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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_pde_1d_parab_dae_fd (d03ph)

Purpose

nag_pde_1d_parab_dae_fd (d03ph) 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).

Syntax

[ts, u, rsave, isave, ind, user, cwsav, lwsav, iwsav, rwsav, ifail] = d03ph(npde, m, ts, tout, pdedef, bndary, u, x, ncode, odedef, xi, rtol, atol, itol, norm_p, laopt, algopt, rsave, isave, itask, itrace, ind, cwsav, lwsav, iwsav, rwsav, 'npts', npts, 'nxi', nxi, 'neqn', neqn, 'lisave', lisave, 'user', user)
[ts, u, rsave, isave, ind, user, cwsav, lwsav, iwsav, rwsav, ifail] = nag_pde_1d_parab_dae_fd(npde, m, ts, tout, pdedef, bndary, u, x, ncode, odedef, xi, rtol, atol, itol, norm_p, laopt, algopt, rsave, isave, itask, itrace, ind, cwsav, lwsav, iwsav, rwsav, 'npts', npts, 'nxi', nxi, 'neqn', neqn, 'lisave', lisave, 'user', user)
Note: the interface to this routine has changed since earlier releases of the toolbox:
Mark 22: lrsave has been removed from the interface
.

Description

nag_pde_1d_parab_dae_fd (d03ph) integrates the system of parabolic-elliptic equations and coupled ODEs
npde
Pi,j(Uj)/(t) + Qi = xm()/(x)(xmRi),  i = 1,2,,npde,  axb,  tt0,
j = 1
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,,ncode,
Fi(t,V,V.,ξ,U*,Ux*,R*,Ut*,Uxt*)=0,  i=1,2,,ncode,
(2)
where (1) defines the PDE part and (2) generalizes the coupled ODE part of the problem.
In (1), Pi,jPi,j and RiRi depend on xx, tt, UU, UxUx and VV; QiQi depends on xx, tt, UU, UxUx, VV and linearly on V.V.. The vector UU is the set of PDE solution values
U (x,t) = [ U1 (x,t) ,, Unpde (x,t) ]T ,
U (x,t) = [ U 1 (x,t) ,, U npde (x,t) ] T ,
and the vector UxUx is the partial derivative with respect to xx. The vector VV is the set of ODE solution values
V (t) = [ V1 (t) ,, Vncode (t) ]T ,
V (t) = [ V 1 ( t ) ,, V ncode ( t ) ] T ,
and V.V. denotes its derivative with respect to time.
In (2), ξξ represents a vector of nξnξ spatial coupling points at which the ODEs are coupled to the PDEs. These points may or may not be equal to some of the PDE spatial mesh points. U*U*, Ux * Ux*, R*R*, Ut * Ut* and Uxt * Uxt* are the functions UU, UxUx, RR, UtUt and UxtUxt evaluated at these coupling points. Each FiFi may only depend linearly on time derivatives. Hence the equation (2) may be written more precisely as
F = GAV.B
(Ut * )
Uxt *
,
F=G-AV.-B Ut* Uxt* ,
(3)
where F = [F1,,Fncode]TF=[F1,,Fncode]T, GG is a vector of length ncode, AA is an ncode by ncode matrix, BB is an ncode by (nξ × npde)(nξ×npde) matrix and the entries in GG, AA and BB may depend on tt, ξξ, U*U*, Ux * Ux* and VV. In practice you only need to supply a vector of information to define the ODEs and not the matrices AA and BB. (See Section [Parameters] for the specification of odedef.)
The integration in time is from t0t0 to touttout, over the space interval axbaxb, where a = x1a=x1 and b = xnptsb=xnpts are the leftmost and rightmost points of a user-defined mesh x1,x2,,xnptsx1,x2,,xnpts. The coordinate system in space is defined by the values of mm; m = 0m=0 for Cartesian coordinates, m = 1m=1 for cylindrical polar coordinates and m = 2m=2 for spherical polar coordinates.
The PDE system which is defined by the functions Pi,jPi,j, QiQi and RiRi must be specified in pdedef.
The initial values of the functions U(x,t)U(x,t) and V(t)V(t) must be given at t = t0t=t0.
The functions RiRi 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,
βi(x,t)Ri(x,t,U,Ux,V)=γi(x,t,U,Ux,V,V.),  i=1,2,,npde,
(4)
where x = ax=a or x = bx=b.
The boundary conditions must be specified in bndary. The function γiγi may depend linearly on V.V..
The problem is subject to the following restrictions:
(i) In (1), V.j(t)V.j(t), for j = 1,2,,ncodej=1,2,,ncode, may only appear linearly in the functions QiQi, for i = 1,2,,npdei=1,2,,npde, with a similar restriction for γγ;
(ii) Pi,jPi,j and the flux RiRi must not depend on any time derivatives;
(iii) t0 < toutt0<tout, so that integration is in the forward direction;
(iv) the evaluation of the terms Pi,jPi,j, QiQi and RiRi is done approximately at the mid-points of the mesh x(i)xi, for i = 1,2,,nptsi=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,,xnptsx1,x2,,xnpts;
(v) at least one of the functions Pi,jPi,j must be nonzero so that there is a time derivative present in the PDE problem;
(vi) if m > 0m>0 and x1 = 0.0x1=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 = 0.0x=0.0 or by specifying a zero flux there, that is βi = 1.0βi=1.0 and γi = 0.0γi=0.0. See also Section [Further Comments] below.
The algebraic-differential equation system which is defined by the functions FiFi must be specified in odedef. You must also specify the coupling points ξξ in the array xi.
The parabolic equations are approximated by a system of ODEs in time for the values of UiUi 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 + ncodenpde×npts+ncode ODEs in the time direction. This system is then integrated forwards in time using a backward differentiation formula (BDF) or a Theta method.

