NAG FL Interface
d03ppf  (dim1_parab_remesh_fd_old)
d03ppa (dim1_parab_remesh_fd)

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1 Purpose

d03ppf/​d03ppa integrates a system of linear or nonlinear parabolic partial differential equations (PDEs) in one space variable, with scope for coupled ordinary differential equations (ODEs), and automatic adaptive spatial remeshing. 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 (BDF) method or a Theta method (switching between Newton's method and functional iteration).
d03ppa is a version of d03ppf that has additional arguments in order to make it safe for use in multithreaded applications (see Section 5).

2 Specification

2.1 Specification for d03ppf

Fortran Interface
Integer, Intent (In) :: npde, m, npts, nv, nxi, neqn, itol, nxfix, nrmesh, ipminf, lrsave, lisave, itask, itrace
Integer, Intent (Inout) :: isave(lisave), ind, ifail
Real (Kind=nag_wp), Intent (In) :: tout, xi(nxi), rtol(*), atol(*), algopt(30), xfix(nxfix), dxmesh, trmesh, xratio, con
Real (Kind=nag_wp), Intent (Inout) :: ts, u(neqn), x(npts), rsave(lrsave)
Logical, Intent (In) :: remesh
Character (1), Intent (In) :: norm, laopt
External :: pdedef, bndary, uvinit, odedef, monitf
C Header Interface
#include <nag.h>
void  d03ppf_ (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),
void (NAG_CALL *uvinit)(const Integer *npde, const Integer *npts, const Integer *nxi, const double x[], const double xi[], double u[], const Integer *nv, double v[]),
double u[], const Integer *npts, 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[], const logical *remesh, const Integer *nxfix, const double xfix[], const Integer *nrmesh, const double *dxmesh, const double *trmesh, const Integer *ipminf, const double *xratio, const double *con,
void (NAG_CALL *monitf)(const double *t, const Integer *npts, const Integer *npde, const double x[], const double u[], const double r[], double fmon[]),
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 d03ppa

Fortran Interface
Integer, Intent (In) :: npde, m, npts, nv, nxi, neqn, itol, nxfix, nrmesh, ipminf, lrsave, lisave, itask, itrace
Integer, Intent (Inout) :: isave(lisave), ind, iuser(*), iwsav(505), ifail
Real (Kind=nag_wp), Intent (In) :: tout, xi(nxi), rtol(*), atol(*), algopt(30), xfix(nxfix), dxmesh, trmesh, xratio, con
Real (Kind=nag_wp), Intent (Inout) :: ts, u(neqn), x(npts), rsave(lrsave), ruser(*), rwsav(1100)
Logical, Intent (In) :: remesh
Logical, Intent (Inout) :: lwsav(100)
Character (1), Intent (In) :: norm, laopt
Character (80), Intent (InOut) :: cwsav(10)
External :: pdedef, bndary, uvinit, odedef, monitf
C Header Interface
#include <nag.h>
void  d03ppa_ (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[]),
void (NAG_CALL *uvinit)(const Integer *npde, const Integer *npts, const Integer *nxi, const double x[], const double xi[], double u[], const Integer *nv, double v[], Integer iuser[], double ruser[]),
double u[], const Integer *npts, 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[], const logical *remesh, const Integer *nxfix, const double xfix[], const Integer *nrmesh, const double *dxmesh, const double *trmesh, const Integer *ipminf, const double *xratio, const double *con,
void (NAG_CALL *monitf)(const double *t, const Integer *npts, const Integer *npde, const double x[], const double u[], const double r[], double fmon[], Integer iuser[], double ruser[]),
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

