NAG FL Interface
d03phf  (dim1_parab_dae_fd_old)
d03pha (dim1_parab_dae_fd)

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1 Purpose

d03phf/​d03pha integrates a system of linear or nonlinear parabolic partial differential equations (PDEs) in one space variable, with scope for coupled ordinary differential equations (ODEs). The spatial discretization is performed using finite differences, 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 method or a Theta method (switching between Newton's method and functional iteration).
d03pha is a version of d03phf that has additional arguments in order to make it safe for use in multithreaded applications (see Section 5).

2 Specification

2.1 Specification for d03phf

Fortran Interface
Subroutine d03phf ( npde, m, ts, tout, pdedef, bndary, u, npts, x, nv, odedef, nxi, xi, neqn, rtol, atol, itol, norm, laopt, algopt, rsave, lrsave, isave, lisave, itask, itrace, ind, ifail)
Integer, Intent (In) :: npde, m, npts, nv, nxi, neqn, itol, lrsave, lisave, itask, itrace
Integer, Intent (Inout) :: isave(lisave), ind, ifail
Real (Kind=nag_wp), Intent (In) :: tout, x(npts), xi(nxi), rtol(*), atol(*), algopt(30)
Real (Kind=nag_wp), Intent (Inout) :: ts, u(neqn), rsave(lrsave)
Character (1), Intent (In) :: norm, laopt
External :: pdedef, bndary, odedef
C Header Interface
#include <nag.h>
void  d03phf_ (const Integer *npde, const Integer *m, double *ts, const double *tout,
void (NAG_CALL *pdedef)(const Integer *npde, const double *t, const double *x, const double u[], const double ux[], const Integer *nv, const double v[], const double vdot[], double p[], double q[], double r[], Integer *ires),
void (NAG_CALL *bndary)(const Integer *npde, const double *t, const double u[], const double ux[], const Integer *nv, const double v[], const double vdot[], const Integer *ibnd, double beta[], double gamma[], Integer *ires),
double u[], const Integer *npts, const double x[], const Integer *nv,
void (NAG_CALL *odedef)(const Integer *npde, const double *t, const Integer *nv, const double v[], const double vdot[], const Integer *nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], Integer *ires),
const Integer *nxi, const double xi[], const Integer *neqn, const double rtol[], const double atol[], const Integer *itol, const char *norm, const char *laopt, const double algopt[], double rsave[], const Integer *lrsave, Integer isave[], const Integer *lisave, const Integer *itask, const Integer *itrace, Integer *ind, Integer *ifail, const Charlen length_norm, const Charlen length_laopt)

2.2 Specification for d03pha

Fortran Interface
Integer, Intent (In) :: npde, m, npts, nv, nxi, neqn, itol, lrsave, lisave, itask, itrace
Integer, Intent (Inout) :: isave(lisave), ind, iuser(*), iwsav(505), ifail
Real (Kind=nag_wp), Intent (In) :: tout, x(npts), xi(nxi), rtol(*), atol(*), algopt(30)
Real (Kind=nag_wp), Intent (Inout) :: ts, u(neqn), rsave(lrsave), ruser(*), rwsav(1100)
Logical, Intent (Inout) :: lwsav(100)
Character (1), Intent (In) :: norm, laopt
Character (80), Intent (InOut) :: cwsav(10)
External :: pdedef, bndary, odedef
C Header Interface
#include <nag.h>
void  d03pha_ (const Integer *npde, const Integer *m, double *ts, const double *tout,
void (NAG_CALL *pdedef)(const Integer *npde, const double *t, const double *x, const double u[], const double ux[], const Integer *nv, const double v[], const double vdot[], double p[], double q[], double r[], Integer *ires, Integer iuser[], double ruser[]),
void (NAG_CALL *bndary)(const Integer *npde, const double *t, const double u[], const double ux[], const Integer *nv, const double v[], const double vdot[], const Integer *ibnd, double beta[], double gamma[], Integer *ires, Integer iuser[], double ruser[]),
double u[], const Integer *npts, const double x[], const Integer *nv,
void (NAG_CALL *odedef)(const Integer *npde, const double *t, const Integer *nv, const double v[], const double vdot[], const Integer *nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], Integer *ires, Integer iuser[], double ruser[]),
const Integer *nxi, const double xi[], const Integer *neqn, const double rtol[], const double atol[], const Integer *itol, const char *norm, const char *laopt, const double algopt[], double rsave[], const Integer *lrsave, Integer isave[], const Integer *lisave, const Integer *itask, const Integer *itrace, Integer *ind, Integer iuser[], double ruser[], char cwsav[], logical lwsav[], Integer iwsav[], double rwsav[], Integer *ifail, const Charlen length_norm, const Charlen length_laopt, const Charlen length_cwsav)