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

Parameters

Compulsory Input Parameters

1:     npde – int64int32nag_int scalar
The number of PDEs to be solved.
Constraint: npde1npde1.
2:     m – int64int32nag_int scalar
The coordinate system used:
m = 0m=0
Indicates Cartesian coordinates.
m = 1m=1
Indicates cylindrical polar coordinates.
m = 2m=2
Indicates spherical polar coordinates.
Constraint: m = 0m=0, 11 or 22.
3:     ts – double scalar
The initial value of the independent variable tt.
Constraint: ts < toutts<tout.
4:     tout – double scalar
The final value of tt to which the integration is to be carried out.
5:     pdedef – function handle or string containing name of m-file
pdedef must evaluate the functions Pi,jPi,j, QiQi and RiRi which define the system of PDEs. The functions may depend on xx, tt, UU, UxUx and VV. QiQi may depend linearly on V.V.. pdedef is called approximately midway between each pair of mesh points in turn by nag_pde_1d_parab_dae_fd (d03ph).
[p, q, r, ires, user] = pdedef(npde, t, x, u, ux, ncode, v, vdot, ires, user)

Input Parameters

1:     npde – int64int32nag_int scalar
The number of PDEs in the system.
2:     t – double scalar
The current value of the independent variable tt.
3:     x – double scalar
The current value of the space variable xx.
4:     u(npde) – double array
u(i)ui contains the value of the component Ui(x,t)Ui(x,t), for i = 1,2,,npdei=1,2,,npde.
5:     ux(npde) – double array
ux(i)uxi contains the value of the component (Ui(x,t))/(x) Ui(x,t) x , for i = 1,2,,npdei=1,2,,npde.
6:     ncode – int64int32nag_int scalar
The number of coupled ODEs in the system.
7:     v(ncode) – double array
If ncode > 0ncode>0, v(i)vi contains the value of the component Vi(t)Vi(t), for i = 1,2,,ncodei=1,2,,ncode.
8:     vdot(ncode) – double array
If ncode > 0ncode>0, vdot(i)vdoti contains the value of component V.i(t)V.i(t), for i = 1,2,,ncodei=1,2,,ncode.
Note:  V.i(t)V.i(t), for i = 1,2,,ncodei=1,2,,ncode, may only appear linearly in QjQj, for j = 1,2,,npdej=1,2,,npde.
9:     ires – int64int32nag_int scalar
Set to 1​ or ​1-1​ or ​1.
10:   user – Any MATLAB object
pdedef is called from nag_pde_1d_parab_dae_fd (d03ph) with the object supplied to nag_pde_1d_parab_dae_fd (d03ph).

Output Parameters

1:     p(npde,npde) – double array
p(i,j)pij must be set to the value of Pi,j(x,t,U,Ux,V)Pi,j(x,t,U,Ux,V), for i = 1,2,,npdei=1,2,,npde and j = 1,2,,npdej=1,2,,npde.
2:     q(npde) – double array
q(i)qi must be set to the value of Qi(x,t,U,Ux,V,V.)Qi(x,t,U,Ux,V,V.), for i = 1,2,,npdei=1,2,,npde.
3:     r(npde) – double array
r(i)ri must be set to the value of Ri(x,t,U,Ux,V)Ri(x,t,U,Ux,V), for i = 1,2,,npdei=1,2,,npde.
4:     ires – int64int32nag_int scalar
Should usually remain unchanged. However, you may set ires to force the integration function to take certain actions as described below:
ires = 2ires=2
Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to ifail = 6ifail=6.
ires = 3ires=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 = 3ires=3 when a physically meaningless input or output value has been generated. If you consecutively set ires = 3ires=3, then nag_pde_1d_parab_dae_fd (d03ph) returns to the calling function with the error indicator set to ifail = 4ifail=4.
5:     user – Any MATLAB object
6:     bndary – function handle or string containing name of m-file
bndary must evaluate the functions βiβi and γiγi which describe the boundary conditions, as given in (4).
[beta, gamma, ires, user] = bndary(npde, t, u, ux, ncode, v, vdot, ibnd, ires, user)

