NAG Library Routine Document

D03PHF/D03PHA

SUBROUTINE D03PHF (	NPDE, M, TS, TOUT, PDEDEF, 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, M, NPTS, NCODE, NXI, NEQN, ITOL, LRSAVE, ISAVE(LISAVE), LISAVE, ITASK, ITRACE, IND, IFAIL
*double precision*	TS, TOUT, U(NEQN), X(NPTS), XI(NXI), RTOL(), ATOL(), ALGOPT(30), RSAVE(LRSAVE)
CHARACTER*1	NORM, LAOPT
EXTERNAL	PDEDEF, BNDARY, ODEDEF

2.2 Specification for D03PHA

SUBROUTINE D03PHA (	NPDE, M, TS, TOUT, PDEDEF, BNDARY, U, NPTS, X, NCODE, ODEDEF, NXI, XI, NEQN, RTOL, ATOL, ITOL, NORM, LAOPT, ALGOPT, RSAVE, LRSAVE, ISAVE, LISAVE, ITASK, ITRACE, IND, IUSER, RUSER, CWSAV, LWSAV, IWSAV, RWSAV, IFAIL)
INTEGER	NPDE, M, NPTS, NCODE, NXI, NEQN, ITOL, LRSAVE, ISAVE(LISAVE), LISAVE, ITASK, ITRACE, IND, IUSER(*), IWSAV(505), IFAIL
*double precision*	TS, TOUT, U(NEQN), X(NPTS), XI(NXI), RTOL(), ATOL(), ALGOPT(30), RSAVE(LRSAVE), RUSER(*), RWSAV(1100)
LOGICAL	LWSAV(100)
CHARACTER*1	NORM, LAOPT
CHARACTER*80	CWSAV(10)
EXTERNAL	PDEDEF, BNDARY, ODEDEF

3 Description

D03PHF/D03PHA integrates the system of parabolic-elliptic equations and coupled ODEs

∑j=1NPDEPi,j ∂Uj ∂t +Qi=x-m ∂∂x xmRi, i=1,2,…,NPDE, a≤x≤b, t≥t0,

(1)

Fit,V,V.,ξ,U*,Ux*,R*,Ut*,Uxt*=0, i=1,2,…,NCODE,

(2)

where (1) defines the PDE part and (2) generalizes the coupled ODE part of the problem.

In (1), Pi,j and Ri depend on x, t, U, Ux and V; Qi depends on x, t, U, Ux, V and linearly on V.. The vector U is the set of PDE solution values

U x,t = U 1 x,t ,…, U NPDE x,t T ,

and the vector Ux is the partial derivative with respect to x. The vector V is the set of ODE solution values

V t = V 1 t ,…, V NCODE t T ,

and V. denotes its derivative with respect to time.

In (2), ξ represents a vector of nξ 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*, Ux*, R*, Ut* and Uxt* are the functions U, Ux, R, Ut and Uxt evaluated at these coupling points. Each Fi may only depend linearly on time derivatives. Hence the equation (2) may be written more precisely as

F=G-AV.-B Ut* Uxt* ,

(3)

where F=F1,…,FNCODET, G is a vector of length NCODE, A is an NCODE by NCODE matrix, B is an NCODE by nξ×NPDE matrix and the entries in G, A and B may depend on t, ξ, U*, Ux* and V. In practice you only need to supply a vector of information to define the ODEs and not the matrices A and B. (See Section 5 for the specification of ODEDEF.)

The integration in time is from t0 to tout, over the space interval a≤x≤b, where a=x1 and b=xNPTS are the leftmost and rightmost points of a user-defined mesh x1,x2,…,xNPTS. The co-ordinate system in space is defined by the values of m; m=0 for Cartesian co-ordinates, m=1 for cylindrical polar co-ordinates and m=2 for spherical polar co-ordinates.

The PDE system which is defined by the functions Pi,j, Qi and Ri must be specified in PDEDEF.

