NAG Library Routine Document
D03PCF/D03PCA
1 Purpose
D03PCF/D03PCA integrates a system of linear or nonlinear parabolic partial differential equations (PDEs) in one space variable. The spatial discretization is performed using finite differences, and the method of lines is employed to reduce the PDEs to a system of ordinary differential equations (ODEs). The resulting system is solved using a backward differentiation formula method.
D03PCA is a version of D03PCF that has additional parameters in order to make it safe for use in multithreaded applications (see
Section 5).
2 Specification
2.1 Specification for D03PCF
| SUBROUTINE D03PCF ( | NPDE, M, TS, TOUT, PDEDEF, BNDARY, U, NPTS, X, ACC, RSAVE, LRSAVE, ISAVE, LISAVE, ITASK, ITRACE, IND, IFAIL) |
| INTEGER | NPDE, M, NPTS, LRSAVE, ISAVE(LISAVE), LISAVE, ITASK, ITRACE, IND, IFAIL |
| double precision | TS, TOUT, U(NPDE,NPTS), X(NPTS), ACC, RSAVE(LRSAVE) |
| EXTERNAL | PDEDEF, BNDARY |
2.2 Specification for D03PCA
| SUBROUTINE D03PCA ( | NPDE, M, TS, TOUT, PDEDEF, BNDARY, U, NPTS, X, ACC, RSAVE, LRSAVE, ISAVE, LISAVE, ITASK, ITRACE, IND, IUSER, RUSER, CWSAV, LWSAV, IWSAV, RWSAV, IFAIL) |
| INTEGER | NPDE, M, NPTS, LRSAVE, ISAVE(LISAVE), LISAVE, ITASK, ITRACE, IND, IUSER(*), IWSAV(505), IFAIL |
| double precision | TS, TOUT, U(NPDE,NPTS), X(NPTS), ACC, RSAVE(LRSAVE), RUSER(*), RWSAV(1100) |
| LOGICAL | LWSAV(100) |
| CHARACTER*80 | CWSAV(10) |
| EXTERNAL | PDEDEF, BNDARY |
3 Description
D03PCF/D03PCA integrates the system of parabolic equations:
where
Pi,j,
Qi and
Ri depend on
x,
t,
U,
Ux and the vector
U is the set of solution values
and the vector
Ux is its partial derivative with respect to
x. Note that
Pi,j,
Qi and
Ri must not depend on
∂U
∂t
.
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 value of m;
m=0 for Cartesian co-ordinates,
m=1 for cylindrical polar co-ordinates and m=2 for spherical polar co-ordinates. The mesh should be chosen in accordance with the expected behaviour of the solution.
The system is defined by the functions
Pi,j,
Qi and
Ri which must be specified in
PDEDEF.
The initial values of the functions
Ux,t must be given at
t=t0. The functions
Ri, for
i=1,2,…,NPDE, which may be thought of as fluxes, are also used in the definition of the boundary conditions for each equation. The boundary conditions must have the form
where
x=a or
x=b.
The boundary conditions must be specified in
BNDARY.
The problem is subject to the following restrictions:
| (i) |
t0<tout, so that integration is in the forward direction; |
| (ii) |
Pi,j,
Qi and the flux Ri must not depend on any time derivatives; |
| (iii) |
the evaluation of the functions Pi,j,
Qi and Ri is done at the mid-points of the mesh intervals 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; |
| (iv) |
at least one of the functions Pi,j must be nonzero so that there is a time derivative present in the problem; and |
| (v) |
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. |
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 ODEs in the time direction. This system is then integrated forwards in time using a backward differentiation formula 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
Dew P M and Walsh J (1981) A set of library routines for solving parabolic equations in one space variable
ACM Trans. Math. Software 7 295–314
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 in the system 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 compute the functions
Pi,j,
Qi and
Ri which define the system of PDEs.
PDEDEF is called approximately midway between each pair of mesh points in turn by D03PCF/D03PCA.
The specification of
PDEDEF
for D03PCF is:
| SUBROUTINE PDEDEF ( | NPDE, T, X, U, UX, P, Q, R, IRES) |
| INTEGER | NPDE, IRES |
| double precision | T, X, U(NPDE), UX(NPDE), P(NPDE,NPDE), Q(NPDE), R(NPDE) |
The specification of
PDEDEF
for D03PCA is:
| SUBROUTINE PDEDEF ( | NPDE, T, X, U, UX, P, Q, R, IRES, IUSER, RUSER) |
| INTEGER | NPDE, IRES, IUSER(*) |
| double precision | T, X, U(NPDE), UX(NPDE), 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: P(NPDE,NPDE) – double precision arrayOutput
-
On exit: Pij must be set to the value of Pi,jx,t,U,Ux, for i,j=1,2,…,NPDE.