Input Parameters

1:     npde – int64int32nag_int scalar
The number of PDEs in the system.
2:     t – double scalar
The current value of the independent variable tt.
3:     u(npde) – double array
u(i)ui contains the value of the component Ui(x,t)Ui(x,t) at the boundary specified by ibnd, for i = 1,2,,npdei=1,2,,npde.
4:     ux(npde) – double array
ux(i)uxi contains the value of the component (Ui(x,t))/(x) Ui(x,t) x  at the boundary specified by ibnd, for i = 1,2,,npdei=1,2,,npde.
5:     ncode – int64int32nag_int scalar
The number of coupled ODEs in the system.
6:     v(ncode) – double array
If ncode > 0ncode>0, v(i)vi contains the value of the component Vi(t)Vi(t), for i = 1,2,,ncodei=1,2,,ncode.
7:     vdot(ncode) – double array
If ncode > 0ncode>0, vdot(i)vdoti contains the value of component V.i(t)V.i(t), for i = 1,2,,ncodei=1,2,,ncode.
Note:  V.i(t)V.i(t), for i = 1,2,,ncodei=1,2,,ncode, may only appear linearly in QjQj, for j = 1,2,,npdej=1,2,,npde.
8:     ibnd – int64int32nag_int scalar
Specifies which boundary conditions are to be evaluated.
ibnd = 0ibnd=0
bndary must set up the coefficients of the left-hand boundary, x = ax=a.
ibnd0ibnd0
bndary must set up the coefficients of the right-hand boundary, x = bx=b.
9:     ires – int64int32nag_int scalar
Set to 1​ or ​1-1​ or ​1.
10:   user – Any MATLAB object
bndary is called from nag_pde_1d_parab_dae_fd (d03ph) with the object supplied to nag_pde_1d_parab_dae_fd (d03ph).

Output Parameters

1:     beta(npde) – double array
beta(i)betai must be set to the value of βi(x,t)βi(x,t) at the boundary specified by ibnd, for i = 1,2,,npdei=1,2,,npde.
2:     gamma(npde) – double array
gamma(i)gammai must be set to the value of γi(x,t,U,Ux,V,V.)γi(x,t,U,Ux,V,V.) at the boundary specified by ibnd, for i = 1,2,,npdei=1,2,,npde.
3:     ires – int64int32nag_int scalar
Should usually remain unchanged. However, you may set ires to force the integration function to take certain actions as described below:
ires = 2ires=2
Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to ifail = 6ifail=6.
ires = 3ires=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 = 3ires=3 when a physically meaningless input or output value has been generated. If you consecutively set ires = 3ires=3, then nag_pde_1d_parab_dae_fd (d03ph) returns to the calling function with the error indicator set to ifail = 4ifail=4.
4:     user – Any MATLAB object
7:     u(neqn) – double array
neqn, the dimension of the array, must satisfy the constraint neqn = npde × npts + ncodeneqn=npde×npts+ncode.
The initial values of the dependent variables defined as follows:
  • u(npde × (j1) + i)unpde×(j-1)+i contain Ui(xj,t0)Ui(xj,t0), for i = 1,2,,npdei=1,2,,npde and j = 1,2,,nptsj=1,2,,npts, and
  • u(npts × npde + i)unpts×npde+i contain Vi(t0)Vi(t0), for i = 1,2,,ncodei=1,2,,ncode.
8:     x(npts) – double array
npts, the dimension of the array, must satisfy the constraint npts3npts3.
The mesh points in the space direction. x(1)x1 must specify the left-hand boundary, aa, and x(npts)xnpts must specify the right-hand boundary, bb.
Constraint: x(1) < x(2) < < x(npts)x1<x2<<xnpts.
9:     ncode – int64int32nag_int scalar
The number of coupled ODE components.
Constraint: ncode0ncode0.
10:   odedef – function handle or string containing name of m-file
odedef must evaluate the functions FF, which define the system of ODEs, as given in (3).
[f, ires, user] = odedef(npde, t, ncode, v, vdot, nxi, xi, ucp, ucpx, rcp, ucpt, ucptx, ires, user)