d03ppf/​d03ppa 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)=[U1(x,t),,Unpde(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)=[V1(t),,Vnv(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 mesh x1,x2,,xnpts defined initially by you and (possibly) adapted automatically during the integration according to user-specified criteria. The coordinate system in space is defined by the following 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 (t=t0) values of the functions U(x,t) and V(t) must be specified in uvinit. Note that uvinit will be called again following any initial remeshing, and so U(x,t0) should be specified for all values of x in the interval axb, and not just the initial mesh points.
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 fixed mesh points specified by xfix;
  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.
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 time direction. This system is then integrated forwards in time using a Backward Differentiation Formula (BDF) or a Theta method.
The adaptive space remeshing can be used to generate meshes that automatically follow the changing time-dependent nature of the solution, generally resulting in a more efficient and accurate solution using fewer mesh points than may be necessary with a fixed uniform or non-uniform mesh. Problems with travelling wavefronts or variable-width boundary layers for example will benefit from using a moving adaptive mesh. The discrete time-step method used here (developed by Furzeland (1984)) automatically creates a new mesh based on the current solution profile at certain time-steps, and the solution is then interpolated onto the new mesh and the integration continues.
The method requires you to supply a monitf which specifies in an analytical or numerical form the particular aspect of the solution behaviour you wish to track. This so-called monitor function is used to choose a mesh which equally distributes the integral of the monitor function over the domain. A typical choice of monitor function is the second space derivative of the solution value at each point (or some combination of the second space derivatives if there is more than one solution component), which results in refinement in regions where the solution gradient is changing most rapidly.
You must specify the frequency of mesh updates together with certain other criteria such as adjacent mesh ratios. Remeshing can be expensive and you are encouraged to experiment with the different options in order to achieve an efficient solution which adequately tracks the desired features of the solution.
Note that unless the monitor function for the initial solution values is zero at all user-specified initial mesh points, a new initial mesh is calculated and adopted according to the user-specified remeshing criteria. uvinit will then be called again to determine the initial solution values at the new mesh points (there is no interpolation at this stage) and the integration proceeds.