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,  axb,  tt0, (1)
Fi(t,V,V.,ξ,U*,Ux*,R*,Ut*,Uxt*)=0,  i=1,2,,nv, (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 nv (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,,Fnv]T, G is a vector of length nv, A is an nv by nv matrix, B is an nv 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 axb, where a=x1 and b=xnpts are the leftmost and rightmost points of a user-defined mesh x1,x2,,xnpts. The coordinate system in space is defined by the values of m; m=0 for Cartesian coordinates, m=1 for cylindrical polar coordinates and m=2 for spherical polar coordinates.
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 U(x,t) and V(t) 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
βi(x,t)Ri(x,t,U,Ux,V)=γi(x,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:
  1. (i)In (1), V.j(t), for j=1,2,,nv, may only appear linearly in the functions Qi, for i=1,2,,npde, with a similar restriction for γ;
  2. (ii)Pi,j and the flux Ri must not depend on any time derivatives;
  3. (iii)t0<tout, so that integration is in the forward direction;
  4. (iv)the evaluation of the terms Pi,j, Qi and Ri is done approximately at the mid-points of the mesh x(i), 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;
  5. (v)at least one of the functions Pi,j must be nonzero so that there is a time derivative present in the PDE problem;
  6. (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 9 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 coordinates, 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+nv 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 Arguments

