NAG Library Routine Document

D03PLF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

D03PLF 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

SUBROUTINE D03PLF (

NPDE, TS, TOUT, PDEDEF, NUMFLX, BNDARY, U, NPTS, X, NCODE, ODEDEF, NXI, XI, NEQN, RTOL, ATOL, ITOL, NORM, LAOPT, ALGOPT, RSAVE, LRSAVE, ISAVE, LISAVE, ITASK, ITRACE, IND, IFAIL)

INTEGER	NPDE, NPTS, NCODE, NXI, NEQN, ITOL, LRSAVE, ISAVE(LISAVE), LISAVE, ITASK, ITRACE, IND, IFAIL
REAL (KIND=nag_wp)	TS, TOUT, U(NEQN), X(NPTS), XI(), RTOL(), ATOL(*), ALGOPT(30), RSAVE(LRSAVE)
CHARACTER(1)	NORM, LAOPT
EXTERNAL	PDEDEF, NUMFLX, BNDARY, ODEDEF

3 Description

D03PLF 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, NCODE,

(3)

where the vector

V

is the set of ODE solution values

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

L

is a vector of length NCODE,

M

is an NCODE by NCODE matrix,

N

is an NCODE 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 D03PCF/D03PCA, D03PHF/D03PHA and D03PPF/D03PPA, 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 D03PLF 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)

An example is given in Section 9 and in the D03PFF documentation.

If the system is nonlinear, Roe's scheme requires that a linearized Jacobian is found (see Roe (1981)).

The functions

P_{i, j}

C_{i}

D_{i}

and

S_{i}

(but not

F_{i}

) must be specified in PDEDEF. The numerical flux

{\hat{F}}_{i}

must be supplied in a separate NUMFLX. For problems in the form (2), the actual argument D03PLP may be used for PDEDEF. D03PLP is included in the NAG Library and sets the matrix with entries

P_{i, j}

to the identity matrix, and the functions

C_{i}

D_{i}

and

S_{i}

to zero.

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 Section 9 and in the D03PFF 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 $j = 1, 2, \dots, NCODE$ , 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 + NCODE

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 Parameters

1: NPDE – INTEGERInput

On entry: the number of PDEs to be solved.

Constraint:

NPDE \geq 1

2: TS – REAL (KIND=nag_wp)Input/Output

On entry: the initial value of the independent variable

t

On exit: the value of

t

corresponding to the solution values in U. Normally

TS = TOUT

Constraint:

TS < TOUT

3: TOUT – REAL (KIND=nag_wp)Input

On entry: the final value of

t

to which the integration is to be carried out.

4: PDEDEF – SUBROUTINE, supplied by the NAG Library or the user.External Procedure

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 D03PLF. The actual argument D03PLP may be used for PDEDEF for problems in the form (2). (D03PLP is included in the NAG Library.)

The specification of PDEDEF is:

SUBROUTINE PDEDEF (

NPDE, T, X, U, UX, NCODE, V, VDOT, P, C, D, S, IRES)

INTEGER	NPDE, NCODE, IRES
REAL (KIND=nag_wp)	T, X, U(NPDE), UX(NPDE), V(NCODE), VDOT(NCODE), P(NPDE,NPDE), C(NPDE), D(NPDE), S(NPDE)

1: NPDE – INTEGERInput

On entry: the number of PDEs in the system.

2: T – REAL (KIND=nag_wp)Input

On entry: the current value of the independent variable

t

3: X – REAL (KIND=nag_wp)Input

On entry: the current value of the space variable

x

4: U(NPDE) – REAL (KIND=nag_wp) arrayInput

On entry:

U (i)

contains the value of the component

U_{i} (x, t)

, for

i = 1, 2, \dots, NPDE

5: UX(NPDE) – REAL (KIND=nag_wp) arrayInput

On entry:

UX (i)

contains the value of the component

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

, for

i = 1, 2, \dots, NPDE

6: NCODE – INTEGERInput

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

7: V(NCODE) – REAL (KIND=nag_wp) arrayInput

On entry: if

NCODE > 0

V (i)

contains the value of the component

V_{i} (t)