The initial values of the functions Ux,t and Vt must be given at t=t0.

The functions Ri which may be thought of as fluxes, are also used in the definition of the boundary conditions. The boundary conditions must have the form

βix,tRix,t,U,Ux,V=γix,t,U,Ux,V,V., i=1,2,…,NPDE,

(4)

where x=a or x=b.

The boundary conditions must be specified in BNDARY. The function γi may depend linearly on V..

The problem is subject to the following restrictions:

(i)	In (1), V.jt, for j=1,2,…,NCODE, may only appear linearly in the functions Qi, for i=1,2,…,NPDE, with a similar restriction for γ;
(ii)	Pi,j and the flux Ri must not depend on any time derivatives;
(iii)	t0<tout, so that integration is in the forward direction;
(iv)	the evaluation of the terms Pi,j, Qi and Ri is done approximately at the mid-points of the mesh Xi, for i=1,2,…,NPTS, by calling the PDEDEF for each mid-point in turn. Any discontinuities in these functions must therefore be at one or more of the mesh points x1,x2,…,xNPTS;
(v)	at least one of the functions Pi,j must be nonzero so that there is a time derivative present in the PDE problem;
(vi)	if m>0 and x1=0.0, which is the left boundary point, then it must be ensured that the PDE solution is bounded at this point. This can be done by either specifying the solution at x=0.0 or by specifying a zero flux there, that is βi=1.0 and γi=0.0. See also Section 8 below.

The algebraic-differential equation system which is defined by the functions Fi 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 Ui at mesh points. For simple problems in Cartesian co-ordinates, this system is obtained by replacing the space derivatives by the usual central, three-point finite-difference formula. However, for polar and spherical problems, or problems with nonlinear coefficients, the space derivatives are replaced by a modified three-point formula which maintains second order accuracy. In total there are NPDE×NPTS+NCODE ODEs in the 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

Skeel R D and Berzins M (1990) A method for the spatial discretization of parabolic equations in one space variable SIAM J. Sci. Statist. Comput. 11 (1) 1–32

5 Parameters

1: NPDE – INTEGERInput

On entry: the number of PDEs to be solved.

Constraint: NPDE≥1.

2: M – INTEGERInput

On entry: the co-ordinate system used:

M=0: Indicates Cartesian co-ordinates.
M=1: Indicates cylindrical polar co-ordinates.
M=2: Indicates spherical polar co-ordinates.

Constraint: M=0, 1 or 2.

3: TS – double precisionInput/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.

4: TOUT – double precisionInput

On entry: the final value of t to which the integration is to be carried out.

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

PDEDEF must evaluate the functions Pi,j, Qi and Ri which define the system of PDEs. The functions may depend on x, t, U, Ux and V. Qi may depend linearly on V.. PDEDEF is called approximately midway between each pair of mesh points in turn by D03PHF/D03PHA.

The specification of PDEDEF for D03PHF is:

SUBROUTINE PDEDEF (	NPDE, T, X, U, UX, NCODE, V, VDOT, P, Q, R, IRES)
INTEGER	NPDE, NCODE, IRES
*double precision*	T, X, U(NPDE), UX(NPDE), V(NCODE), VDOT(NCODE), P(NPDE,NPDE), Q(NPDE), R(NPDE)

The specification of PDEDEF for D03PHA is:

SUBROUTINE PDEDEF (	NPDE, T, X, U, UX, NCODE, V, VDOT, P, Q, R, IRES, IUSER, RUSER)
INTEGER	NPDE, NCODE, IRES, IUSER(*)
*double precision*	T, X, U(NPDE), UX(NPDE), V(NCODE), VDOT(NCODE), P(NPDE,NPDE), Q(NPDE), R(NPDE), RUSER(*)

1: NPDE – INTEGERInput

On entry: the number of PDEs in the system.

2: T – double precisionInput

On entry: the current value of the independent variable t.

3: X – double precisionInput

On entry: the current value of the space variable x.