1: npde Integer Input
On entry: the number of PDEs to be solved.
Constraint: npde1.
2: m Integer Input
On entry: the coordinate system used:
m=0
Indicates Cartesian coordinates.
m=1
Indicates cylindrical polar coordinates.
m=2
Indicates spherical polar coordinates.
Constraint: m=0, 1 or 2.
3: ts Real (Kind=nag_wp) Input/Output
On entry: the initial value of the independent variable t.
On exit: the value of t corresponding to the solution values in u. Normally ts=tout.
Constraint: ts<tout.
4: tout Real (Kind=nag_wp) Input
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:
Fortran Interface
Subroutine pdedef ( npde, t, x, u, ux, nv, v, vdot, p, q, r, ires)
Integer, Intent (In) :: npde, nv
Integer, Intent (Inout) :: ires
Real (Kind=nag_wp), Intent (In) :: t, x, u(npde), ux(npde), v(nv), vdot(nv)
Real (Kind=nag_wp), Intent (Out) :: p(npde,npde), q(npde), r(npde)
C Header Interface
void  pdedef (const Integer *npde, const double *t, const double *x, const double u[], const double ux[], const Integer *nv, const double v[], const double vdot[], double p[], double q[], double r[], Integer *ires)
The specification of pdedef for d03pha is:
Fortran Interface
Subroutine pdedef ( npde, t, x, u, ux, nv, v, vdot, p, q, r, ires, iuser, ruser)
Integer, Intent (In) :: npde, nv
Integer, Intent (Inout) :: ires, iuser(*)
Real (Kind=nag_wp), Intent (In) :: t, x, u(npde), ux(npde), v(nv), vdot(nv)
Real (Kind=nag_wp), Intent (Inout) :: ruser(*)
Real (Kind=nag_wp), Intent (Out) :: p(npde,npde), q(npde), r(npde)
C Header Interface
void  pdedef (const Integer *npde, const double *t, const double *x, const double u[], const double ux[], const Integer *nv, const double v[], const double vdot[], double p[], double q[], double r[], Integer *ires, Integer iuser[], double ruser[])
1: npde Integer Input
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) array Input
On entry: u(i) contains the value of the component Ui(x,t), for i=1,2,,npde.
5: ux(npde) Real (Kind=nag_wp) array Input
On entry: ux(i) contains the value of the component Ui(x,t) x , for i=1,2,,npde.
6: nv Integer Input
On entry: the number of coupled ODEs in the system.
7: v(nv) Real (Kind=nag_wp) array Input
On entry: if nv>0, v(i) contains the value of the component Vi(t), for i=1,2,,nv.
8: vdot(nv) Real (Kind=nag_wp) array Input
On entry: if nv>0, vdot(i) contains the value of component V.i(t), for i=1,2,,nv.
Note: V.i(t), for i=1,2,,nv, may only appear linearly in Qj, for j=1,2,,npde.
9: p(npde,npde) Real (Kind=nag_wp) array Output
On exit: p(i,j) must be set to the value of Pi,j(x,t,U,Ux,V), for i=1,2,,npde and j=1,2,,npde.
10: q(npde) Real (Kind=nag_wp) array Output
On exit: q(i) must be set to the value of Qi(x,t,U,Ux,V,V.), for i=1,2,,npde.
11: r(npde) Real (Kind=nag_wp) array Output
On exit: r(i) must be set to the value of Ri(x,t,U,Ux,V), for i=1,2,,npde.
12: ires Integer Input/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 (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, d03phf/​d03pha returns to the calling subroutine with the error indicator set to ifail=4.
Note: the following are additional arguments for specific use with d03pha. Users of d03phf therefore need not read the remainder of this description.
13: iuser(*) Integer array User Workspace
14: ruser(*) Real (Kind=nag_wp) array User Workspace
pdedef is called with the arguments iuser and ruser as supplied to d03phf/​d03pha. You should use the arrays iuser and ruser to supply information to pdedef.
pdedef must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03phf/​d03pha is called. Arguments denoted as Input must not be changed by this procedure.
Note: pdedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03phf/​d03pha. If your code inadvertently does return any NaNs or infinities, d03phf/​d03pha is likely to produce unexpected results.
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:
Fortran Interface
Subroutine bndary ( npde, t, u, ux, nv, v, vdot, ibnd, beta, gamma, ires)
Integer, Intent (In) :: npde, nv, ibnd
Integer, Intent (Inout) :: ires
Real (Kind=nag_wp), Intent (In) :: t, u(npde), ux(npde), v(nv), vdot(nv)
Real (Kind=nag_wp), Intent (Out) :: beta(npde), gamma(npde)
C Header Interface
void  bndary (const Integer *npde, const double *t, const double u[], const double ux[], const Integer *nv, const double v[], const double vdot[], const Integer *ibnd, double beta[], double gamma[], Integer *ires)
The specification of bndary for d03pha is:
Fortran Interface
Subroutine bndary ( npde, t, u, ux, nv, v, vdot, ibnd, beta, gamma, ires, iuser, ruser)
Integer, Intent (In) :: npde, nv, ibnd
Integer, Intent (Inout) :: ires, iuser(*)