, for

i = 1, 2, \dots, NCODE

8: VDOT(NCODE) – REAL (KIND=nag_wp) arrayInput

On entry: if

NCODE > 0

VDOT (i)

contains the value of component

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

, for

i = 1, 2, \dots, NCODE

Note:

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

, for

i = 1, 2, \dots, NCODE

, may only appear linearly in

S_{j}

, for

j = 1, 2, \dots, NPDE

9: P(NPDE,NPDE) – REAL (KIND=nag_wp) arrayOutput

On exit:

P (i, j)

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) – REAL (KIND=nag_wp) arrayOutput

On exit:

C (i)

must be set to the value of

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

, for

i = 1, 2, \dots, NPDE

11: D(NPDE) – REAL (KIND=nag_wp) arrayOutput

On exit:

D (i)

must be set to the value of

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

, for

i = 1, 2, \dots, NPDE

12: S(NPDE) – REAL (KIND=nag_wp) arrayOutput

On exit:

S (i)

must be set to the value of

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

, for

i = 1, 2, \dots, NPDE

13: IRES – INTEGERInput/Output

On entry: set to

- 1 ​ or ​ 1

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

$IRES = 2$: Indicates to the integrator that control should be passed back immediately to the calling subroutine with the error indicator set to $IFAIL = 6$ .
$IRES = 3$: 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 $IRES = 3$ , then D03PLF returns to the calling subroutine with the error indicator set to $IFAIL = 4$ .

PDEDEF must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D03PLF is called. Parameters denoted as Input must not be changed by this procedure.

5: NUMFLX – SUBROUTINE, supplied by the user.External Procedure

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 D03PLF.

The specification of NUMFLX is:

SUBROUTINE NUMFLX (

NPDE, T, X, NCODE, V, ULEFT, URIGHT, FLUX, IRES)

INTEGER	NPDE, NCODE, IRES
REAL (KIND=nag_wp)	T, X, V(NCODE), ULEFT(NPDE), URIGHT(NPDE), FLUX(NPDE)

1: NPDE – INTEGERInput

On entry: the number of PDEs in the system.

2: T – REAL (KIND=nag_wp)Input

On entry: the current value of the independent variable

t

3: X – REAL (KIND=nag_wp)Input

On entry: the current value of the space variable

x

4: NCODE – INTEGERInput

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

5: V(NCODE) – REAL (KIND=nag_wp) arrayInput

On entry: if

NCODE > 0

V (i)

contains the value of the component

V_{i} (t)

, for

i = 1, 2, \dots, NCODE

6: ULEFT(NPDE) – REAL (KIND=nag_wp) arrayInput

On entry:

ULEFT (i)

contains the left value of the component

U_{i} (x)

, for

i = 1, 2, \dots, NPDE

7: URIGHT(NPDE) – REAL (KIND=nag_wp) arrayInput

On entry:

URIGHT (i)

contains the right value of the component

U_{i} (x)

, for

i = 1, 2, \dots, NPDE

8: FLUX(NPDE) – REAL (KIND=nag_wp) arrayOutput

On exit:

FLUX (i)

must be set to the numerical flux

{\hat{F}}_{i}

, for

i = 1, 2, \dots, NPDE

9: IRES – INTEGERInput/Output

On entry: set to

- 1 ​ or ​ 1

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

$IRES = 2$: Indicates to the integrator that control should be passed back immediately to the calling subroutine with the error indicator set to $IFAIL = 6$ .
$IRES = 3$: 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 $IRES = 3$ , then D03PLF returns to the calling subroutine with the error indicator set to $IFAIL = 4$ .

NUMFLX must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D03PLF is called. Parameters denoted as Input must not be changed by this procedure.

6: BNDARY – SUBROUTINE, supplied by the user.External Procedure

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:

SUBROUTINE BNDARY (

NPDE, NPTS, T, X, U, NCODE, V, VDOT, IBND, G, IRES)

INTEGER	NPDE, NPTS, NCODE, IBND, IRES
REAL (KIND=nag_wp)	T, X(NPTS), U(NPDE,NPTS), V(NCODE), VDOT(NCODE), G(NPDE)

1: NPDE – INTEGERInput

On entry: the number of PDEs in the system.