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
Furzeland R M (1984) The construction of adaptive space meshes TNER.85.022 Thornton Research Centre, Chester
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 d03ppf/​d03ppa.
The specification of pdedef for d03ppf 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 d03ppa 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, d03ppf/​d03ppa returns to the calling subroutine with the error indicator set to ifail=4.
Note: the following are additional arguments for specific use with d03ppa. Users of d03ppf 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 d03ppf/​d03ppa. 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 d03ppf/​d03ppa 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 d03ppf/​d03ppa. If your code inadvertently does return any NaNs or infinities, d03ppf/​d03ppa 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 d03ppf 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 d03ppa 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: 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 γj, 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, d03ppf/​d03ppa returns to the calling subroutine with the error indicator set to ifail=4.
Note: the following are additional arguments for specific use with d03ppa. Users of d03ppf 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 d03ppf/​d03ppa. 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 d03ppf/​d03ppa 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 d03ppf/​d03ppa. If your code inadvertently does return any NaNs or infinities, d03ppf/​d03ppa is likely to produce unexpected results.
7: uvinit Subroutine, supplied by the user. External Procedure
uvinit must supply the initial (t=t0) values of U(x,t) and V(t) for all values of x in the interval axb.
The specification of uvinit for d03ppf is:
Fortran Interface
Subroutine uvinit ( npde, npts, nxi, x, xi, u, nv, v)
Integer, Intent (In) :: npde, npts, nxi, nv
Real (Kind=nag_wp), Intent (In) :: x(npts), xi(nxi)
Real (Kind=nag_wp), Intent (Out) :: u(npde,npts), v(nv)
C Header Interface
void  uvinit (const Integer *npde, const Integer *npts, const Integer *nxi, const double x[], const double xi[], double u[], const Integer *nv, double v[])
The specification of uvinit for d03ppa is:
Fortran Interface
Subroutine uvinit ( npde, npts, nxi, x, xi, u, nv, v, iuser, ruser)
Integer, Intent (In) :: npde, npts, nxi, nv
Integer, Intent (Inout) :: iuser(*)
Real (Kind=nag_wp), Intent (In) :: x(npts), xi(nxi)
Real (Kind=nag_wp), Intent (Inout) :: ruser(*)
Real (Kind=nag_wp), Intent (Out) :: u(npde,npts), v(nv)
C Header Interface
void  uvinit (const Integer *npde, const Integer *npts, const Integer *nxi, const double x[], const double xi[], double u[], const Integer *nv, double v[], Integer iuser[], double ruser[])
1: npde Integer Input
On entry: the number of PDEs in the system.
2: npts Integer Input
On entry: the number of mesh points in the interval [a,b].
3: nxi Integer Input
On entry: the number of ODE/PDE coupling points.
4: x(npts) Real (Kind=nag_wp) array Input
On entry: the current mesh. x(i) contains the value of xi, for i=1,2,,npts.
5: xi(nxi) Real (Kind=nag_wp) array Input
On entry: if nxi>0, xi(i) contains the value of the ODE/PDE coupling point, ξi, for i=1,2,,nxi.
6: u(npde,npts) Real (Kind=nag_wp) array Output
On exit: u(i,j) contains the value of the component Ui(xj,t0), for i=1,2,,npde and j=1,2,,npts.
7: nv Integer Input
On entry: the number of coupled ODEs in the system.
8: v(nv) Real (Kind=nag_wp) array Output
On exit: v(i) contains the value of component Vi(t0), for i=1,2,,nv.
Note: the following are additional arguments for specific use with d03ppa. Users of d03ppf therefore need not read the remainder of this description.
9: iuser(*) Integer array User Workspace
10: ruser(*) Real (Kind=nag_wp) array User Workspace
uvinit is called with the arguments iuser and ruser as supplied to d03ppf/​d03ppa. You should use the arrays iuser and ruser to supply information to uvinit.
uvinit must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03ppf/​d03ppa is called. Arguments denoted as Input must not be changed by this procedure.
Note: uvinit should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03ppf/​d03ppa. If your code inadvertently does return any NaNs or infinities, d03ppf/​d03ppa is likely to produce unexpected results.
8: u(neqn) Real (Kind=nag_wp) array Input/Output
On entry: if ind=1 the value of u must be unchanged from the previous call.
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.
9: npts Integer Input
On entry: the number of mesh points in the interval [a,b].
Constraint: npts3.
10: x(npts) Real (Kind=nag_wp) array Input/Output
On entry: the initial 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).
On exit: the final values of the mesh points.
11: nv Integer Input
On entry: the number of coupled ODE in the system.
Constraint: nv0.
12: 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 d03ppf (or d53pck for d03ppa). d03pck and d53pck are included in the NAG Library.
The specification of odedef for d03ppf 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 d03ppa 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, d03ppf/​d03ppa returns to the calling subroutine with the error indicator set to ifail=4.
Note: the following are additional arguments for specific use with d03ppa. Users of d03ppf 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 d03ppf/​d03ppa. 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 d03ppf/​d03ppa 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 d03ppf/​d03ppa. If your code inadvertently does return any NaNs or infinities, d03ppf/​d03ppa is likely to produce unexpected results.
13: nxi Integer Input
On entry: the number of ODE/PDE coupling points.
Constraints:
  • if nv=0, nxi=0;
  • if nv>0, nxi0.
14: 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).
15: neqn Integer Input
On entry: the number of ODEs in the time direction.
Constraint: neqn=npde×npts+nv.
16: 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.
17: 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.