4: U(NPDE) – double precision arrayInput

On entry: Ui contains the value of the component Uix,t, for i=1,2,…,NPDE.

5: UX(NPDE) – double precision arrayInput

On entry: UXi contains the value of the component ∂Uix,t ∂x , for i=1,2,…,NPDE.

6: NCODE – INTEGERInput

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

7: V(NCODE) – double precision arrayInput

On entry: if NCODE>0, Vi contains the value of component Vit, for i=1,2,…,NCODE.

8: VDOT(NCODE) – double precision arrayInput

On entry: if NCODE>0, VDOTi contains the value of component V.it, for i=1,2,…,NCODE.

Note: V.it, for i=1,2,…,NCODE, may only appear linearly in Qj, for j=1,2,…,NPDE.

9: P(NPDE,NPDE) – double precision arrayOutput

On exit: Pij must be set to the value of Pi,jx,t,U,Ux,V, for i,j=1,2,…,NPDE.

10: Q(NPDE) – double precision arrayOutput

On exit: Qi must be set to the value of Qix,t,U,Ux,V,V., for i=1,2,…,NPDE.

11: R(NPDE) – double precision arrayOutput

On exit: Ri must be set to the value of Rix,t,U,Ux,V, for i=1,2,…,NPDE.

12: 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 D03PHF/D03PHA returns to the calling subroutine with the error indicator set to IFAIL=4.

Note: the following are additional parameters for specific use with D03PHA. Users of D03PHF therefore need not read the remainder of this description.

13: IUSER(*) – INTEGER arrayUser Workspace

14: RUSER(*) – double precision arrayUser Workspace

PDEDEF is called from D03PHA with the parameters IUSER and RUSER as supplied to D03PHA. You are free to use the arrays IUSER and RUSER to supply information to PDEDEF.

PDEDEF must be declared as EXTERNAL in the (sub)program from which D03PHF/D03PHA 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 βi and γi which describe the boundary conditions, as given in (4).

The specification of BNDARY for D03PHF is:

SUBROUTINE BNDARY (	NPDE, T, U, UX, NCODE, V, VDOT, IBND, BETA, GAMMA, IRES)
INTEGER	NPDE, NCODE, IBND, IRES
*double precision*	T, U(NPDE), UX(NPDE), V(NCODE), VDOT(NCODE), BETA(NPDE), GAMMA(NPDE)

The specification of BNDARY for D03PHA is:

SUBROUTINE BNDARY (	NPDE, T, U, UX, NCODE, V, VDOT, IBND, BETA, GAMMA, IRES, IUSER, RUSER)
INTEGER	NPDE, NCODE, IBND, IRES, IUSER(*)
*double precision*	T, U(NPDE), UX(NPDE), V(NCODE), VDOT(NCODE), BETA(NPDE), GAMMA(NPDE), RUSER(*)

1: NPDE – INTEGERInput

On entry: the number of PDEs in the system.

2: T – double precisionInput

On entry: the current value of the independent variable t.

3: U(NPDE) – double precision arrayInput

On entry: Ui contains the value of the component Uix,t at the boundary specified by IBND, for i=1,2,…,NPDE.

4: UX(NPDE) – double precision arrayInput

On entry: UXi contains the value of the component ∂Uix,t ∂x at the boundary specified by IBND, for i=1,2,…,NPDE.

5: NCODE – INTEGERInput

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

6: V(NCODE) – double precision arrayInput

On entry: if NCODE>0, Vi contains the value of component Vit, for i=1,2,…,NCODE.

7: VDOT(NCODE) – double precision arrayInput

On entry: if NCODE>0, VDOTi contains the value of component V.it, for i=1,2,…,NCODE.

Note: V.it, for i=1,2,…,NCODE, may only appear linearly in Qj, for j=1,2,…,NPDE.