2: NPTS – INTEGERInput

On entry: the number of mesh points in the interval

[a, b]

3: T – REAL (KIND=nag_wp)Input

On entry: the current value of the independent variable

t

4: X(NPTS) – REAL (KIND=nag_wp) arrayInput

On entry: the mesh points in the spatial direction.

X (1)

corresponds to the left-hand boundary,

a

, and

X (NPTS)

corresponds to the right-hand boundary,

b

5: U(NPDE,NPTS) – REAL (KIND=nag_wp) arrayInput

On entry:

U (i, j)

contains the value of the component

U_{i} (x, t)

x = X (j)

, 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: NCODE – INTEGERInput

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

7: V(NCODE) – REAL (KIND=nag_wp) arrayInput

On entry: if

NCODE > 0

V (i)

contains the value of the component

V_{i} (t)

, for

i = 1, 2, \dots, NCODE

8: VDOT(NCODE) – REAL (KIND=nag_wp) arrayInput

On entry: if

NCODE > 0

VDOT (i)

contains the value of component

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

, for

i = 1, 2, \dots, NCODE

Note:

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

, for

i = 1, 2, \dots, NCODE

, may only appear linearly in

G_{j}^{L}

and

G_{j}^{R}

, for

j = 1, 2, \dots, NPDE

9: IBND – INTEGERInput

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) – REAL (KIND=nag_wp) arrayOutput

On exit:

G (i)

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 – INTEGERInput/Output

On entry: set to

- 1 ​ or ​ 1

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

$IRES = 2$: Indicates to the integrator that control should be passed back immediately to the calling subroutine with the error indicator set to $IFAIL = 6$ .
$IRES = 3$: 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 $IRES = 3$ , then D03PLF returns to the calling subroutine with the error indicator set to $IFAIL = 4$ .

BNDARY must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D03PLF is called. Parameters denoted as Input must not be changed by this procedure.

7: U(NEQN) – REAL (KIND=nag_wp) arrayInput/Output

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

$U (NPDE \times (j - 1) + i)$ contain $U_{i} (x_{j}, t_{0})$ , for $i = 1, 2, \dots, NPDE$ and $j = 1, 2, \dots, NPTS$ , and
$U (NPTS \times NPDE + k)$ contain $V_{k} (t_{0})$ , for $k = 1, 2, \dots, NCODE$ .

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, NCODE

, all evaluated at

t = TS

8: NPTS – INTEGERInput

On entry: the number of mesh points in the interval

[a, b]

Constraint:

NPTS \geq 3

9: X(NPTS) – REAL (KIND=nag_wp) arrayInput

On entry: the mesh points in the space direction.

X (1)

must specify the left-hand boundary,

a

, and

X (NPTS)

must specify the right-hand boundary,

b

Constraint:

X (1) < X (2) < \dots < X (NPTS)

10: NCODE – INTEGERInput

On entry: the number of coupled ODE components.

Constraint:

NCODE \geq 0

11: ODEDEF – SUBROUTINE, supplied by the NAG Library or the user.External Procedure

ODEDEF must evaluate the functions

R

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

If you wish to compute the solution of a system of PDEs only (i.e.,

NCODE = 0

), ODEDEF must be the dummy routine D03PEK. (D03PEK is included in the NAG Library.)

The specification of ODEDEF is:

SUBROUTINE ODEDEF (

NPDE, T, NCODE, V, VDOT, NXI, XI, UCP, UCPX, UCPT, R, IRES)

INTEGER	NPDE, NCODE, NXI, IRES
REAL (KIND=nag_wp)	T, V(NCODE), VDOT(NCODE), XI(NXI), UCP(NPDE,), UCPX(NPDE,), UCPT(NPDE,*), R(NCODE)

1: NPDE – INTEGERInput

On entry: the number of PDEs in the system.