Constraints:
  • atol(i)0.0 for all relevant i;
  • Corresponding elements of atol and rtol cannot both be 0.0.
18: itol Integer Input
On entry: a value to indicate the form of the local error test. itol indicates to d03ppf/​d03ppa 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.
19: 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'.
20: 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).
21: 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.
22: remesh Logical Input
On entry: indicates whether or not spatial remeshing should be performed.
remesh=.TRUE.
Indicates that spatial remeshing should be performed as specified.
remesh=.FALSE.
Indicates that spatial remeshing should be suppressed.
Note: remesh should not be changed between consecutive calls to d03ppf/​d03ppa. Remeshing can be switched off or on at specified times by using appropriate values for the arguments nrmesh and trmesh at each call.
23: nxfix Integer Input
On entry: the number of fixed mesh points.
Constraint: 0nxfixnpts-2.
Note: the end points x(1) and x(npts) are fixed automatically and hence should not be specified as fixed points.
24: xfix(nxfix) Real (Kind=nag_wp) array Input
On entry: xfix(i), for i=1,2,,nxfix, must contain the value of the x coordinate at the ith fixed mesh point.
Constraints:
  • xfix(i)<xfix(i+1), for i=1,2,,nxfix-1;
  • each fixed mesh point must coincide with a user-supplied initial mesh point, that is xfix(i)=x(j) for some j, 2jnpts-1.
Note: the positions of the fixed mesh points in the array x remain fixed during remeshing, and so the number of mesh points between adjacent fixed points (or between fixed points and end points) does not change. You should take this into account when choosing the initial mesh distribution.
25: nrmesh Integer Input
On entry: specifies the spatial remeshing frequency and criteria for the calculation and adoption of a new mesh.
nrmesh<0
Indicates that a new mesh is adopted according to the argument dxmesh. The mesh is tested every |nrmesh| timesteps.
nrmesh=0
Indicates that remeshing should take place just once at the end of the first time step reached when t>trmesh.
nrmesh>0
Indicates that remeshing will take place every nrmesh time steps, with no testing using dxmesh.
Note: nrmesh may be changed between consecutive calls to d03ppf/​d03ppa to give greater flexibility over the times of remeshing.
26: dxmesh Real (Kind=nag_wp) Input
On entry: determines whether a new mesh is adopted when nrmesh is set less than zero. A possible new mesh is calculated at the end of every |nrmesh| time steps, but is adopted only if
xi(new)>xi (old) +dxmesh×(xi+1 (old) -xi (old) )  
or
xi(new)<xi (old) -dxmesh×(xi (old) -xi- 1 (old) )  
dxmesh thus imposes a lower limit on the difference between one mesh and the next.
Constraint: dxmesh0.0.
27: trmesh Real (Kind=nag_wp) Input
On entry: specifies when remeshing will take place when nrmesh is set to zero. Remeshing will occur just once at the end of the first time step reached when t is greater than trmesh.
Note: trmesh may be changed between consecutive calls to d03ppf/​d03ppa to force remeshing at several specified times.
28: ipminf Integer Input
On entry: the level of trace information regarding the adaptive remeshing. Details are directed to the current advisory message unit (see x04abf).
ipminf=0
No trace information.
ipminf=1
Brief summary of mesh characteristics.
ipminf=2
More detailed information, including old and new mesh points, mesh sizes and monitor function values.
Constraint: ipminf=0, 1 or 2.
29: xratio Real (Kind=nag_wp) Input
On entry: an input bound on the adjacent mesh ratio (greater than 1.0 and typically in the range 1.5 to 3.0). The remeshing routines will attempt to ensure that
(xi-xi-1)/xratio<xi+1-xi<xratio×(xi-xi-1). 
Suggested value: xratio=1.5.
Constraint: xratio>1.0.
30: con Real (Kind=nag_wp) Input
On entry: an input bound on the sub-integral of the monitor function Fmon(x) over each space step. The remeshing routines will attempt to ensure that
xixi+1Fmon(x)dxconx1xnptsFmon(x)dx,  
(see Furzeland (1984)). con gives you more control over the mesh distribution e.g., decreasing con allows more clustering. A typical value is 2/(npts-1), but you are encouraged to experiment with different values. Its value is not critical and the mesh should be qualitatively correct for all values in the range given below.
Suggested value: con=2.0/(npts-1).
Constraint: 0.1/(npts-1)con10.0/(npts-1).
31: monitf Subroutine, supplied by the NAG Library or the user. External Procedure
monitf must supply and evaluate a remesh monitor function to indicate the solution behaviour of interest.
If you specify remesh=.FALSE., i.e., no remeshing, monitf will not be called and the dummy routine d03pcl for d03ppf (or d53pcl for d03ppa) may be used for monitf. (d03pcl and d53pcl are included in the NAG Library.)
The specification of monitf for d03ppf is:
Fortran Interface
Subroutine monitf ( t, npts, npde, x, u, r, fmon)
Integer, Intent (In) :: npts, npde
Real (Kind=nag_wp), Intent (In) :: t, x(npts), u(npde,npts), r(npde,npts)
Real (Kind=nag_wp), Intent (Out) :: fmon(npts)
C Header Interface
void  monitf (const double *t, const Integer *npts, const Integer *npde, const double x[], const double u[], const double r[], double fmon[])
The specification of monitf for d03ppa is:
Fortran Interface
Subroutine monitf ( t, npts, npde, x, u, r, fmon, iuser, ruser)
Integer, Intent (In) :: npts, npde
Integer, Intent (Inout) :: iuser(*)
Real (Kind=nag_wp), Intent (In) :: t, x(npts), u(npde,npts), r(npde,npts)
Real (Kind=nag_wp), Intent (Inout) :: ruser(*)
Real (Kind=nag_wp), Intent (Out) :: fmon(npts)
C Header Interface
void  monitf (const double *t, const Integer *npts, const Integer *npde, const double x[], const double u[], const double r[], double fmon[], Integer iuser[], double ruser[])
1: t Real (Kind=nag_wp) Input
On entry: the current value of the independent variable t.
2: npts Integer Input
On entry: the number of mesh points in the interval [a,b].
3: npde Integer Input
On entry: the number of PDEs in the system.
4: x(npts) Real (Kind=nag_wp) array Input