Real (Kind=nag_wp), Intent (In) :: t, u(npde), ux(npde), v(nv), vdot(nv)
Real (Kind=nag_wp), Intent (Inout) :: ruser(*)
Real (Kind=nag_wp), Intent (Out) :: beta(npde), gamma(npde)
C Header Interface
void  bndary (const Integer *npde, const double *t, const double u[], const double ux[], const Integer *nv, const double v[], const double vdot[], const Integer *ibnd, double beta[], double gamma[], Integer *ires, Integer iuser[], double ruser[])
1: npde Integer Input
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: u(npde) Real (Kind=nag_wp) array Input
On entry: u(i) contains the value of the component Ui(x,t) at the boundary specified by ibnd, for i=1,2,,npde.
4: ux(npde) Real (Kind=nag_wp) array Input
On entry: ux(i) contains the value of the component Ui(x,t) x at the boundary specified by ibnd, for i=1,2,,npde.
5: nv Integer Input
On entry: the number of coupled ODEs in the system.
6: v(nv) Real (Kind=nag_wp) array Input
On entry: if nv>0, v(i) contains the value of the component Vi(t), for i=1,2,,nv.
7: vdot(nv) Real (Kind=nag_wp) array Input
On entry: if nv>0, vdot(i) contains the value of component V.i(t), for i=1,2,,nv.
Note: V.i(t), for i=1,2,,nv, may only appear linearly in Qj, for j=1,2,,npde.
8: ibnd Integer Input
On entry: specifies which boundary conditions are to be evaluated.
ibnd=0
bndary must set up the coefficients of the left-hand boundary, x=a.
ibnd0
bndary must set up the coefficients of the right-hand boundary, x=b.
9: beta(npde) Real (Kind=nag_wp) array Output
On exit: beta(i) must be set to the value of βi(x,t) at the boundary specified by ibnd, for i=1,2,,npde.
10: gamma(npde) Real (Kind=nag_wp) array Output
On exit: gamma(i) must be set to the value of γi(x,t,U,Ux,V,V.) at the boundary specified by ibnd, for i=1,2,,npde.
11: ires Integer Input/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 (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, d03phf/​d03pha returns to the calling subroutine with the error indicator set to ifail=4.
Note: the following are additional arguments for specific use with d03pha. Users of d03phf therefore need not read the remainder of this description.
12: iuser(*) Integer array User Workspace
13: ruser(*) Real (Kind=nag_wp) array User Workspace
bndary is called with the arguments iuser and ruser as supplied to d03phf/​d03pha. You should use the arrays iuser and ruser to supply information to bndary.
bndary must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03phf/​d03pha is called. Arguments denoted as Input must not be changed by this procedure.
Note: bndary should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03phf/​d03pha. If your code inadvertently does return any NaNs or infinities, d03phf/​d03pha is likely to produce unexpected results.
7: u(neqn) Real (Kind=nag_wp) array Input/Output
On entry: the initial values of the dependent variables defined as follows:
  • u(npde×(j-1)+i) contain Ui(xj,t0), for i=1,2,,npde and j=1,2,,npts, and
  • u(npts×npde+i) contain Vi(t0), for i=1,2,,nv.
On exit: the computed solution Ui(xj,t), for i=1,2,,npde and j=1,2,,npts, and Vk(t), for k=1,2,,nv, evaluated at t=ts, as follows:
  • u(npde×(j-1)+i) contain Ui(xj,t), for i=1,2,,npde and j=1,2,,npts, and
  • u(npts×npde+i) contain Vi(t), for i=1,2,,nv.
8: npts Integer Input
On entry: the number of mesh points in the interval [a,b].
Constraint: npts3.
9: x(npts) Real (Kind=nag_wp) array Input
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)<<x(npts).
10: nv Integer Input
On entry: the number of coupled ODE components.
Constraint: nv0.
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 (nv=0), odedef must be the dummy routine d03pck for d03phf (or d53pck for d03pha). d03pck and d53pck are included in the NAG Library.
The specification of odedef for d03phf is:
Fortran Interface
Subroutine odedef ( npde, t, nv, v, vdot, nxi, xi, ucp, ucpx, rcp, ucpt, ucptx, f, ires)
Integer, Intent (In) :: npde, nv, nxi
Integer, Intent (Inout) :: ires
Real (Kind=nag_wp), Intent (In) :: t, v(nv), vdot(nv), xi(nxi), ucp(npde,nxi), ucpx(npde,nxi), rcp(npde,nxi), ucpt(npde,nxi), ucptx(npde,nxi)
Real (Kind=nag_wp), Intent (Out) :: f(nv)
C Header Interface
void  odedef (const Integer *npde, const double *t, const Integer *nv, const double v[], const double vdot[], const Integer *nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], Integer *ires)
The specification of odedef for d03pha is:
Fortran Interface
Subroutine odedef ( npde, t, nv, v, vdot, nxi, xi, ucp, ucpx, rcp, ucpt, ucptx, f, ires, iuser, ruser)
Integer, Intent (In) :: npde, nv, nxi
Integer, Intent (Inout) :: ires, iuser(*)