2: T – REAL (KIND=nag_wp)Input

On entry: the current value of the independent variable

t

3: NCODE – INTEGERInput

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

4: V(NCODE) – REAL (KIND=nag_wp) arrayInput

On entry: if

NCODE > 0

V (i)

contains the value of the component

V_{i} (t)

, for

i = 1, 2, \dots, NCODE

5: VDOT(NCODE) – REAL (KIND=nag_wp) arrayInput

On entry: if

NCODE > 0

VDOT (i)

contains the value of component

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

, for

i = 1, 2, \dots, NCODE

6: NXI – INTEGERInput

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

7: XI(NXI) – REAL (KIND=nag_wp) arrayInput

On entry: if

NXI > 0

XI (i)

contains the ODE/PDE coupling point,

ξ_{i}

, for

i = 1, 2, \dots, NXI

8: UCP(NPDE, $*$ ) – REAL (KIND=nag_wp) arrayInput

On entry: if

NXI > 0

UCP (i, j)

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, $*$ ) – REAL (KIND=nag_wp) arrayInput

On entry: if

NXI > 0

UCPX (i, j)

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, $*$ ) – REAL (KIND=nag_wp) arrayInput

On entry: if

NXI > 0

UCPT (i, j)

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(NCODE) – REAL (KIND=nag_wp) arrayOutput

On exit:

R (i)

must contain the

i

th component of

R

, for

i = 1, 2, \dots, NCODE

, 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 – INTEGERInput/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 routine to take certain actions, as described below:

$IRES = 2$: Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to $IFAIL = 6$ .
$IRES = 3$: 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 $IRES = 3$ , then D03PLF returns to the calling subroutine with the error indicator set to $IFAIL = 4$ .

ODEDEF must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D03PLF is called. Parameters denoted as Input must not be changed by this procedure.

12: NXI – INTEGERInput

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

Constraints:

if $NCODE = 0$ , $NXI = 0$ ;
if $NCODE > 0$ , $NXI \geq 0$ .

13: XI( $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array XI must be at least

\max (1, NXI)

On entry:

XI (i)

, for

i = 1, 2, \dots, NXI

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

Constraint:

X (1) \leq XI (1) < XI (2) < \dots < XI (NXI) \leq X (NPTS)

14: NEQN – INTEGERInput

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

Constraint:

NEQN = NPDE \times NPTS + NCODE

15: RTOL( $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array RTOL must be at least

1

ITOL = 1

2

and at least

NEQN

ITOL = 3

4

On entry: the relative local error tolerance.

Constraint:

RTOL (i) \geq 0.0

for all relevant

i

16: ATOL( $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array ATOL must be at least

1

ITOL = 1

3

and at least

NEQN

ITOL = 2

4

On entry: the absolute local error tolerance.

Constraint:

ATOL (i) \geq 0.0

for all relevant

i

Note: corresponding elements of RTOL and ATOL cannot both be

0.0

17: ITOL – INTEGERInput

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

e_{i}

is the estimated local error for

U (i)

, for

i = 1, 2, \dots, NEQN

, and

‖‖

denotes the norm, then the error test to be satisfied is

‖e_{i}‖ < 1.0

. ITOL indicates to D03PLF 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	$RTOL (1) \times \|U (i)\| + ATOL (1)$
2	scalar	vector	$RTOL (1) \times \|U (i)\| + ATOL (i)$
3	vector	scalar	$RTOL (i) \times \|U (i)\| + ATOL (1)$
4	vector	vector	$RTOL (i) \times \|U (i)\| + ATOL (i)$

Constraint:

1 \leq ITOL \leq 4

18: NORM – CHARACTER(1)Input

On entry: the type of norm to be used.

$NORM ='1'$: Averaged $L_{1}$ norm.
$NORM ='2'$: 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) / w_{i},

and for the averaged

L_{2}

norm

U_{norm} = \sqrt{\frac{1}{NEQN} \sum_{i = 1}^{NEQN} {(U (i) / w_{i})}^{2}} .

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

w

Constraint:

NORM ='1'

'2'

19: LAOPT – CHARACTER(1)Input

On entry: the type of matrix algebra required.

$LAOPT ='F'$: Full matrix methods to be used.
$LAOPT ='B'$: Banded matrix methods to be used.
$LAOPT ='S'$: Sparse matrix methods to be used.

Constraint:

LAOPT ='F'

'B'

'S'

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

NCODE = 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( $30$ ) – REAL (KIND=nag_wp) arrayInput

On entry: may be set to control various options available in the integrator. If you wish to employ all the default options, then

ALGOPT (1)

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:

$ALGOPT (1)$: Selects the ODE integration method to be used. If $ALGOPT (1) = 1.0$ , a BDF method is used and if $ALGOPT (1) = 2.0$ , a Theta method is used. The default is $ALGOPT (1) = 1.0$ .