Input Parameters

1:     npde – int64int32nag_int scalar
The number of PDEs in the system.
2:     t – double scalar
The current value of the independent variable tt.
3:     ncode – int64int32nag_int scalar
The number of coupled ODEs in the system.
4:     v(ncode) – double array
If ncode > 0ncode>0, v(i)vi contains the value of the component Vi(t)Vi(t), for i = 1,2,,ncodei=1,2,,ncode.
5:     vdot(ncode) – double array
If ncode > 0ncode>0, vdot(i)vdoti contains the value of component V.i(t)V.i(t), for i = 1,2,,ncodei=1,2,,ncode.
6:     nxi – int64int32nag_int scalar
The number of ODE/PDE coupling points.
7:     xi(nxi) – double array
If nxi > 0nxi>0, xi(i)xii contains the ODE/PDE coupling points, ξiξi, for i = 1,2,,nxii=1,2,,nxi.
8:     ucp(npde, : :) – double array
The second dimension of the array must be at least max (1,nxi)max(1,nxi)
If nxi > 0nxi>0, ucp(i,j)ucpij contains the value of Ui(x,t)Ui(x,t) at the coupling point x = ξjx=ξj, for i = 1,2,,npdei=1,2,,npde and j = 1,2,,nxij=1,2,,nxi.
9:     ucpx(npde, : :) – double array
The second dimension of the array must be at least max (1,nxi)max(1,nxi)
If nxi > 0nxi>0, ucpx(i,j)ucpxij contains the value of (Ui(x,t))/(x) Ui(x,t) x  at the coupling point x = ξjx=ξj, for i = 1,2,,npdei=1,2,,npde and j = 1,2,,nxij=1,2,,nxi.
10:   rcp(npde, : :) – double array
The second dimension of the array must be at least max (1,nxi)max(1,nxi)
rcp(i,j)rcpij contains the value of the flux RiRi at the coupling point x = ξjx=ξj, for i = 1,2,,npdei=1,2,,npde and j = 1,2,,nxij=1,2,,nxi.
11:   ucpt(npde, : :) – double array
The second dimension of the array must be at least max (1,nxi)max(1,nxi)
If nxi > 0nxi>0, ucpt(i,j)ucptij contains the value of (Ui)/(t) Ui t  at the coupling point x = ξjx=ξj, for i = 1,2,,npdei=1,2,,npde and j = 1,2,,nxij=1,2,,nxi.
12:   ucptx(npde, : :) – double array
The second dimension of the array must be at least max (1,nxi)max(1,nxi)
ucptx(i,j)ucptxij contains the value of (2Ui)/( x t ) 2Ui x t  at the coupling point x = ξjx=ξj, for i = 1,2,,npdei=1,2,,npde and j = 1,2,,nxij=1,2,,nxi.
13:   ires – int64int32nag_int scalar
The form of FF that must be returned in the array f.
ires = 1ires=1
Equation (5) must be used.
ires = 1ires=-1
Equation (6) must be used.
14:   user – Any MATLAB object
odedef is called from nag_pde_1d_parab_dae_fd (d03ph) with the object supplied to nag_pde_1d_parab_dae_fd (d03ph).