On entry: the current mesh. x(i) contains the value of xi, for i=1,2,,npts.
5: u(npde,npts) Real (Kind=nag_wp) array Input
On entry: u(i,j) contains the value of Ui(x,t) at x=x(j) and time t, for i=1,2,,npde and j=1,2,,npts.
6: r(npde,npts) Real (Kind=nag_wp) array Input
On entry: r(i,j) contains the value of Ri(x,t,U,Ux,V) at x=x(j) and time t, for i=1,2,,npde and j=1,2,,npts.
7: fmon(npts) Real (Kind=nag_wp) array Output
On exit: fmon(i) must contain the value of the monitor function Fmon(x) at mesh point x=x(i).
Constraint: fmon(i)0.0.
Note: the following are additional arguments for specific use with d03ppa. Users of d03ppf therefore need not read the remainder of this description.
8: iuser(*) Integer array User Workspace
9: ruser(*) Real (Kind=nag_wp) array User Workspace
monitf is called with the arguments iuser and ruser as supplied to d03ppf/​d03ppa. You should use the arrays iuser and ruser to supply information to monitf.
monitf must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d03ppf/​d03ppa is called. Arguments denoted as Input must not be changed by this procedure.
Note: monitf should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d03ppf/​d03ppa. If your code inadvertently does return any NaNs or infinities, d03ppf/​d03ppa is likely to produce unexpected results.
32: 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.
33: lrsave Integer Input
On entry: the dimension of the array rsave as declared in the (sub)program from which d03ppf/​d03ppa is called. Its size depends on the type of matrix algebra selected.
If laopt='F', lrsaveneqn×neqn+neqn+nwkres+lenode.
If laopt='B', lrsave(3×mlu+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,
mlu=2npde-1;
for coupled PDE/ODE problems,
mlu=neqn-1.
Where nwkres is defined by
if nv>0​ and ​nxi>0,
nwkres=npde(3npde+6nxi+npts+15)+nxi+nv+7npts+nxfix+1;
if nv>0​ and ​nxi=0,
nwkres=npde(3npde+npts+21)+nv+7npts+nxfix+2;
if nv=0,
nwkres=npde(3npde+npts+21)+7npts+nxfix+3.
Where lenode is defined by
if the BDF method is used,
lenode=(6+int(algopt(2)))×neqn+50;
if the Theta method is used,
lenode=9neqn+50.
Note: when using the sparse option, the value of lrsave may be too small when supplied to the integrator. An estimate of the minimum size of lrsave is printed on the current error message unit if itrace>0 and the routine returns with ifail=15.
34: 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 required for subsequent calls. 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 ODE method last used in the time integration.
isave(5)
Contains the number of Newton iterations performed by the time integrator. Each iteration involves residual evaluation of the resulting ODE system followed by a back-substitution using the LU decomposition of the Jacobian matrix.
The rest of the array is used as workspace.
35: lisave Integer Input
On entry: the dimension of the array isave as declared in the (sub)program from which d03ppf/​d03ppa is called.
Its size depends on the type of matrix algebra selected:
  • if laopt='B', lisaveneqn+25+nxfix;
  • if laopt='F', lisave25+nxfix;
  • if laopt='S', lisave25×neqn+25+nxfix.
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.
36: 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.
37: itrace Integer Input
On entry: the level of trace information required from d03ppf/​d03ppa and the underlying ODE solver:
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=1
Output from the underlying ODE solver is printed on the current advisory message unit (see x04abf). This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.
itrace=2
Output from the underlying ODE solver is similar to that produced when itrace=1, except that the advisory messages are given in greater detail.
itrace3
Output from the underlying ODE solver is similar to that produced when itrace=2, except that the advisory messages are given in greater detail.
You are advised to set itrace=0, unless you are experienced with Sub-chapter D02M–N.
38: ind Integer Input/Output
On entry: must be set to 0 or 1.
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 and the remeshing arguments nrmesh, dxmesh, trmesh, xratio and con may be reset between calls to d03ppf/​d03ppa.
Constraint: 0ind1.
On exit: ind=1.
39: ifail Integer Input/Output
Note: for d03ppa, 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 d03ppa. Users of d03ppf therefore need not read the remainder of this description.
39: iuser(*) Integer array User Workspace
40: ruser(*) Real (Kind=nag_wp) array User Workspace
iuser and ruser are not used by d03ppf/​d03ppa, but are passed directly to pdedef, bndary, uvinit, odedef and monitf and may be used to pass information to these routines.
41: cwsav(10) Character(80) array Communication Array
42: lwsav(100) Logical array Communication Array
43: iwsav(505) Integer array Communication Array
44: 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 d03ppf/​d03ppa.
45: 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, con=value, npts=value.
Constraint: con10.0/(npts-1).
On entry, con=value, npts=value.
Constraint: con0.1/(npts-1).
On entry, dxmesh=value.
Constraint: dxmesh0.0.
On entry, i=value, xfix(i+1)=value and xfix(i)=value.
Constraint: xfix(i+1)>xfix(i).
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, ipminf=value.
Constraint: ipminf=0, 1 or 2.
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, nxfix=value, npts=value.
Constraint: nxfixnpts-2.
On entry, nxfix=value.
Constraint: nxfix0.
On entry, on initial entry ind=1.
Constraint: on initial entry ind=0.
On entry, the point xfix(i) does not coincide with any x(j): i=value and xfix(i)=value.
On entry, tout=value and ts=value.
Constraint: tout>ts.
On entry, tout-ts is too small: tout=value and ts=value.
On entry, xratio=value.
Constraint: xratio>1.0.
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.
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 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.
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 itask2 or 5.)
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=16
remesh has been changed between calls to d03ppf/​d03ppa.
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