ALGOPT (1) = 2.0

, then

ALGOPT (i)

, for

i = 2, 3, 4

, are not used.

$ALGOPT (2)$: Specifies the maximum order of the BDF integration formula to be used. $ALGOPT (2)$ may be $1.0$ , $2.0$ , $3.0$ , $4.0$ or $5.0$ . The default value is $ALGOPT (2) = 5.0$ .
$ALGOPT (3)$: Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the BDF method. If $ALGOPT (3) = 1.0$ a modified Newton iteration is used and if $ALGOPT (3) = 2.0$ a functional iteration method is used. If functional iteration is selected and the integrator encounters difficulty, then there is an automatic switch to the modified Newton iteration. The default value is $ALGOPT (3) = 1.0$ .
$ALGOPT (4)$: 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 $ALGOPT (4) = 1.0$ , then the Petzold test is used. If $ALGOPT (4) = 2.0$ , then the Petzold test is not used. The default value is $ALGOPT (4) = 1.0$ .

ALGOPT (1) = 1.0

, then

ALGOPT (i)

, for

i = 5, 6, 7

, are not used.

$ALGOPT (5)$: Specifies the value of Theta to be used in the Theta integration method. $0.51 \leq ALGOPT (5) \leq 0.99$ . The default value is $ALGOPT (5) = 0.55$ .
$ALGOPT (6)$: Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the Theta method. If $ALGOPT (6) = 1.0$ , a modified Newton iteration is used and if $ALGOPT (6) = 2.0$ , a functional iteration method is used. The default value is $ALGOPT (6) = 1.0$ .
$ALGOPT (7)$: 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 $ALGOPT (7) = 1.0$ , then switching is allowed and if $ALGOPT (7) = 2.0$ , then switching is not allowed. The default value is $ALGOPT (7) = 1.0$ .
$ALGOPT (11)$: 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 parameter ITASK. If $ALGOPT (1) \neq 0.0$ , a value of $0.0$ for $ALGOPT (11)$ , say, should be specified even if ITASK subsequently specifies that $t_{crit}$ will not be used.
$ALGOPT (12)$: Specifies the minimum 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 maximum absolute step size to be allowed in the time integration. If this option is not required, $ALGOPT (13)$ should be set to $0.0$ .
$ALGOPT (14)$: Specifies the initial step size to be attempted by the integrator. If $ALGOPT (14) = 0.0$ , then the initial step size is calculated internally.
$ALGOPT (15)$: Specifies the maximum number of steps to be attempted by the integrator in any one call. If $ALGOPT (15) = 0.0$ , then no limit is imposed.
$ALGOPT (23)$: 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 $ALGOPT (23) = 1.0$ , a modified Newton iteration is used and if $ALGOPT (23) = 2.0$ , functional iteration is used. The default value is $ALGOPT (23) = 1.0$ .

ALGOPT (29)

and

ALGOPT (30)

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

LAOPT ='S'

$ALGOPT (29)$: Governs the choice of pivots during the decomposition of the first Jacobian matrix. It should lie in the range $0.0 < ALGOPT (29) < 1.0$ , with smaller values biasing the algorithm towards maintaining sparsity at the expense of numerical stability. If $ALGOPT (29)$ lies outside the range then the default value is used. If the routines regard the Jacobian matrix as numerically singular, then increasing $ALGOPT (29)$ towards $1.0$ may help, but at the cost of increased fill-in. The default value is $ALGOPT (29) = 0.1$ .
$ALGOPT (30)$: Used as the relative pivot threshold during subsequent Jacobian decompositions (see $ALGOPT (29)$ ) below which an internal error is invoked. $ALGOPT (30)$ must be greater than zero, otherwise the default value is used. If $ALGOPT (30)$ 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 $ALGOPT (29)$ ). The default value is $ALGOPT (30) = 0.0001$ .

21: RSAVE(LRSAVE) – REAL (KIND=nag_wp) arrayCommunication Array

IND = 0

, RSAVE need not be set on entry.