Output Parameters

1:     f(ncode) – double array
f(i)fi must contain the iith component of FF, for i = 1,2,,ncodei=1,2,,ncode, where FF is defined as
F = GAV.B
(Ut * )
Uxt *
,
F=G-AV.-B Ut* Uxt* ,
(5)
or
F = AV.B
(Ut * )
Uxt *
.
F=-AV.-B Ut* Uxt* .
(6)
The definition of FF is determined by the input value of ires.
2:     ires – int64int32nag_int scalar
Should usually remain unchanged. However, you may reset ires to force the integration function to take certain actions as described below:
ires = 2ires=2
Indicates to the integrator that control should be passed back immediately to the calling (sub)routine with the error indicator set to ifail = 6ifail=6.
ires = 3ires=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 = 3ires=3 when a physically meaningless input or output value has been generated. If you consecutively set ires = 3ires=3, then nag_pde_1d_parab_dae_fd (d03ph) returns to the calling function with the error indicator set to ifail = 4ifail=4.
3:     user – Any MATLAB object
11:   xi(nxi) – double array
nxi, the dimension of the array, must satisfy the constraint
If nxi > 0nxi>0, xi(i)xii, for i = 1,2,,nxii=1,2,,nxi, must be set to the ODE/PDE coupling points.
Constraint: x(1)xi(1) < xi(2) < < xi(nxi)x(npts)x1xi1<xi2<<xinxixnpts.
12:   rtol( : :) – double array
Note: the dimension of the array rtol must be at least 11 if itol = 1itol=1 or 22 and at least neqnneqn if itol = 3itol=3 or 44.
The relative local error tolerance.
Constraint: rtol(i)0.0rtoli0.0 for all relevant ii.
13:   atol( : :) – double array
Note: the dimension of the array atol must be at least 11 if itol = 1itol=1 or 33 and at least neqnneqn if itol = 2itol=2 or 44.
The absolute local error tolerance.
Constraint: atol(i)0.0atoli0.0 for all relevant ii.
Note: corresponding elements of rtol and atol cannot both be 0.00.0.
14:   itol – int64int32nag_int scalar
A value to indicate the form of the local error test. itol indicates to nag_pde_1d_parab_dae_fd (d03ph) whether to interpret either or both of rtol or atol as a vector or scalar. The error test to be satisfied is ei / wi < 1.0ei/wi<1.0, where wiwi is defined as follows:
itolrtolatolwiwi
1scalarscalarrtol(1) × |Ui| + atol(1)rtol1×|Ui|+atol1
2scalarvectorrtol(1) × |Ui| + atol(i)rtol1×|Ui|+atoli
3vectorscalarrtol(i) × |Ui| + atol(1)rtoli×|Ui|+atol1
4vectorvectorrtol(i) × |Ui| + atol(i)rtoli×|Ui|+atoli
In the above, eiei denotes the estimated local error for the iith component of the coupled PDE/ODE system in time, u(i)ui, for i = 1,2,,neqni=1,2,,neqn.
The choice of norm used is defined by the parameter norm_p.
Constraint: 1itol41itol4.
15:   norm_p – string (length ≥ 1)
The type of norm to be used.
norm = 'M'norm='M'
Maximum norm.
norm = 'A'norm='A'
Averaged L2L2 norm.
If unormunorm denotes the norm of the vector u of length neqn, then for the averaged L2L2 norm
unorm = sqrt(1/(neqn)i = 1neqn(u(i) / wi)2),
unorm=1neqni=1neqn(ui/wi)2,
while for the maximum norm
unorm = maxi |u(i) / wi| .
u norm = maxi| ui / wi | .
See the description of itol for the formulation of the weight vector ww.
Constraint: norm = 'M'norm='M' or 'A''A'.
16:   laopt – string (length ≥ 1)
The type of matrix algebra required.
laopt = 'F'laopt='F'
Full matrix methods to be used.
laopt = 'B'laopt='B'
Banded matrix methods to be used.
laopt = 'S'laopt='S'
Sparse matrix methods to be used.
Constraint: laopt = 'F'laopt='F', 'B''B' or 'S''S'.
Note: you are recommended to use the banded option when no coupled ODEs are present (i.e., ncode = 0ncode=0).
17:   algopt(3030) – double array
May be set to control various options available in the integrator. If you wish to employ all the default options, then algopt(1)algopt1 should be set to 0.00.0. 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)algopt1
Selects the ODE integration method to be used. If algopt(1) = 1.0algopt1=1.0, a BDF method is used and if algopt(1) = 2.0algopt1=2.0, a Theta method is used. The default value is algopt(1) = 1.0algopt1=1.0.
If algopt(1) = 2.0algopt1=2.0, then algopt(i)algopti, for i = 2,3,4i=2,3,4 are not used.
algopt(2)algopt2
Specifies the maximum order of the BDF integration formula to be used. algopt(2)algopt2 may be 1.01.0, 2.02.0, 3.03.0, 4.04.0 or 5.05.0. The default value is algopt(2) = 5.0algopt2=5.0.
algopt(3)algopt3
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) = 1.0algopt3=1.0 a modified Newton iteration is used and if algopt(3) = 2.0algopt3=2.0 a functional iteration method is used. If functional iteration is selected and the integrator encounters difficulty, then there is an automatic switch to the modified Newton iteration. The default value is algopt(3) = 1.0algopt3=1.0.
algopt(4)algopt4
Specifies whether or not the Petzold error test is to be employed. The Petzold error test results in extra overhead but is more suitable when algebraic equations are present, such as Pi,j = 0.0Pi,j=0.0, for j = 1,2,,npdej=1,2,,npde, for some ii or when there is no V.i(t)V.i(t) dependence in the coupled ODE system. If algopt(4) = 1.0algopt4=1.0, then the Petzold test is used. If algopt(4) = 2.0algopt4=2.0, then the Petzold test is not used. The default value is algopt(4) = 1.0algopt4=1.0.
If algopt(1) = 1.0algopt1=1.0, then algopt(i)algopti, for i = 5,6,7i=5,6,7, are not used.
algopt(5)algopt5
Specifies the value of Theta to be used in the Theta integration method. 0.51algopt(5)0.990.51algopt50.99. The default value is algopt(5) = 0.55algopt5=0.55.
algopt(6)algopt6
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) = 1.0algopt6=1.0, a modified Newton iteration is used and if algopt(6) = 2.0algopt6=2.0, a functional iteration method is used. The default value is algopt(6) = 1.0algopt6=1.0.
algopt(7)algopt7
Specifies whether or not the integrator is allowed to switch automatically between modified Newton and functional iteration methods in order to be more efficient. If algopt(7) = 1.0algopt7=1.0, then switching is allowed and if algopt(7) = 2.0algopt7=2.0, then switching is not allowed. The default value is algopt(7) = 1.0algopt7=1.0.