8: IBND – INTEGERInput

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

IBND=0

9: BETA(NPDE) – double precision arrayOutput

On exit: BETAi must be set to the value of βix,t at the boundary specified by IBND, for i=1,2,…,NPDE.

10: GAMMA(NPDE) – double precision arrayOutput

On exit: GAMMAi must be set to the value of γix,t,U,Ux,V,V. at the boundary specified by IBND, for i=1,2,…,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 D03PHF/D03PHA returns to the calling subroutine with the error indicator set to IFAIL=4.

Note: the following are additional parameters for specific use with D03PHA. Users of D03PHF therefore need not read the remainder of this description.

12: IUSER(*) – INTEGER arrayUser Workspace

13: RUSER(*) – double precision arrayUser Workspace

BNDARY is called from D03PHA with the parameters IUSER and RUSER as supplied to D03PHA. You are free to use the arrays IUSER and RUSER to supply information to BNDARY.

BNDARY must be declared as EXTERNAL in the (sub)program from which D03PHF/D03PHA is called. Parameters denoted as Input must not be changed by this procedure.

7: U(NEQN) – double precision arrayInput/Output

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

UNPDE×j-1+i contain Uixj,t0, for i=1,2,…,NPDE and j=1,2,…,NPTS, and
UNPTS×NPDE+i contain Vit0, for i=1,2,…,NCODE.

On exit: the computed solution Uixj,t, for i=1,2,…,NPDE and j=1,2,…,NPTS, and Vkt, for k=1,2,…,NCODE, evaluated at t=TS.

8: NPTS – INTEGERInput

On entry: the number of mesh points in the interval a,b.

Constraint: NPTS≥3.

9: X(NPTS) – double precision arrayInput

On entry: the mesh points in the space direction. X1 must specify the left-hand boundary, a, and XNPTS must specify the right-hand boundary, b.

Constraint: X1<X2<⋯<XNPTS.

10: NCODE – INTEGERInput

On entry: the number of coupled ODE components.

Constraint: NCODE≥0.

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

ODEDEF must evaluate the functions F, which define the system of ODEs, as given in (3). If you wish to compute the solution of a system of PDEs only (NCODE=0), ODEDEF must be the dummy routine D03PCK/D53PCK for D03PHF (or D53PCK for D03PHA). D03PCK/D53PCK and D53PCK are included in the NAG Library.

The specification of ODEDEF for D03PHF is:

SUBROUTINE ODEDEF (	NPDE, T, NCODE, V, VDOT, NXI, XI, UCP, UCPX, RCP, UCPT, UCPTX, F, IRES)
INTEGER	NPDE, NCODE, NXI, IRES
*double precision*	T, V(NCODE), VDOT(NCODE), XI(NXI), UCP(NPDE,), UCPX(NPDE,), RCP(NPDE,), UCPT(NPDE,), UCPTX(NPDE,*), F(NCODE)

The specification of ODEDEF for D03PHA is:

SUBROUTINE ODEDEF (	NPDE, T, NCODE, V, VDOT, NXI, XI, UCP, UCPX, RCP, UCPT, UCPTX, F, IRES, IUSER, RUSER)
INTEGER	NPDE, NCODE, NXI, IRES, IUSER(*)
*double precision*	T, V(NCODE), VDOT(NCODE), XI(NXI), UCP(NPDE,), UCPX(NPDE,), RCP(NPDE,), UCPT(NPDE,), UCPTX(NPDE,), F(NCODE), RUSER()

1: NPDE – INTEGERInput

On entry: the number of PDEs in the system.

2: T – double precisionInput

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) – double precision arrayInput

On entry: if NCODE>0, Vi contains the value of component Vit, for i=1,2,…,NCODE.

5: VDOT(NCODE) – double precision arrayInput

On entry: if NCODE>0, VDOTi contains the value of component V.it, for i=1,2,…,NCODE.

6: NXI – INTEGERInput

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

7: XI(NXI) – double precision arrayInput

On entry: if NXI>0, XIi contains the ODE/PDE coupling points, ξi, for i=1,2,…,NXI.