Real (Kind=nag_wp), Intent (In) :: t, v(nv), vdot(nv), xi(nxi), ucp(npde,nxi), ucpx(npde,nxi), rcp(npde,nxi), ucpt(npde,nxi), ucptx(npde,nxi)
Real (Kind=nag_wp), Intent (Inout) :: ruser(*)
Real (Kind=nag_wp), Intent (Out) :: f(nv)
C Header Interface
void  odedef (const Integer *npde, const double *t, const Integer *nv, const double v[], const double vdot[], const Integer *nxi, const double xi[], const double ucp[], const double ucpx[], const double rcp[], const double ucpt[], const double ucptx[], double f[], Integer *ires, Integer iuser[], double ruser[])
1: npde Integer Input
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: nv Integer Input
On entry: the number of coupled ODEs in the system.
4: v(nv) Real (Kind=nag_wp) array Input
On entry: if nv>0, v(i) contains the value of the component Vi(t), for i=1,2,,nv.
5: vdot(nv) Real (Kind=nag_wp) array Input
On entry: if nv>0, vdot(i) contains the value of component V.i(t), for i=1,2,,nv.
6: nxi Integer Input
On entry: the number of ODE/PDE coupling points.
7: xi(nxi) Real (Kind=nag_wp) array Input
On entry: if nxi>0, xi(i) contains the ODE/PDE coupling points, ξi, for i=1,2,,nxi.
8: ucp(npde,nxi) Real (Kind=nag_wp) array Input
On entry: if nxi>0, ucp(i,j) contains the value of Ui(x,t) at the coupling point x=ξj, for i=1,2,,npde and j=1,2,,nxi.
9: ucpx(npde,nxi) Real (Kind=nag_wp) array Input
On entry: if nxi>0, ucpx(i,j) contains the value of Ui(x,t) x at the coupling point x=ξj, for i=1,2,,npde and j=1,2,,nxi.
10: rcp(npde,nxi) Real (Kind=nag_wp) array Input
On entry: rcp(i,j) 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,nxi) Real (Kind=nag_wp) array Input
On entry: if nxi>0, ucpt(i,j) 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,nxi) Real (Kind=nag_wp) array Input
On entry: ucptx(i,j) 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(nv) Real (Kind=nag_wp) array Output
On exit: f(i) must contain the ith component of F, for i=1,2,,nv, where F is defined as
F=G-AV.-B ( Ut* Uxt* ) , (5)
or
F=-AV.-B ( Ut* Uxt* ) . (6)
The definition of F is determined by the input value of ires.
14: ires Integer Input/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 (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, d03phf/​d03pha returns to the calling subroutine with the error indicator set to ifail=4.
Note: the following are additional arguments for specific use with d03pha. Users of d03phf therefore need not read the remainder of this description.
15: iuser(*) Integer array User Workspace
16: ruser(*) Real (Kind=nag_wp) array User Workspace
odedef is called with the arguments iuser and ruser as supplied to d03phf/​d03pha. You should use the arrays iuser and ruser to supply information to odedef.
odedef must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03phf/​d03pha is called. Arguments denoted as Input must not be changed by this procedure.
Note: odedef should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03phf/​d03pha. If your code inadvertently does return any NaNs or infinities, d03phf/​d03pha is likely to produce unexpected results.
12: nxi Integer Input
On entry: the number of ODE/PDE coupling points.
Constraints:
  • if nv=0, nxi=0;
  • if nv>0, nxi0.
13: xi(nxi) Real (Kind=nag_wp) array Input
On entry: if nxi>0, xi(i), for i=1,2,,nxi, must be set to the ODE/PDE coupling points.
Constraint: x(1)xi(1)<xi(2)<<xi(nxi)x(npts).
14: neqn Integer Input
On entry: the number of ODEs in the time direction.
Constraint: neqn=npde×npts+nv.
15: rtol(*) Real (Kind=nag_wp) array Input
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: rtol(i)0.0 for all relevant i.
16: atol(*) Real (Kind=nag_wp) array Input
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: atol(i)0.0 for all relevant i.
Note: corresponding elements of rtol and atol cannot both be 0.0.
17: itol Integer Input
On entry: a value to indicate the form of the local error test. 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 rtol(1)×|Ui|+atol(1)
2 scalar vector rtol(1)×|Ui|+atol(i)
3 vector scalar rtol(i)×|Ui|+atol(1)
4 vector vector rtol(i)×|Ui|+atol(i)
In the above, ei denotes the estimated local error for the ith component of the coupled PDE/ODE system in time, u(i), for i=1,2,,neqn.
The choice of norm used is defined by the argument norm.
Constraint: 1itol4.
18: norm Character(1) Input
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=1neqni=1neqn(u(i)/wi)2,  
while for the maximum norm
u norm = maxi|u(i)/wi| .  
See the description of itol for the formulation of the weight vector w.
Constraint: norm='M' or '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' or 'S'.
Note: you are recommended to use the banded option when no coupled ODEs are present (i.e., nv=0).