- 7: Q(NPDE) – double precision arrayOutput
-
On exit: Qi must be set to the value of Qix,t,U,Ux, for i=1,2,…,NPDE.
- 8: R(NPDE) – double precision arrayOutput
-
On exit: Ri must be set to the value of Rix,t,U,Ux, for i=1,2,…,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 D03PCF/D03PCA returns to the calling subroutine with the error indicator set to IFAIL=4.
- Note: the following are additional parameters for specific use with D03PCA. Users of D03PCF therefore need not read the remainder of this description.
- 10: IUSER(*) – INTEGER arrayUser Workspace
- 11: RUSER(*) – double precision arrayUser Workspace
PDEDEF is called from D03PCA with the parameters
IUSER and
RUSER as supplied to D03PCA. 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 D03PCF/D03PCA is called. Parameters denoted as
Input must
not be changed by this procedure.
- 6: BNDARY – SUBROUTINE, supplied by the user.External Procedure
BNDARY must compute the functions
βi and
γi which define the boundary conditions as in equation
(3).
The specification of
BNDARY
for D03PCF is:
| SUBROUTINE BNDARY ( | NPDE, T, U, UX, IBND, BETA, GAMMA, IRES) |
| INTEGER | NPDE, IBND, IRES |
| double precision | T, U(NPDE), UX(NPDE), BETA(NPDE), GAMMA(NPDE) |
The specification of
BNDARY
for D03PCA is:
| SUBROUTINE BNDARY ( | NPDE, T, U, UX, IBND, BETA, GAMMA, IRES, IUSER, RUSER) |
| INTEGER | NPDE, IBND, IRES, IUSER(*) |
| double precision | T, U(NPDE), UX(NPDE), 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: IBND – INTEGERInput
-
On entry: determines the position of the boundary conditions.
- IBND=0
- BNDARY must set up the coefficients of the left-hand boundary, x=a.
- IBND≠0
- Indicates that BNDARY must set up the coefficients of the right-hand boundary, x=b.
- 6: 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.
- 7: GAMMA(NPDE) – double precision arrayOutput
-
On exit:
GAMMAi must be set to the value of
γix,t,U,Ux at the boundary specified by
IBND, for
i=1,2,…,NPDE.
- 8: 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 D03PCF/D03PCA returns to the calling subroutine with the error indicator set to IFAIL=4.
- Note: the following are additional parameters for specific use with D03PCA. Users of D03PCF therefore need not read the remainder of this description.
- 9: IUSER(*) – INTEGER arrayUser Workspace
- 10: RUSER(*) – double precision arrayUser Workspace
BNDARY is called from D03PCA with the parameters
IUSER and
RUSER as supplied to D03PCA. 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 D03PCF/D03PCA is called. Parameters denoted as
Input must
not be changed by this procedure.
- 7: U(NPDE,NPTS) – double precision arrayInput/Output
-
On entry: the initial values of Ux,t at t=TS and the mesh points Xj, for j=1,2,…,NPTS.
On exit: Uij will contain the computed solution 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 spatial direction. X1 must specify the left-hand boundary, a, and XNPTS must specify the right-hand boundary, b.
Constraint:
X1<X2<⋯<XNPTS.
- 10: ACC – double precisionInput
-
On entry: a positive quantity for controlling the local error estimate in the time integration. If
Ei,j is the estimated error for
Ui at the
jth mesh point, the error test is:
Constraint:
ACC>0.0.
- 11: 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.
- 12: LRSAVE – INTEGERInput
On entry: the dimension of the array
RSAVE as declared in the (sub)program from which D03PCF/D03PCA is called.
Constraint:
LRSAVE≥6×NPDE+10×NPDE×NPTS+3×NPDE+21×NPDE+7×NPTS+54.
- 13: 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.
- 14: LISAVE – INTEGERInput
-
On entry: the dimension of the array
ISAVE as declared in the (sub)program from which D03PCF/D03PCA is called.
Constraint:
LISAVE≥NPDE×NPTS+24.
- 15: ITASK – INTEGERInput
-
On entry: specifies the task to be performed by the ODE integrator.
- ITASK=1
- Normal computation of output values U at t=TOUT.
- ITASK=2
- One step and return.
