NAG Library Routine Document

D03PKF

+− 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

D03PKF integrates a system of linear or nonlinear, first-order, time-dependent partial differential equations (PDEs) in one space variable, with scope for coupled ordinary differential equations (ODEs). The spatial discretization is performed using the Keller box scheme and the method of lines is employed to reduce the PDEs to a system of ODEs. The resulting system is solved using a Backward Differentiation Formula (BDF) method or a Theta method (switching between Newton's method and functional iteration).

2 Specification

SUBROUTINE D03PKF (

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

INTEGER	NPDE, NPTS, NLEFT, 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, BNDARY, ODEDEF

3 Description

D03PKF integrates the system of first-order PDEs and coupled ODEs

G_{i} (x, t, U, U_{x}, U_{t}, V, \dot{V}) = 0, i = 1, 2, \dots, NPDE, a \leq x \leq b, t \geq t_{0},

(1)

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

(2)

In the PDE part of the problem given by (1), the functions

G_{i}

must have the general form

G_{i} = \sum_{j = 1}^{NPDE} P_{i, j} \frac{\partial U_{j}}{\partial t} + \sum_{j = 1}^{NCODE} Q_{i, j} {\dot{V}}_{j} + S_{i} = 0, i = 1, 2, \dots, NPDE,

(3)

where

P_{i, j}

Q_{i, j}

and

S_{i}

depend on

x, t, U, U_{x}

and

V

The vector

U

is the set of PDE solution values

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

and the vector

U_{x}

is the partial derivative with respect to

x

. The vector

V

is the set of ODE solution values

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

and

\dot{V}

denotes its derivative with respect to time.

In the ODE part given by (2),

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 some of the 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 only depend linearly on time derivatives. Hence equation (2) may be written more precisely as

R = A - B \dot{V} - C U_{t}^{*},

(4)

where

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

A

is a vector of length NCODE,

B

is an NCODE by NCODE matrix,

C

is an NCODE by

(n_{ξ} \times NPDE)

matrix. The entries in

A

B

and

C

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

B

and

C

. (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 PDE system which is defined by the functions

G_{i}

must be specified in PDEDEF.

The initial values of the functions

U (x, t)

and

V (t)

must be given at

t = t_{0}

For a first-order system of PDEs, only one boundary condition is required for each PDE component

U_{i}

. The NPDE boundary conditions are separated into

n_{a}

at the left-hand boundary

x = a

, and

n_{b}

at the right-hand boundary

x = b

, such that

n_{a} + n_{b} = NPDE

. The position of the boundary condition for each component should be chosen with care; the general rule is that if the characteristic direction of

U_{i}

at the left-hand boundary (say) points into the interior of the solution domain, then the boundary condition for

U_{i}

should be specified at the left-hand boundary. Incorrect positioning of boundary conditions generally results in initialization or integration difficulties in the underlying time integration routines.

The boundary conditions have the form:

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

(5)

at the left-hand boundary, and

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

(6)

at the right-hand boundary.

Note that the functions

G_{i}^{L}

and

G_{i}^{R}

must not depend on

U_{x}

, since spatial derivatives are not determined explicitly in the Keller box scheme. If the problem involves derivative (Neumann) boundary conditions then it is generally possible to restate such boundary conditions in terms of permissible variables. Also note that

G_{i}^{L}

and

G_{i}^{R}

must be linear with respect to time derivatives, so that the boundary conditions have the general form:

\sum_{j = 1}^{NPDE} E_{i, j}^{L} \frac{\partial U_{j}}{\partial t} + \sum_{j = 1}^{NCODE} H_{i, j}^{L} {\dot{V}}_{j} + K_{i}^{L} = 0, i = 1, 2, \dots, n_{a},

(7)

at the left-hand boundary, and

\sum_{j = 1}^{NPDE} E_{i, j}^{R} \frac{\partial U_{j}}{\partial t} + \sum_{j = 1}^{NCODE} H_{i, j}^{R} {\dot{V}}_{j} + K_{i}^{R} = 0, i = 1, 2, \dots, n_{b},

(8)

at the right-hand boundary, where

E_{i, j}^{L}

E_{i, j}^{R}

H_{i, j}^{L}

H_{i, j}^{R}

K_{i}^{L}

and

K_{i}^{R}

depend on

x, t, U

and

V

only.

The boundary conditions must be specified in BNDARY.