20: algopt(30) Real (Kind=nag_wp) array Input
On entry: may be set to control various options available in the integrator. If you wish to employ all the default options, 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.
If algopt(1)=2.0, 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, 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 Pi,j=0.0, for j=1,2,,npde, for some i or when there is no V.i(t) dependence in the coupled ODE system. If algopt(4)=1.0, the Petzold test is used. If algopt(4)=2.0, the Petzold test is not used. The default value is algopt(4)=1.0.
If algopt(1)=1.0, 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. 0.51algopt(5)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, switching is allowed and if algopt(7)=2.0, switching is not allowed. The default value is algopt(7)=1.0.
algopt(11)
Specifies a point in the time direction, tcrit, beyond which integration must not be attempted. The use of tcrit is described under the argument itask. If algopt(1)0.0, a value of 0.0 for algopt(11), say, should be specified even if itask subsequently specifies that tcrit 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, 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, 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, Ut, V and 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, 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)
Is used as a relative pivot threshold during subsequent Jacobian decompositions (see algopt(29)) below which an internal error is invoked. 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) array Communication 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 Integer Input
On entry: the dimension of the array rsave 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', lrsaveneqn×neqn+neqn+nwkres+lenode.
If laopt='B', lrsave(3mlu+1)×neqn+nwkres+lenode.
If laopt='S', lrsave4neqn+11neqn/2+1+nwkres+lenode.
Where mlu is the lower or upper half bandwidths such that
for PDE problems only (no coupled ODEs),
mlu=3npde-1;
for coupled PDE/ODE problems,
mlu=neqn-1.
Where nwkres is defined by
if nv>0​ and ​nxi>0,
nwkres=npde(2npts+6nxi+3npde+26)+nxi+nv+7npts+2;
if nv>0​ and ​nxi=0,
nwkres=npde(2npts+3npde+32)+nv+7npts+3;
if nv=0,
nwkres=npde(2npts+3npde+32)+7npts+4.
Where lenode is defined by
the BDF method is used,
lenode=(6+int(algopt(2)))×neqn+50;
the Theta method is used,
lenode=9neqn+50.
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 array Communication 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:
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 computing 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 last backward differentiation formula method used.
isave(5)
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 Integer Input
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', lisave24;
  • if laopt='B', lisaveneqn+24;
  • if laopt='S', lisave25×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 Integer Input
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.
itask=4
Normal computation of output values u at t=tout but without overshooting t=tcrit where tcrit is described under the argument algopt.
itask=5
Take one step in the time direction and return, without passing tcrit, where tcrit is described under the argument algopt.
Constraint: itask=1, 2, 3, 4 or 5.
26: itrace Integer Input
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, -1 is assumed and similarly if itrace>3, 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 Integer Input/Output
On entry: indicates whether this is a continuation call or a new integration.
ind=0
Starts or restarts the integration in time.
ind=1
Continues the integration after an earlier exit from the routine. In this case, only the arguments tout and ifail should be reset between calls to d03phf/​d03pha.
Constraint: ind=0 or 1.
On exit: ind=1.
28: ifail Integer Input/Output
Note: for d03pha, ifail does not occur in this position in the argument list. See the additional arguments described below.
On entry: ifail must be set to 0, -1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of -1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value -1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. 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).
Note: the following are additional arguments for specific use with d03pha. Users of d03phf therefore need not read the remainder of this description.
28: iuser(*) Integer array User Workspace
29: ruser(*) Real (Kind=nag_wp) array User Workspace
iuser and ruser are not used by d03phf/​d03pha, but are passed directly to pdedef, bndary and odedef and may be used to pass information to these routines.