8: UCP(NPDE,*) – double precision arrayInput

Note: the second dimension of the array UCP is max1,NXI.

On entry: if NXI>0, UCPij contains the value of Uix,t at the coupling point x=ξj, for i=1,2,…,NPDE and j=1,2,…,NXI.

9: UCPX(NPDE,*) – double precision arrayInput

Note: the second dimension of the array UCPX is max1,NXI.

On entry: if NXI>0, UCPXij contains the value of ∂Uix,t ∂x at the coupling point x=ξj, for i=1,2,…,NPDE and j=1,2,…,NXI.

10: RCP(NPDE,*) – double precision arrayInput

Note: the second dimension of the array RCP is max1,NXI.

On entry: RCPij contains the value of the flux Ri at the coupling point x=ξj, for i=1,2,…,NPDE and j=1,2,…,NXI.

11: UCPT(NPDE,*) – double precision arrayInput

Note: the second dimension of the array UCPT is max1,NXI.

On entry: if NXI>0, UCPTij contains the value of ∂Ui ∂t at the coupling point x=ξj, for i=1,2,…,NPDE and j=1,2,…,NXI.

12: UCPTX(NPDE,*) – double precision arrayInput

Note: the second dimension of the array UCPTX is max1,NXI.

On entry: UCPTXij contains the value of ∂2Ui ∂x ∂t at the coupling point x=ξj, for i=1,2,…,NPDE and j=1,2,…,NXI.

13: F(NCODE) – double precision arrayOutput

On exit: Fi must contain the ith component of F, for i=1,2,…,NCODE, where F is defined as

The definition of F is determined by the input value of IRES.

14: IRES – INTEGERInput/Output

On entry: the form of F that must be returned in the array F.

IRES=1: Equation (5) must be used.
IRES=-1: Equation (6) 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 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 D03PHF/D03PHA returns to the calling subroutine with the error indicator set to IFAIL=4.

Note: the following are additional parameters for specific use with D03PHA. Users of D03PHF therefore need not read the remainder of this description.

15: IUSER(*) – INTEGER arrayUser Workspace

16: RUSER(*) – double precision arrayUser Workspace

ODEDEF is called from D03PHA with the parameters IUSER and RUSER as supplied to D03PHA. You are free to use the arrays IUSER and RUSER to supply information to ODEDEF.

ODEDEF must be declared as EXTERNAL in the (sub)program from which D03PHF/D03PHA 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≥0.

13: XI(NXI) – double precision arrayInput

On entry: if NXI>0, XIi, for i=1,2,…,NXI, must be set to the ODE/PDE coupling points.

Constraint: X1≤XI1<XI2<⋯<XINXI≤XNPTS.

14: NEQN – INTEGERInput

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

Constraint: NEQN=NPDE×NPTS+NCODE.

15: RTOL(*) – double precision arrayInput

Note: the dimension of the array RTOL must be at least 1 if ITOL=1 or 2 and at least NEQN if ITOL=3 or 4.

On entry: the relative local error tolerance.

Constraint: RTOLi≥0 for all relevant i.

16: ATOL(*) – double precision arrayInput

Note: the dimension of the array ATOL must be at least 1 if ITOL=1 or 3 and at least NEQN if ITOL=2 or 4.

On entry: the absolute local error tolerance.

Constraint: ATOLi≥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 D03PHF/D03PHA whether to interpret either or both of RTOL or ATOL as a vector or scalar. The error test to be satisfied is ei/wi<1.0, where wi is defined as follows:

ITOL	RTOL	ATOL	wi
1	scalar	scalar	RTOL1×Ui+ATOL1
2	scalar	vector	RTOL1×Ui+ATOLi
3	vector	scalar	RTOLi×Ui+ATOL1
4	vector	vector	RTOLi×Ui+ATOLi

In the above, ei denotes the estimated local error for the ith component of the coupled PDE/ODE system in time, Ui, for i=1,2,…,NEQN.