The problem is subject to the following restrictions:

(i)	$P_{i, j}$ , $Q_{i, j}$ and $S_{i}$ must not depend on any time derivatives;
(ii)	$t_{0} < t_{out}$ , so that integration is in the forward direction;
(iii)	The evaluation of the function $G_{i}$ is done approximately at the mid-points of the mesh $X (i)$ , for $i = 1, 2, \dots, NPTS$ , by calling the PDEDEF for each mid-point in turn. Any discontinuities in the function must therefore be at one or more of the mesh points $x_{1}, x_{2}, \dots, x_{NPTS}$ ;
(iv)	At least one of the functions $P_{i, j}$ must be nonzero so that there is a time derivative present in the PDE problem.

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

in the array XI.

The parabolic equations are approximated by a system of ODEs in time for the values of

U_{i}

at mesh points. In this method of lines approach the Keller box scheme (see Keller (1970)) is applied to each PDE in the space variable only, resulting in a system of ODEs in time for the values of

U_{i}

at each mesh point. In total there are

NPDE \times NPTS + NCODE

ODEs in time direction. This system is then integrated forwards in time using a Backward Differentiation Formula (BDF) or a Theta method.

4 References

Berzins M (1990) Developments in the NAG Library software for parabolic equations Scientific Software Systems (eds J C Mason and M G Cox) 59–72 Chapman and Hall

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

Berzins M and Furzeland R M (1992) An adaptive theta method for the solution of stiff and nonstiff differential equations Appl. Numer. Math. 9 1–19

Keller H B (1970) A new difference scheme for parabolic problems Numerical Solutions of Partial Differential Equations (ed J Bramble) 2 327–350 Academic Press

Pennington S V and Berzins M (1994) New NAG Library software for first-order partial differential equations ACM Trans. Math. Softw. 20 63–99

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

Constraint:

TS < TOUT

On exit: the value of

t

corresponding to the solution in U. Normally

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 user.External Procedure

PDEDEF must evaluate the functions

G_{i}

which define the system of PDEs. PDEDEF is called approximately midway between each pair of mesh points in turn by D03PKF.

The specification of PDEDEF is:

SUBROUTINE PDEDEF (

NPDE, T, X, U, UT, UX, NCODE, V, VDOT, RES, IRES)

INTEGER	NPDE, NCODE, IRES
REAL (KIND=nag_wp)	T, X, U(NPDE), UT(NPDE), UX(NPDE), V(NCODE), VDOT(NCODE), RES(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: UT(NPDE) – REAL (KIND=nag_wp) arrayInput

On entry:

UT (i)

contains the value of the component

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

, for

i = 1, 2, \dots, NPDE

6: 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

7: NCODE – INTEGERInput

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

8: 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

9: 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

10: RES(NPDE) – REAL (KIND=nag_wp) arrayOutput

On exit:

RES (i)

must contain the

i

th component of

G

, for

i = 1, 2, \dots, NPDE

, where

G

is defined as

G_{i} = \sum_{j = 1}^{NPDE} P_{i, j} \frac{\partial U_{j}}{\partial t} + \sum_{j = 1}^{NCODE} Q_{i, j} {\dot{V}}_{j},

(9)

i.e., only terms depending explicitly on time derivatives, or

G_{i} = \sum_{j = 1}^{NPDE} P_{i, j} \frac{\partial U_{j}}{\partial t} + \sum_{j = 1}^{NCODE} Q_{i, j} {\dot{V}}_{j} + S_{i},

(10)

i.e., all terms in equation (3).

The definition of

G

is determined by the input value of IRES.

11: IRES – INTEGERInput/Output

On entry: the form of

G_{i}

that must be returned in the array RES.

$IRES = - 1$: Equation (9) must be used.
$IRES = 1$: Equation (10) must be used.

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 (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 D03PKF 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 D03PKF is called. Parameters denoted as Input must not be changed by this procedure.

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

BNDARY must evaluate the functions

G_{i}^{L}

and

G_{i}^{R}

which describe the boundary conditions, as given in (5) and (6).

The specification of BNDARY is:

SUBROUTINE BNDARY (

NPDE, T, IBND, NOBC, U, UT, NCODE, V, VDOT, RES, IRES)

INTEGER	NPDE, IBND, NOBC, NCODE, IRES
REAL (KIND=nag_wp)	T, U(NPDE), UT(NPDE), V(NCODE), VDOT(NCODE), RES(NOBC)

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: IBND – INTEGERInput

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

$IBND = 0$: BNDARY must compute the left-hand boundary condition at $x = a$ .
$IBND \neq 0$: BNDARY must compute the right-hand boundary condition at $x = b$ .

4: NOBC – INTEGERInput

On entry: specifies the number of boundary conditions at the boundary specified by IBND.