algopt(11)algopt11
Specifies a point in the time direction, tcrittcrit, beyond which integration must not be attempted. The use of tcrittcrit is described under the parameter itask. If algopt(1)0.0algopt10.0, a value of 0.00.0 for algopt(11)algopt11, say, should be specified even if itask subsequently specifies that tcrittcrit will not be used.
algopt(12)algopt12
Specifies the minimum absolute step size to be allowed in the time integration. If this option is not required, algopt(12)algopt12 should be set to 0.00.0.
algopt(13)algopt13
Specifies the maximum absolute step size to be allowed in the time integration. If this option is not required, algopt(13)algopt13 should be set to 0.00.0.
algopt(14)algopt14
Specifies the initial step size to be attempted by the integrator. If algopt(14) = 0.0algopt14=0.0, then the initial step size is calculated internally.
algopt(15)algopt15
Specifies the maximum number of steps to be attempted by the integrator in any one call. If algopt(15) = 0.0algopt15=0.0, then no limit is imposed.
algopt(23)algopt23
Specifies what method is to be used to solve the nonlinear equations at the initial point to initialize the values of UU, UtUt, VV and V.V.. If algopt(23) = 1.0algopt23=1.0, a modified Newton iteration is used and if algopt(23) = 2.0algopt23=2.0, functional iteration is used. The default value is algopt(23) = 1.0algopt23=1.0.
algopt(29)algopt29 and algopt(30)algopt30 are used only for the sparse matrix algebra option, laopt = 'S'laopt='S'.
algopt(29)algopt29
Governs the choice of pivots during the decomposition of the first Jacobian matrix. It should lie in the range 0.0 < algopt(29) < 1.00.0<algopt29<1.0, with smaller values biasing the algorithm towards maintaining sparsity at the expense of numerical stability. If algopt(29)algopt29 lies outside this range then the default value is used. If the functions regard the Jacobian matrix as numerically singular then increasing algopt(29)algopt29 towards 1.01.0 may help, but at the cost of increased fill-in. The default value is algopt(29) = 0.1algopt29=0.1.
algopt(30)algopt30
Is used as a relative pivot threshold during subsequent Jacobian decompositions (see algopt(29)algopt29) below which an internal error is invoked. If algopt(30)algopt30 is greater than 1.01.0 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)algopt29). The default value is algopt(30) = 0.0001algopt30=0.0001.
18:   rsave(lrsave) – double array
lrsave, the dimension of the array, must satisfy the constraint
If laopt = 'F'laopt='F', lrsaveneqn × neqn + neqn + nwkres + lenodelrsaveneqn×neqn+neqn+nwkres+lenode.
If laopt = 'B'laopt='B', lrsave(3 × mlu + 1) × neqn + nwkres + lenodelrsave(3×mlu+1)×neqn+nwkres+lenode.
If laopt = 'S'laopt='S', lrsave4 × neqn + 11 × neqn / 2 + 1 + nwkres + lenodelrsave4×neqn+11×neqn/2+1+nwkres+lenode.
Note: when laopt = 'S'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 > 0itrace>0 and the function returns with ifail = 15ifail=15.
.
If ind = 0ind=0, rsave need not be set on entry.
If ind = 1ind=1, rsave must be unchanged from the previous call to the function because it contains required information about the iteration.
19:   isave(lisave) – int64int32nag_int array
If ind = 0ind=0, isave need not be set on entry.
If ind = 1ind=1, isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular:
isave(1)isave1
Contains the number of steps taken in time.
isave(2)isave2
Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.
isave(3)isave3
Contains the number of Jacobian evaluations performed by the time integrator.
isave(4)isave4
Contains the order of the last backward differentiation formula method used.
isave(5)isave5
Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the LULU decomposition of the Jacobian matrix.
20:   itask – int64int32nag_int scalar
Specifies the task to be performed by the ODE integrator.
itask = 1itask=1
Normal computation of output values u at t = toutt=tout.
itask = 2itask=2
One step and return.
itask = 3itask=3
Stop at first internal integration point at or beyond t = toutt=tout.
itask = 4itask=4
Normal computation of output values u at t = toutt=tout but without overshooting t = tcritt=tcrit where tcrittcrit is described under the parameter algopt.
itask = 5itask=5
Take one step in the time direction and return, without passing tcrittcrit, where tcrittcrit is described under the parameter algopt.
Constraint: itask = 1itask=1, 22, 33, 44 or 55.
21:   itrace – int64int32nag_int scalar
The level of trace information required from nag_pde_1d_parab_dae_fd (d03ph) and the underlying ODE solver. itrace may take the value 1-1, 00, 11, 22 or 33.
itrace = 1itrace=-1
No output is generated.
itrace = 0itrace=0
Only warning messages from the PDE solver are printed on the current error message unit (see nag_file_set_unit_error (x04aa)).
itrace > 0itrace>0
Output from the underlying ODE solver is printed on the current advisory message unit (see nag_file_set_unit_advisory (x04ab)). This output contains details of Jacobian entries, the nonlinear iteration and the time integration during the computation of the ODE system.
If itrace < 1itrace<-1, then 1-1 is assumed and similarly if itrace > 3itrace>3, then 33 is assumed.
The advisory messages are given in greater detail as itrace increases. You are advised to set itrace = 0itrace=0, unless you are experienced with sub-chapter D02M–N.
22:   ind – int64int32nag_int scalar
Indicates whether this is a continuation call or a new integration.
ind = 0ind=0
Starts or restarts the integration in time.
ind = 1ind=1
Continues the integration after an earlier exit from the function. In this case, only the parameters tout and ifail should be reset between calls to nag_pde_1d_parab_dae_fd (d03ph).
Constraint: ind = 0ind=0 or 11.
23:   cwsav(1010) – cell array of strings
24:   lwsav(100100) – logical array
25:   iwsav(505505) – int64int32nag_int array
26:   rwsav(11001100) – double array