d03ppf/​d03ppa 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

d03ppf/​d03ppa is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d03ppf/​d03ppa 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 d03prf.
The time taken depends on the complexity of the parabolic system, the accuracy requested, and the frequency of the mesh updates. For a given system with fixed accuracy and mesh-update frequency it is approximately proportional to neqn.

10 Example

This example uses Burgers Equation, a common test problem for remeshing algorithms, given by
U t =-U U x +E 2U x2 ,  
for x[0,1] and t[0,1], where E is a small constant.
The initial and boundary conditions are given by the exact solution
U(x,t)=0.1exp(-A)+0.5exp(-B)+exp(-C) exp(-A)+exp(-B)+exp(-C) ,  
where
A = 50E(x-0.5+4.95t), B = 250E(x-0.5+0.75t), C = 500E(x-0.375).  

10.1 Program Text

Note: the following programs illustrate the use of d03ppf and d03ppa.
Program Text (d03ppfe.f90)
Program Text (d03ppae.f90)

10.2 Program Data

Program Data (d03ppfe.d)
Program Data (d03ppae.d)

10.3 Program Results

Program Results (d03ppfe.r)
Program Results (d03ppae.r)
GnuplotProduced by GNUPLOT 5.0 patchlevel 0 Example Program Solution of Burgers Equation using Moving Mesh U(x,t) gnuplot_plot_1