IND = 1

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

22: LRSAVE – INTEGERInput

On entry: the dimension of the array RSAVE as declared in the (sub)program from which D03PLF is called. Its size depends on the type of matrix algebra selected.

LAOPT ='F'

LRSAVE \geq NEQN \times NEQN + NEQN + nwkres + lenode

LAOPT ='B'

LRSAVE \geq (3 \times mlu + 1) \times NEQN + nwkres + lenode

LAOPT ='S'

LRSAVE \geq 4 \times NEQN + 11 \times NEQN / 2 + 1 + nwkres + lenode

Where

	$mlu$ is the lower or upper half bandwidths such that $mlu = 3 \times NPDE - 1$ , for PDE problems only (no coupled ODEs); or $mlu = NEQN - 1$ , for coupled PDE/ODE problems.
	$nwkres = \{\begin{cases} NPDE \times (2 \times NPTS + 6 \times NXI + 3 \times NPDE + 26) + NXI + NCODE + 7 \times NPTS + 2, & when NCODE > 0 and NXI > 0; \\ NPDE \times (2 \times NPTS + 3 \times NPDE + 32) + NCODE + 7 \times NPTS + 3, & when NCODE > 0 and NXI = 0; \\ NPDE \times (2 \times NPTS + 3 \times NPDE + 32) + 7 \times NPTS + 4, & when NCODE = 0 . \end{cases}$
	$lenode = \{\begin{cases} (6 + int (ALGOPT (2))) \times NEQN + 50, & when the BDF method is used; or \\ 9 \times NEQN + 50, & when the Theta method is used. \end{cases}$

Note: when

LAOPT ='S'

, the value of LRSAVE may be too small when supplied to the integrator. An estimate of the minimum size of LRSAVE is printed on the current error message unit if

ITRACE > 0

and the routine returns with

IFAIL = 15

23: ISAVE(LISAVE) – INTEGER arrayCommunication Array

IND = 0

, ISAVE need not be set.

IND = 1

, ISAVE must be unchanged from the previous call to the routine because it contains required information about the iteration. In particular the following components of the array ISAVE concern the efficiency of the integration:

$ISAVE (1)$: Contains the number of steps taken in time.
$ISAVE (2)$: 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.
$ISAVE (3)$: Contains the number of Jacobian evaluations performed by the time integrator.
$ISAVE (4)$: Contains the order of the BDF method last used in the time integration, if applicable. When the Theta method is used, $ISAVE (4)$ contains no useful information.
$ISAVE (5)$: 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 – INTEGERInput

On entry: the dimension of the array ISAVE as declared in the (sub)program from which D03PLF is called. Its size depends on the type of matrix algebra selected:

if $LAOPT ='F'$ , $LISAVE \geq 24$ ;
if $LAOPT ='B'$ , $LISAVE \geq NEQN + 24$ ;
if $LAOPT ='S'$ , $LISAVE \geq 25 \times NEQN + 24$ .

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 on the current error message unit if

ITRACE > 0

and the routine returns with

IFAIL = 15

25: ITASK – INTEGERInput

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 parameter ALGOPT.
$ITASK = 5$: Take one step in the time direction and return, without passing $t_{crit}$ , where $t_{crit}$ is described under the parameter ALGOPT.

Constraint:

ITASK = 1

2

3

4

5

26: ITRACE – INTEGERInput

On entry: the level of trace information required from D03PLF 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 on the current error message unit (see X04AAF).
$ITRACE > 0$: Output from the underlying ODE solver is printed on the current advisory message unit (see X04ABF). This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.

ITRACE < - 1

, then

- 1

is assumed and similarly if

ITRACE > 3

, then

3

is assumed.

The advisory messages are given in greater detail as ITRACE increases. You are advised to set

ITRACE = 0

, unless you are experienced with sub-chapter D02M–N.

27: IND – INTEGERInput/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 routine. In this case, only the parameters TOUT and IFAIL should be reset between calls to D03PLF.