Optional Input Parameters

1:     npts – int64int32nag_int scalar
Default: The dimension of the array x.
The number of mesh points in the interval [a,b][a,b].
Constraint: npts3npts3.
2:     nxi – int64int32nag_int scalar
Default: The dimension of the array xi.
The number of ODE/PDE coupling points.
Constraints:
3:     neqn – int64int32nag_int scalar
Default: The dimension of the array u.
The number of ODEs in the time direction.
Constraint: neqn = npde × npts + ncodeneqn=npde×npts+ncode.
4:     lisave – int64int32nag_int scalar
Default: The dimension of the array isave.
The dimension of the array isave as declared in the (sub)program from which nag_pde_1d_parab_dae_fd (d03ph) is called. Its size depends on the type of matrix algebra selected:
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 > 0itrace>0 and the function returns with ifail = 15ifail=15.
5:     user – Any MATLAB object
user is not used by nag_pde_1d_parab_dae_fd (d03ph), but is passed to pdedef, bndary and odedef. Note that for large objects it may be more efficient to use a global variable which is accessible from the m-files than to use user.

Input Parameters Omitted from the MATLAB Interface

lrsave iuser ruser

Output Parameters

1:     ts – double scalar
The value of tt corresponding to the solution values in u. Normally ts = toutts=tout.
2:     u(neqn) – double array
The computed solution Ui(xj,t)Ui(xj,t), for i = 1,2,,npdei=1,2,,npde and j = 1,2,,nptsj=1,2,,npts, and Vk(t)Vk(t), for k = 1,2,,ncodek=1,2,,ncode, evaluated at t = tst=ts.
3:     rsave(lrsave) – double array
If ind = 1ind=1, rsave must be unchanged from the previous call to the function because it contains required information about the iteration.
4:     isave(lisave) – int64int32nag_int array
If ind = 1ind=1, isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular:
isave(1)isave1
Contains the number of steps taken in time.
isave(2)isave2
Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.
isave(3)isave3
Contains the number of Jacobian evaluations performed by the time integrator.
isave(4)isave4
Contains the order of the last backward differentiation formula method used.
isave(5)isave5
Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the LULU decomposition of the Jacobian matrix.
5:     ind – int64int32nag_int scalar
ind = 1ind=1.
6:     user – Any MATLAB object
7:     cwsav(1010) – cell array of strings
8:     lwsav(100100) – logical array
9:     iwsav(505505) – int64int32nag_int array
10:   rwsav(11001100) – double array
11:   ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

  ifail = 1ifail=1
On entry,touttstout-ts is too small,
oritask1itask1, 22, 33, 44 or 55,
orm0m0, 11 or 22,
orat least one of the coupling points defined in array xi is outside the interval [x(1),x(npts)x1,xnpts],
orm > 0m>0 and x(1) < 0.0x1<0.0,
ornpts < 3npts<3,
ornpde < 1npde<1,
ornorm'A'norm'A' or 'M''M',
orlaopt'F'laopt'F', 'B''B' or 'S''S',
oritol1itol1, 22, 33 or 44,
orind0ind0 or 11,
ormesh points x(i)xi are badly ordered,
orlrsave is too small,
orlisave is too small,
orncode and nxi are incorrectly defined,
orneqnnpde × npts + ncodeneqnnpde×npts+ncode,
oreither an element of rtol or atol < 0.0atol<0.0,
orall the elements of rtol and atol are zero.
W ifail = 2ifail=2
The underlying ODE solver cannot make any further progress, with the values of atol and rtol, across the integration range from the current point t = tst=ts. The components of u contain the computed values at the current point t = tst=ts.
W ifail = 3ifail=3
In the underlying ODE solver, there were repeated error test failures on an attempted step, before completing the requested task, but the integration was successful as far as t = tst=ts. The problem may have a singularity, or the error requirement may be inappropriate.
  ifail = 4ifail=4
In setting up the ODE system, the internal initialization function was unable to initialize the derivative of the ODE system. This could be due to the fact that ires was repeatedly set to 33 in at least pdedef, bndary or odedef, when the residual in the underlying ODE solver was being evaluated.
  ifail = 5ifail=5
In solving the ODE system, a singular Jacobian has been encountered. You should check your problem formulation.
W ifail = 6ifail=6
When evaluating the residual in solving the ODE system, ires was set to 22 in at least pdedef, bndary or odedef. Integration was successful as far as t = tst=ts.
  ifail = 7ifail=7
The values of atol and rtol are so small that the function is unable to start the integration in time.
  ifail = 8ifail=8
In one of pdedef, bndary or odedef, ires was set to an invalid value.
  ifail = 9ifail=9 (nag_ode_ivp_stiff_imp_revcom (d02nn))
A serious error has occurred in an internal call to the specified function. Check the problem specification and all parameters and array dimensions. Setting itrace = 1itrace=1 may provide more information. If the problem persists, contact NAG.
W ifail = 10ifail=10
The required task has been completed, but it is estimated that a small change in atol and rtol is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when itask2itask2 or 55.)
  ifail = 11ifail=11
An error occurred during Jacobian formulation of the ODE system (a more detailed error description may be directed to the current error message unit). If using the sparse matrix algebra option, the values of algopt(29)algopt29 and algopt(30)algopt30 may be inappropriate.
  ifail = 12ifail=12
In solving the ODE system, the maximum number of steps specified in algopt(15)algopt15 have been taken.
W ifail = 13ifail=13
Some error weights wiwi became zero during the time integration (see the description of itol). Pure relative error control (atol(i) = 0.0atoli=0.0) was requested on a variable (the iith) which has become zero. The integration was successful as far as t = tst=ts.
  ifail = 14ifail=14
The flux function RiRi was detected as depending on time derivatives, which is not permissible.
  ifail = 15ifail=15
When using the sparse option, the value of lisave or lrsave was not sufficient (more detailed information may be directed to the current error message unit).

Accuracy

nag_pde_1d_parab_dae_fd (d03ph) controls the accuracy of the integration in the time direction but not the accuracy of the approximation in space. The spatial accuracy depends on both the number of mesh points and on their distribution in space. In the time integration only the local error over a single step is controlled and so the accuracy over a number of steps cannot be guaranteed. You should therefore test the effect of varying the accuracy parameters atol and rtol.