- ITASK=3
- Stop at first internal integration point at or beyond t=TOUT.
Constraint:
ITASK=1, 2 or 3.
- 16: ITRACE – INTEGERInput
-
On entry: the level of trace information required from D03PCF/D03PCA 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.
- 17: 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 D03PCF/D03PCA.
Constraint:
IND=0 or 1.
On exit: IND=1.
- 18: IFAIL – INTEGERInput/Output
-
Note: for D03PCA, 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 D03PCA. Users of D03PCF therefore need not read the remainder of this description.
- 18: IUSER(*) – INTEGER arrayUser Workspace
Note: the dimension of the array
IUSER
must be at least
1.
IUSER is not used by D03PCA, but is passed directly to
PDEDEF and
BNDARY and may be used to pass information to these routines.
- 19: RUSER(*) – double precision arrayUser Workspace
Note: the dimension of the array
RUSER
must be at least
1.
RUSER is not used by D03PCA, but is passed directly to
PDEDEF and
BNDARY and may be used to pass information to these routines.
- 20: CWSAV(10) – CHARACTER*80 arrayCommunication Array
If
IND=0,
CWSAV need not be set on entry.
If
IND=1,
CWSAV must be unchanged from the previous call to the routine.
- 21: LWSAV(100) – LOGICAL arrayCommunication Array
If
IND=0,
LWSAV need not be set on entry.
If
IND=1,
LWSAV must be unchanged from the previous call to the routine.
- 22: IWSAV(505) – INTEGER arrayCommunication Array
If
IND=0,
IWSAV need not be set on entry.
If
IND=1,
IWSAV must be unchanged from the previous call to the routine.
- 23: RWSAV(1100) – double precision arrayCommunication Array
If
IND=0,
RWSAV need not be set on entry.
If
IND=1,
RWSAV must be unchanged from the previous call to the routine.
- 24: 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, |
| or | TOUT-TS is too small, |
| or | ITASK≠1, 2 or 3, |
| or | M≠0, 1 or 2, |
| or | M>0 and X1<0.0, |
| or | the mesh points Xi
are not ordered, |
| or | NPTS<3, |
| or | NPDE<1, |
| or | ACC≤0.0, |
| or | IND≠0 or 1, |
| or | LRSAVE is too small, |
| or | LISAVE is too small. |
- IFAIL=2
-
The underlying ODE solver cannot make any further progress across the integration range from the current point
t=TS with the supplied value of
ACC. The components of
U contain the computed values at the current point
t=TS.
- IFAIL=3
-
In the underlying ODE solver, there were repeated errors or corrector convergence test failures on an attempted step, before completing the requested task. The problem may have a singularity or
ACC is too small for the integration to continue. Integration was successful as far as
t=TS.
- 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 or
BNDARY, 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 or
BNDARY. Integration was successful as far as
t=TS.
- IFAIL=7
-
The value of
ACC is so small that the routine is unable to start the integration in time.
- IFAIL=8
-
In one of
PDEDEF or
BNDARY,
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
ACC is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when
ITASK≠2.)
- 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).
- IFAIL=12
-
Not applicable.
- IFAIL=13
-
Not applicable.
- IFAIL=14
-
The flux function Ri was detected as depending on time derivatives, which is not permissible.
7 Accuracy
D03PCF/D03PCA 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 parameter,
ACC.
8 Further Comments
D03PCF/D03PCA is designed to solve parabolic systems (possibly including some elliptic equations) with second-order derivatives in space. 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
D03PEF.
The time taken depends on the complexity of the parabolic system and on the accuracy requested.
9 Example
We use the example given in
Dew and Walsh (1981) which consists of an elliptic-parabolic pair of PDEs. The problem was originally derived from a single third-order in space PDE. The elliptic equation is
and the parabolic equation is
where
r,t∈0,1×0,1. The boundary conditions are given by
and
The first of these boundary conditions implies that the flux term in the second PDE,
∂U2
∂r
-
U2
U1
, is zero at r=0.
The initial conditions at
t=0 are given by
The value
α=1 was used in the problem definition. A mesh of
20 points was used with a circular mesh spacing to cluster the points towards the right-hand side of the spatial interval,
r=1.
9.1 Program Text
Note: the following programs illustrate the use of D03PCF and D03PCA.
Program Text (d03pcfe.f)
Program Text (d03pcae.f)
9.2 Program Data
None.
9.3 Program Results
Program Results (d03pcfe.r)
Program Results (d03pcae.r)