Constraint:

IND = 0

1

On exit:

IND = 1

28: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$TS \geq TOUT$ ,
or	$TOUT - TS$ is too small,
or	$ITASK \neq 1$ , $2$ , $3$ , $4$ or $5$ ,
or	at least one of the coupling points defined in array XI is outside the interval [ $X (1), X (NPTS)$ ],
or	the coupling points are not in strictly increasing order,
or	$NPTS < 3$ ,
or	$NPDE < 1$ ,
or	$LAOPT \neq'F'$ , 'B' or 'S',
or	$ITOL \neq 1$ , $2$ , $3$ or $4$ ,
or	$IND \neq 0$ or $1$ ,
or	mesh points $X (i)$ are badly ordered,
or	LRSAVE or LISAVE are too small,
or	NCODE and NXI are incorrectly defined,
or	$IND = 1$ on initial entry to D03PLF,
or	$NEQN \neq NPDE \times NPTS + NCODE$ ,
or	an element of RTOL or $ATOL < 0.0$ ,
or	corresponding elements of RTOL and ATOL are both $0.0$ ,
or	$NORM \neq 1$ or $2$ .

$IFAIL = 2$: The underlying ODE solver cannot make any further progress, with the values of ATOL and RTOL, across the integration range from the current point $t = TS$ . The components of U contain the computed values at the current point $t = TS$ .

$IFAIL = 3$: In the underlying ODE solver, there were repeated error test failures on an attempted step, before completing the requested task, but the integration was successful as far as $t = TS$ . The problem may have a singularity, or the error requirement may be inappropriate. Incorrect specification of boundary conditions may also result in this error.

$IFAIL = 4$: In setting up the ODE system, the internal initialization routine was unable to initialize the derivative of the ODE system. This could be due to the fact that IRES was repeatedly set to $3$ in one of PDEDEF, NUMFLX, BNDARY or ODEDEF, when the residual in the underlying ODE solver was being evaluated. Incorrect specification of boundary conditions may also result in this error.

$IFAIL = 5$: In solving the ODE system, a singular Jacobian has been encountered. Check the problem formulation.

$IFAIL = 6$: When evaluating the residual in solving the ODE system, IRES was set to $2$ in at least one of PDEDEF, NUMFLX, BNDARY or ODEDEF. Integration was successful as far as $t = TS$ .

$IFAIL = 7$: The values of ATOL and RTOL are so small that the routine is unable to start the integration in time.

$IFAIL = 8$: In either, PDEDEF, NUMFLX, BNDARY or ODEDEF, IRES was set to an invalid value.

$IFAIL = 9$ (D02NNF): A serious error has occurred in an internal call to the specified routine. Check the problem specification and all parameters and array dimensions. Setting $ITRACE = 1$ may provide more information. If the problem persists, contact NAG.

$IFAIL = 10$: The required task has been completed, but it is estimated that a small change in ATOL and RTOL is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when $ITASK \neq 2$ or $5$ .)

$IFAIL = 11$: An error occurred during Jacobian formulation of the ODE system (a more detailed error description may be directed to the current advisory message unit when $ITRACE \geq 1$ ). If using the sparse matrix algebra option, the values of $ALGOPT (29)$ and $ALGOPT (30)$ may be inappropriate.

$IFAIL = 12$: In solving the ODE system, the maximum number of steps specified in $ALGOPT (15)$ has been taken.

$IFAIL = 13$: Some error weights $w_{i}$ became zero during the time integration (see the description of ITOL). Pure relative error control ( $ATOL (i) = 0.0$ ) was requested on a variable (the $i$ th) which has become zero. The integration was successful as far as $t = TS$ .

$IFAIL = 14$: One or more of the functions $P_{i, j}$ , $D_{i}$ or $C_{i}$ was detected as depending on time derivatives, which is not permissible.

$IFAIL = 15$: When using the sparse option, the value of LISAVE or LRSAVE was not sufficient (more detailed information may be directed to the current error message unit).

7 Accuracy

D03PLF 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 parameters, ATOL and RTOL.

8 Further Comments

D03PLF 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 routine 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 (13)

. It is worth experimenting with this parameter, 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.

9 Example

For this routine 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 parameter-vector method (see Roe (1981)), and this is provided in the utility routine D03PUF. An alternative Approxiate Riemann Solver using Osher's scheme is provided in D03PVF. Either D03PUF or D03PVF can be called from NUMFLX.

NAG Library Routine DocumentD03PLF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

D03PLF