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

On entry:

U (i)

contains the value of the component

U_{i} (x, t)

at the boundary specified by IBND, for

i = 1, 2, \dots, NPDE

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

On entry:

UT (i)

contains the value of the component

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

at the boundary specified by IBND, for

i = 1, 2, \dots, NPDE

7: NCODE – INTEGERInput

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

8: 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

9: 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:

VDOT (i)

, for

i = 1, 2, \dots, NCODE

, may only appear linearly as in (7) and (8).

10: RES(NOBC) – REAL (KIND=nag_wp) arrayOutput

On exit:

RES (i)

must contain the

i

th component of

G^{L}

G^{R}

, depending on the value of IBND, for

i = 1, 2, \dots, NOBC

, where

G^{L}

is defined as

G_{i}^{L} = \sum_{j = 1}^{NPDE} E_{i, j}^{L} \frac{\partial U_{j}}{\partial t} + \sum_{j = 1}^{NCODE} H_{i, j}^{L} {\dot{V}}_{j},

(11)

i.e., only terms depending explicitly on time derivatives, or

G_{i}^{L} = \sum_{j = 1}^{NPDE} E_{i, j}^{L} \frac{\partial U_{j}}{\partial t} + \sum_{j = 1}^{NCODE} H_{i, j}^{L} {\dot{V}}_{j} + K_{i}^{L},

(12)

i.e., all terms in equation (7), and similarly for

G_{i}^{R}

The definitions of

G^{L}

and

G^{R}

are determined by the input value of IRES.

11: IRES – INTEGERInput/Output

On entry: the form of

G_{i}^{L}

(or

G_{i}^{R}

) that must be returned in the array RES.

$IRES = - 1$: Equation (11) must be used.
$IRES = 1$: Equation (12) must be used.

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 (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 D03PKF 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 D03PKF is called. Parameters denoted as Input must not be changed by this procedure.

6: 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
$U (NPTS \times NPDE + i)$ contain $V_{i} (t_{0})$ , for $i = 1, 2, \dots, NCODE$ .

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

, evaluated at

t = TS

7: NPTS – INTEGERInput

On entry: the number of mesh points in the interval

[a, b]

Constraint:

NPTS \geq 3

8: 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)

9: NLEFT – INTEGERInput

On entry: the number

n_{a}

of boundary conditions at the left-hand mesh point

X (1)

Constraint:

0 \leq NLEFT \leq NPDE

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 points,

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: if

NCODE > 0

R (i)

must contain the

i

th component of

R

, for

i = 1, 2, \dots, NCODE

, where

R

is defined as

R = - B \dot{V} - C U_{t}^{*},

(13)

i.e., only terms depending explicitly on time derivatives, or

R = A - B \dot{V} - C U_{t}^{*},

(14)

i.e., all terms in equation (4). 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 (13) must be used.
$IRES = 1$: Equation (14) 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 D03PKF 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 D03PKF 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,

ξ_{i}

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. ITOL indicates to D03PKF whether to interpret either or both of RTOL or ATOL as a vector or scalar. The error test to be satisfied is

‖e_{i} / w_{i}‖ < 1.0

, where

w_{i}

is defined as follows:

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	$RTOL (i) \times \|U (i)\| + ATOL (i)$

In the above,

e_{i}

denotes the estimated local error for the

i

th component of the coupled PDE/ODE system in time,

U (i)

, for

i = 1, 2, \dots, NEQN

The choice of norm used is defined by the parameter NORM.

Constraint:

1 \leq ITOL \leq 4

18: NORM – CHARACTER(1)Input

On entry: the type of norm to be used.

$NORM ='M'$: Maximum norm.
$NORM ='A'$: Averaged $L_{2}$ norm.

U_{norm}

denotes the norm of the vector U of length NEQN, then for the averaged

L_{2}

norm

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

while for the maximum norm

U_{norm} = \max_{i} |U (i) / w_{i}| .