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

Constraint: 1≤ITOL≤4.

18: NORM – CHARACTER*1Input

On entry: the type of norm to be used.

NORM='M': Maximum norm.
NORM='A': Averaged L2 norm.

If Unorm denotes the norm of the vector U of length NEQN, then for the averaged L2 norm

Unorm=1NEQN∑i=1NEQNUi/wi2,

while for the maximum norm

U norm = maxi Ui / wi .

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

Constraint: NORM='M' or 'A'.

19: LAOPT – CHARACTER*1Input

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' or 'S'.

Note: you are recommended to use the banded option when no coupled ODEs are present (i.e., NCODE=0).

20: ALGOPT(30) – double precision arrayInput

On entry: may be set to control various options available in the integrator. If you wish to employ all the default options, then ALGOPT1 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:

ALGOPT1: Selects the ODE integration method to be used. If ALGOPT1=1.0, a BDF method is used and if ALGOPT1=2.0, a Theta method is used. The default value is ALGOPT1=1.0.

If ALGOPT1=2.0, then ALGOPTi, for i=2,3,4 are not used.

ALGOPT2: Specifies the maximum order of the BDF integration formula to be used. ALGOPT2 may be 1.0, 2.0, 3.0, 4.0 or 5.0. The default value is ALGOPT2=5.0.
ALGOPT3: Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the BDF method. If ALGOPT3=1.0 a modified Newton iteration is used and if ALGOPT3=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 ALGOPT3=1.0.
ALGOPT4: 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 Pi,j=0.0, for j=1,2,…,NPDE for some i or when there is no V.it dependence in the coupled ODE system. If ALGOPT4=1.0, then the Petzold test is used. If ALGOPT4=2.0, then the Petzold test is not used. The default value is ALGOPT4=1.0.

If ALGOPT1=1.0, then ALGOPTi, for i=5,6,7 are not used.

ALGOPT5: Specifies the value of Theta to be used in the Theta integration method. 0.51≤ALGOPT5≤0.99. The default value is ALGOPT5=0.55.
ALGOPT6: Specifies what method is to be used to solve the system of nonlinear equations arising on each step of the Theta method. If ALGOPT6=1.0, a modified Newton iteration is used and if ALGOPT6=2.0, a functional iteration method is used. The default value is ALGOPT6=1.0.
ALGOPT7: 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 ALGOPT7=1.0, then switching is allowed and if ALGOPT7=2.0, then switching is not allowed. The default value is ALGOPT7=1.0.
ALGOPT11: Specifies a point in the time direction, tcrit, beyond which integration must not be attempted. The use of tcrit is described under the parameter ITASK. If ALGOPT1≠0.0, a value of 0.0 for ALGOPT11, say, should be specified even if ITASK subsequently specifies that tcrit will not be used.
ALGOPT12: Specifies the minimum absolute step size to be allowed in the time integration. If this option is not required, ALGOPT12 should be set to 0.0.
ALGOPT13: Specifies the maximum absolute step size to be allowed in the time integration. If this option is not required, ALGOPT13 should be set to 0.0.
ALGOPT14: Specifies the initial step size to be attempted by the integrator. If ALGOPT14=0.0, then the initial step size is calculated internally.
ALGOPT15: Specifies the maximum number of steps to be attempted by the integrator in any one call. If ALGOPT15=0.0, then no limit is imposed.
ALGOPT23: Specifies what method is to be used to solve the nonlinear equations at the initial point to initialize the values of U, Ut, V and V.. If ALGOPT23=1.0, a modified Newton iteration is used and if ALGOPT23=2.0, functional iteration is used. The default value is ALGOPT23=1.0.

ALGOPT29 and ALGOPT30 are used only for the sparse matrix algebra option, LAOPT='S'.