30: cwsav(10) Character(80) array Communication Array
31: lwsav(100) Logical array Communication Array
32: iwsav(505) Integer array Communication Array
33: rwsav(1100) Real (Kind=nag_wp) array Communication Array
If ind=0, cwsav, lwsav, iwsav and rwsav need not be set on entry.
If ind=1, cwsav, lwsav, iwsav and rwsav must be unchanged from the previous call to d03phf/​d03pha.
34: ifail Integer Input/Output
Note: see the argument 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, at least one point in xi lies outside [x(1),x(npts)]: x(1)=value and x(npts)=value.
On entry, i=value, x(i)=value, j=value and x(j)=value.
Constraint: x(1)<x(2)<<x(npts).
On entry, i=value, xi(i+1)=value and xi(i)=value.
Constraint: xi(i+1)>xi(i).
On entry, i=value and atol(i)=value.
Constraint: atol(i)0.0.
On entry, i=value and j=value.
Constraint: corresponding elements atol(i) and rtol(j) cannot both be 0.0.
On entry, i=value and rtol(i)=value.
Constraint: rtol(i)0.0.
On entry, ind=value.
Constraint: ind=0 or 1.
On entry, itask=value.
Constraint: itask=1, 2, 3, 4 or 5.
On entry, itol=value.
Constraint: itol=1, 2, 3 or 4.
On entry, laopt=value.
Constraint: laopt='F', 'B' or 'S'.
On entry, lisave=value.
Constraint: lisavevalue.
On entry, lrsave=value.
Constraint: lrsavevalue.
On entry, m=value.
Constraint: m=0, 1 or 2.
On entry, m=value and x(1)=value.
Constraint: m0 or x(1)0.0
On entry, neqn=value, npde=value, npts=value and nv=value.
Constraint: neqn=npde×npts+nv.
On entry, norm=value.
Constraint: norm='A' or 'M'.
On entry, npde=value.
Constraint: npde1.
On entry, npts=value.
Constraint: npts3.
On entry, nv=value.
Constraint: nv0.
On entry, nv=value and nxi=value.
Constraint: nxi=0 when nv=0.
On entry, nv=value and nxi=value.
Constraint: nxi0 when nv>0.
On entry, on initial entry ind=1.
Constraint: on initial entry ind=0.
On entry, tout=value and ts=value.
Constraint: tout>ts.
On entry, tout-ts is too small: tout=value and ts=value.
ifail=2
Underlying ODE solver cannot make further progress from the point ts with the supplied values of atol and rtol. ts=value.
ifail=3
Repeated errors in an attempted step of underlying ODE solver. Integration was successful as far as ts: ts=value.
ifail=4
In setting up the ODE system an internal auxiliary was unable to initialize the derivative. This could be due to your setting ires=3 in pdedef or bndary.
ifail=5
Singular Jacobian of ODE system. Check problem formulation.
ifail=6
In evaluating residual of ODE system, ires=2 has been set in pdedef, bndary, or odedef. Integration is successful as far as ts: ts=value.
ifail=7
atol and rtol were too small to start integration.
ifail=8
ires set to an invalid value in call to pdedef, bndary, or odedef.
ifail=9
Serious error in internal call to an auxiliary. Increase itrace for further details.
ifail=10
Integration completed, but small changes in atol or rtol are unlikely to result in a changed solution.
ifail=11
Error during Jacobian formulation for ODE system. Increase itrace for further details.
ifail=12
In solving ODE system, the maximum number of steps algopt(15) has been exceeded. algopt(15)=value.
ifail=13
Zero error weights encountered during time integration.
Some error weights wi 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 ith) which has become zero. The integration was successful as far as t=ts.
ifail=14
Flux function appears to depend on time derivatives.
ifail=15
When using the sparse option lisave or lrsave is too small: lisave=value, lrsave=value.
ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
ifail=-399
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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 arguments atol and rtol.

8 Parallelism and Performance

d03phf/​d03pha is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d03phf/​d03pha makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The argument 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.

10 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 =-V1expt.  
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.

10.1 Program Text

Note: the following programs illustrate the use of d03phf and d03pha.
Program Text (d03phfe.f90)
Program Text (d03phae.f90)

10.2 Program Data

Program Data (d03phfe.d)
Program Data (d03phae.d)

10.3 Program Results

Program Results (d03phfe.r)
Program Results (d03phae.r)
GnuplotProduced by GNUPLOT 5.0 patchlevel 0 Example Program Parabolic PDE Coupled with ODE using Finite-differences and BDF U(x,t) gnuplot_plot_1 gnuplot_plot_2 0.1 0.5 1 2 3 Time (logscale) 0 0.2 0.4 0.6 0.8 1 x 0 5 10 15 20 25