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

w

Constraint:

NORM ='M'

'A'

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 (i.e.,

NCODE = 0

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 value 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 this 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 a 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 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 D03PKF 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 (2 \times ml + mu + 2) \times NEQN + nwkres + lenode

LAOPT ='S'

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

Where

	$ml$ and $mu$ are the lower and upper half bandwidths given by $ml = NPDE + NLEFT - 1$ such that $mu = 2 \times NPDE - NLEFT - 1$ , for problems involving PDEs only; or $ml = mu = NEQN - 1$ , for coupled PDE/ODE problems.
	$nwkres = \{\begin{cases} NPDE \times (3 \times NPDE + 6 \times NXI + NPTS + 15) + NXI + NCODE + 7 \times NPTS + 2, & when NCODE > 0 and NXI > 0; or \\ NPDE \times (3 \times NPDE + NPTS + 21) + NCODE + 7 \times NPTS + 3, & when NCODE > 0 and NXI = 0; or \\ NPDE \times (3 \times NPDE + NPTS + 21) + 7 \times NPTS + 4, & when NCODE = 0 . \end{cases}$
	$lenode = \{\begin{cases} (6 + int (ALGOPT (2))) \times NEQN + 50, & when the BDF method is used; or \\ 9 \times NEQN + 50, & when the Theta method is used. \end{cases}$

Note: when using the sparse option, 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 ODE method last used in the time integration.
$ISAVE (5)$: 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 D03PKF 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 D03PKF and the underlying ODE solver as follows:

$ITRACE \leq - 1$: No output is generated.
$ITRACE = 0$: Only warning messages from the PDE solver are printed on the current error message unit (see X04AAF).
$ITRACE = 1$: 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.
$ITRACE = 2$: Output from the underlying ODE solver is similar to that produced when $ITRACE = 1$ , except that the advisory messages are given in greater detail.
$ITRACE \geq 3$: Output from the underlying ODE solver is similar to that produced when $ITRACE = 2$ , except that the advisory messages are given in greater detail.

You 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 D03PKF.

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,	$(TOUT - TS)$ 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	$NPTS < 3$ ,
or	$NPDE < 1$ ,
or	NLEFT not in range $0$ to NPDE,
or	$NORM \neq'A'$ or $'M'$ ,
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 D03PKF,
or	an element of RTOL or $ATOL < 0.0$ ,
or	corresponding elements of ATOL and RTOL are both $0.0$ ,
or	$NEQN \neq NPDE \times NPTS + NCODE$ .

$IFAIL = 2$: 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 positioning 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, BNDARY or ODEDEF, when the residual in the underlying ODE solver was being evaluated. Incorrect positioning of boundary conditions may also result in this error.

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

$IFAIL = 6$: When evaluating the residual in solving the ODE system, IRES was set to $2$ in one of PDEDEF, 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, 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). 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 $t = TS$ .

$IFAIL = 14$: Not applicable.

$IFAIL = 15$: When using the sparse option, the value of LISAVE or LRSAVE was insufficient (more detailed information may be directed to the current error message unit).

7 Accuracy

D03PKF 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

The Keller box scheme can be used to solve higher-order problems which have been reduced to first-order by the introduction of new variables (see the example in Section 9). In general, a second-order problem can be solved with slightly greater accuracy using the Keller box scheme instead of a finite difference scheme (see D03PCF/D03PCA or D03PHF/D03PHA for example), but at the expense of increased CPU time due to the larger number of function evaluations required.

It should be noted that the Keller box scheme, in common with other central-difference schemes, may be unsuitable for some hyperbolic first-order problems such as the apparently simple linear advection equation

U_{t} + a U_{x} = 0

, where

a

is a constant, resulting in spurious oscillations due to the lack of dissipation. This type of problem requires a discretization scheme with upwind weighting (D03PLF for example), or the addition of a second-order artificial dissipation term.

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

This example provides a simple coupled system of two PDEs and one ODE.

\begin{array}{l} {(V_{1})}^{2} \frac{\partial U_{1}}{\partial t} - x V_{1} {\dot{V}}_{1} U_{2} - \frac{\partial U_{2}}{\partial x} = 0, \\ U_{2} - \frac{\partial U_{1}}{\partial x} = 0, \\ {\dot{V}}_{1} - V_{1} U_{1} - U_{2} - 1 - t = 0, \end{array}

for

t \in [10^{- 4}, 0.1 \times 2^{i}]

, for

i = 1, 2, \dots, 5, x \in [0, 1]

. The left boundary condition at

x = 0

U_{2} = - V_{1} \exp t,

and the right boundary condition at

x = 1

U_{2} = - V_{1} {\dot{V}}_{1} .

The initial conditions at

t = 10^{- 4}

are defined by the exact solution:

V_{1} = t, U_{1} (x, t) = \exp \{t (1 - x)\} - 1.0 and U_{2} (x, t) = - t \exp \{t (1 - x)\}, ​ x \in [0, 1],

and the coupling point is at

ξ_{1} = 1.0

This problem is exactly the same as the D03PHF/D03PHA example problem, but reduced to first-order by the introduction of a second PDE variable (as mentioned in Section 8).

NAG Library Routine DocumentD03PKF

+− 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

D03PKF