = ⟨ value ⟩$ and $ts = ⟨ value ⟩$ .
Constraint: $tout > ts$ .

On entry, $tout - ts$ is too small: $tout = ⟨ value ⟩$ and $ts = ⟨ value ⟩$ .
NE_REAL_ARRAY: On entry, $i = ⟨ value ⟩$ and $atol [i - 1] = ⟨ value ⟩$ .
Constraint: $atol [i - 1] \geq 0.0$ .

On entry, $i = ⟨ value ⟩$ and $rtol [i - 1] = ⟨ value ⟩$ .
Constraint: $rtol [i - 1] \geq 0.0$ .
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, $ires = 2$ has been set in user-supplied functions pdedef, numflx, bndary or odedef. Integration is successful as far as ts: $ts = ⟨ value ⟩$ .
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 ( $atol [i - 1] = 0.0$ ) was requested on a variable (the $i$ th) which has become zero. The integration was successful as far as $t = ts$ .

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 implementation-specific information.

9 Further Comments

d03plc 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

algopt [12]

. 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. 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 first-order system with coupled ODEs arising from the use of the characteristic equations for the numerical boundary conditions.

The PDEs are

\begin{matrix} \frac{\partial U_{1}}{\partial t} + \frac{\partial U_{1}}{\partial x} + 2 \frac{\partial U_{2}}{\partial x} = 0, \\ \frac{\partial U_{2}}{\partial t} + 2 \frac{\partial U_{1}}{\partial x} + \frac{\partial U_{2}}{\partial x} = 0, \end{matrix}

for

x \in [0, 1]

and

t \geq 0

The PDEs have an exact solution given by

U_{1} (x, t) = f (x - 3 t) + g (x + t), U_{2} (x, t) = f (x - 3 t) - g (x + t),

where

f (z) = \exp (π z) \sin (2 π z)

g (z) = \exp (- 2 π z) \cos (2 π z)

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

d x / d t = −1

and

d x / d t = 3

respectively. Hence we require a physical boundary condition for

W_{2}

at the left-hand boundary and for

W_{1}

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

W_{1}

at the left-hand boundary and for

W_{2}

at the right-hand 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

\frac{\partial W_{1}}{\partial t} - \frac{\partial W_{1}}{\partial x} = 0

at the left-hand boundary, and

\frac{\partial W_{2}}{\partial t} + 3 \frac{\partial W_{2}}{\partial x} = 0

at the right-hand boundary.

In order to specify these boundary conditions, two ODE variables

V_{1}

and

V_{2}

are introduced, defined by

\begin{matrix} V_{1} (t) = W_{1} (0, t) = U_{1} (0, t) - U_{2} (0, t), \\ V_{2} (t) = W_{2} (1, t) = U_{1} (1, t) + U_{2} (1, t) . \end{matrix}

The coupling points are, therefore, at

x = 0

and

x = 1

The numerical boundary conditions are now

{\dot{V}}_{1} - \frac{\partial W_{1}}{\partial x} = 0

at the left-hand boundary, and

{\dot{V}}_{2} + 3 \frac{\partial W_{2}}{\partial x} = 0

at the right-hand boundary.

The spatial derivatives are evaluated at the appropriate boundary points in bndary using one-sided 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

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

Example 2 (ex2)

This example is the standard shock-tube 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

\begin{array}{rcl} \frac{\partial ρ}{\partial t} + \frac{\partial m}{\partial x} & = & 0, \\ \frac{\partial m}{\partial t} + \frac{\partial}{\partial x} (\frac{m^{2}}{ρ} + (γ - 1) (e - \frac{m^{2}}{2 ρ})) & = & 0, \\ \frac{\partial e}{\partial t} + \frac{\partial}{\partial x} (\frac{m e}{ρ} + \frac{m}{ρ} (γ - 1) (e - \frac{m^{2}}{2 ρ})) & = & 0, \end{array}

where

ρ

is the density;

m

is the momentum, such that

m = ρ u

, where

u

is the velocity;

e

is the specific energy; and

γ

is the (constant) ratio of specific heats. The pressure

p

is given by

p = (γ - 1) (e - \frac{ρ u^{2}}{2}) .

The solution domain is

0 \leq x \leq 1

for

0 < t \leq 0.2

, with the initial discontinuity at

x = 0.5

, and initial conditions

\begin{array}{l} ρ (x, 0) = 1, & m (x, 0) = 0, & e (x, 0) = 2.5, & for ​ x < 0.5, \\ ρ (x, 0) = 0.125, & m (x, 0) = 0, & e (x, 0) = 0.25, & for ​ x > 0.5 . \end{array}

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 argument-vector 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.