ALGOPT29: Governs the choice of pivots during the decomposition of the first Jacobian matrix. It should lie in the range 0.0<ALGOPT29<1.0, with smaller values biasing the algorithm towards maintaining sparsity at the expense of numerical stability. If ALGOPT29 lies outside this range then the default value is used. If the routines regard the Jacobian matrix as numerically singular then increasing ALGOPT29 towards 1.0 may help, but at the cost of increased fill-in. The default value is ALGOPT29=0.1.
ALGOPT30: Is used as a relative pivot threshold during subsequent Jacobian decompositions (see ALGOPT29) below which an internal error is invoked. If ALGOPT30 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 ALGOPT29). The default value is ALGOPT30=0.0001.

21: RSAVE(LRSAVE) – double precision arrayCommunication Array

If IND=0, RSAVE need not be set on entry.

If 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 D03PHF/D03PHA is called.

If LAOPT='F', LRSAVE≥NEQN×NEQN+NEQN+nwkres+lenode.

If LAOPT='B', LRSAVE≥3×mlu+1×NEQN+nwkres+lenode.

If LAOPT='S', LRSAVE≥4×NEQN+11×NEQN/2+1+nwkres+lenode.

Where

	mlu is the lower or upper half bandwidths such that mlu=3×NPDE-1, for PDE problems only (no coupled ODEs); or mlu=NEQN-1, for coupled PDE/ODE problems.
	nwkres= NPDE×2×NPTS+6×NXI+3×NPDE+26+NXI+NCODE+7×NPTS+2, when NCODE>0 and NXI>0; NPDE×2×NPTS+3×NPDE+32+NCODE+7×NPTS+3, when NCODE>0 and NXI=0; NPDE×2×NPTS+3×NPDE+32+7×NPTS+4, when NCODE=0.
	lenode= 6+intALGOPT2×NEQN+50, when the BDF method is used; or 9×NEQN+50, when the Theta method is used.

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

If IND=0, ISAVE need not be set on entry.

If IND=1, ISAVE must be unchanged from the previous call to the routine because it contains required information about the iteration. In particular:

ISAVE1: Contains the number of steps taken in time.
ISAVE2: Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.
ISAVE3: Contains the number of Jacobian evaluations performed by the time integrator.
ISAVE4: Contains the order of the last backward differentiation formula method used.
ISAVE5: Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the LU decomposition of the Jacobian matrix.

24: LISAVE – INTEGERInput

On entry: the dimension of the array ISAVE as declared in the (sub)program from which D03PHF/D03PHA is called. Its size depends on the type of matrix algebra selected:

if LAOPT='F', LISAVE≥24;
if LAOPT='B', LISAVE≥NEQN+24;
if LAOPT='S', LISAVE≥25×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: specifies the task to be performed by the ODE integrator.

ITASK=1

Constraint: ITASK=1, 2, 3, 4 or 5.

26: ITRACE – INTEGERInput

On entry: the level of trace information required from D03PHF/D03PHA and the underlying ODE solver. ITRACE may take the value -1, 0, 1, 2 or 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.

If 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

Constraint: IND=0 or 1.

On exit: IND=1.

28: IFAIL – INTEGERInput/Output

Note: for D03PHA, IFAIL does not occur in this position in the parameter list. See the additional parameters described below.

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.

On exit: IFAIL=0 unless the routine detects an error (see Section 6).

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.

Note: the following are additional parameters for specific use with D03PHA. Users of D03PHF therefore need not read the remainder of this description.

28: IUSER(*) – INTEGER arrayUser Workspace

Note: the dimension of the array IUSER must be at least 1.

IUSER is not used by D03PHA, but is passed directly to PDEDEF, BNDARY and ODEDEF and may be used to pass information to these routines.

29: RUSER(*) – double precision arrayUser Workspace

Note: the dimension of the array RUSER must be at least 1.

RUSER is not used by D03PHA, but is passed directly to PDEDEF, BNDARY and ODEDEF and may be used to pass information to these routines.

30: CWSAV(10) – CHARACTER*80 arrayCommunication Array

31: LWSAV(100) – LOGICAL arrayCommunication Array

32: IWSAV(505) – INTEGER arrayCommunication Array

33: RWSAV(1100) – double precision arrayCommunication Array

34: IFAIL – INTEGERInput/Output

Note: see the parameter description for IFAIL above.

6 Error Indicators and Warnings

If on entry IFAIL=0 or -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≠1, 2, 3, 4 or 5,
or M≠0, 1 or 2,
or at least one of the coupling points defined in array XI is outside the interval [X1,XNPTS],
or M>0 and X1<0.0,
or NPTS<3,
or NPDE<1,
or NORM≠'A' or 'M',
or LAOPT≠'F', 'B' or 'S',
or ITOL≠1, 2, 3 or 4,
or IND≠0 or 1,
or mesh points Xi are badly ordered,
or LRSAVE is too small,
or LISAVE is too small,
or NCODE and NXI are incorrectly defined,
or NEQN≠NPDE×NPTS+NCODE,
or either an element of RTOL or ATOL<0.0,
or all the elements of RTOL and ATOL are zero.

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.

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 at least PDEDEF, BNDARY or ODEDEF, when the residual in the underlying ODE solver was being evaluated.

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

IFAIL=6: When evaluating the residual in solving the ODE system, IRES was set to 2 in at least 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 one of 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≠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 error message unit). If using the sparse matrix algebra option, the values of ALGOPT29 and ALGOPT30 may be inappropriate.

IFAIL=12: In solving the ODE system, the maximum number of steps specified in ALGOPT15 have been taken.

IFAIL=13: Some error weights wi became zero during the time integration (see the description of ITOL). Pure relative error control (ATOLi=0.0) was requested on a variable (the ith) which has become zero. The integration was successful as far as t=TS.

IFAIL=14: The flux function Ri 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

D03PHF/D03PHA 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 parameter specification allows you to include equations with only first-order derivatives in the space direction but there is no guarantee that the method of integration will be satisfactory for such systems. The position and nature of the boundary conditions in particular are critical in defining a stable problem. It may be advisable in such cases to reduce the whole system to first-order and to use the Keller box scheme routine D03PKF.

The time taken depends on the complexity of the parabolic 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 one PDE and one ODE.

V 1 2 ∂ U 1 ∂ t - x V 1 V . 1 ∂ U 1 ∂ x = ∂ 2 y U 1 ∂ x 2 V . 1 = V 1 U 1 + ∂ U 1 ∂ x + 1 + t ,

for t∈10-4,0.1×2i ; i=1,2,…,5 ; x∈0,1 .

The left boundary condition at x=0 is

∂U1 ∂x =-V1exp⁡t.

The right boundary condition at x=1 is

∂U1 ∂x = -V1 V.1 .

The initial conditions at t=10-4 are defined by the exact solution:

V1 = t , and U1 x,t = exp t 1-x - 1.0 , x∈0,1 ,

and the coupling point is at ξ1=1.0.

On entry,	TOUT-TS is too small,
or	ITASK≠1, 2, 3, 4 or 5,
or	M≠0, 1 or 2,
or	at least one of the coupling points defined in array XI is outside the interval [X1,XNPTS],
or	M>0 and X1<0.0,
or	NPTS<3,
or	NPDE<1,
or	NORM≠'A' or 'M',
or	LAOPT≠'F', 'B' or 'S',
or	ITOL≠1, 2, 3 or 4,
or	IND≠0 or 1,
or	mesh points Xi are badly ordered,
or	LRSAVE is too small,
or	LISAVE is too small,
or	NCODE and NXI are incorrectly defined,
or	NEQN≠NPDE×NPTS+NCODE,
or	either an element of RTOL or ATOL<0.0,
or	all the elements of RTOL and ATOL are zero.

NAG Library Routine DocumentD03PHF/D03PHA

+− Contents

NAG Library Routine Document

D03PHF/D03PHA