Further Comments

The parameter specification allows you to include equations with only first-order derivatives in the space direction but there is no guarantee that the method of integration will be satisfactory for such systems. The position and nature of the boundary conditions in particular are critical in defining a stable problem. It may be advisable in such cases to reduce the whole system to first-order and to use the Keller box scheme function nag_pde_1d_parab_dae_keller (d03pk).
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.

Example

function nag_pde_1d_parab_dae_fd_example
npde = int64(1);
m = int64(0);
ts = 0.0001;
tout = 0.2;
u = [0.0001000050001666708;
     9.500451264289925e-05;
     9.000405012150273e-05;
     8.500361260235637e-05;
     8.000320008533506e-05;
     7.500281257031378e-05;
     7.000245005716764e-05;
     6.500211254577162e-05;
     6.000180003600051e-05;
     5.500151252772951e-05;
     5.000125002083361e-05;
     4.50010125151877e-05;
     4.000080001066676e-05;
     3.50006125071459e-05;
     3.00004500045e-05;
     2.500031250260415e-05;
     2.000020000133336e-05;
     1.500011250056251e-05;
     1.000005000016669e-05;
     5.000012500020888e-06;
     0;
     0.0001];
x = [0;
     0.05;
     0.1;
     0.15;
     0.2;
     0.25;
     0.3;
     0.35;
     0.4;
     0.45;
     0.5;
     0.55;
     0.6;
     0.65;
     0.7;
     0.75;
     0.8;
     0.85;
     0.9;
     0.95;
     1];
ncode = int64(1);
xi = [1];
rtol = [0.0001];
atol = [0.0001];
itol = int64(1);
normtype = 'A';
laopt = 'F';
algopt = [0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0];
rsave = zeros(994, 1);
isave = zeros(24, 1, 'int64');
itask = int64(1);
itrace = int64(0);
ind = int64(0);
cwsav = {''; ''; ''; ''; ''; ''; ''; ''; ''; ''};
lwsav = false(100, 1);
iwsav = zeros(505, 1, 'int64');
rwsav = zeros(1100, 1);
[tsOut, uOut, rsaveOut, isaveOut, indOut, user, ...
 cwsavOut, lwsavOut, iwsavOut, rwsavOut, ifail] = ...
    nag_pde_1d_parab_dae_fd(npde, m, ts, tout, @pdedef, @bndary, u, x, ncode, ...
    @odedef, xi, rtol, atol, itol, normtype, laopt, algopt, ...
    rsave, isave, itask, itrace, ind, cwsav, lwsav, iwsav, rwsav);
 tsOut, uOut, indOut, ifail

function [p, q, r, ires, user] = pdedef(npde, t, x, u, ux, ncode, v, vdot, ires, user)
      p = zeros(npde, npde);
      q = zeros(npde, 1);
      r = zeros(npde, 1);

      p(1,1) = v(1)*v(1);
      r(1) = ux(1);
      q(1) = -x*ux(1)*v(1)*vdot(1);
function [beta, gamma, ires, user] = ...
    bndary(npde, t, u, ux, ncode, v, vdot, ibnd, ires, user)
      beta = zeros(npde, 1);
      gamma = zeros(npde, 1);
      beta(1) = 1.0d0;
      if (ibnd == 0)
         gamma(1) = -v(1)*exp(t);
      else
         gamma(1) = -v(1)*vdot(1);
      end
function [f, ires, user] = ...
    odedef(npde, t, ncode, v, vdot, nxi, xi, ucp, ucpx, rcp, ucpt, ucptx, ires, user)
      f = zeros(ncode,1);
      if (ires == 1)
         f(1) = vdot(1) - v(1)*ucp(1,1) - ucpx(1,1) - 1.0d0 - t;
      elseif (ires == -1)
         f(1) = vdot(1);
      end
 

tsOut =

    0.2000


uOut =

    0.2221
    0.2099
    0.1979
    0.1860
    0.1742
    0.1625
    0.1510
    0.1396
    0.1282
    0.1170
    0.1059
    0.0949
    0.0840
    0.0733
    0.0626
    0.0520
    0.0416
    0.0312
    0.0210
    0.0109
    0.0008
    0.2000


indOut =

                    1


ifail =

                    0


function d03ph_example
npde = int64(1);
m = int64(0);
ts = 0.0001;
tout = 0.2;
u = [0.0001000050001666708;
     9.500451264289925e-05;
     9.000405012150273e-05;
     8.500361260235637e-05;
     8.000320008533506e-05;
     7.500281257031378e-05;
     7.000245005716764e-05;
     6.500211254577162e-05;
     6.000180003600051e-05;
     5.500151252772951e-05;
     5.000125002083361e-05;
     4.50010125151877e-05;
     4.000080001066676e-05;
     3.50006125071459e-05;
     3.00004500045e-05;
     2.500031250260415e-05;
     2.000020000133336e-05;
     1.500011250056251e-05;
     1.000005000016669e-05;
     5.000012500020888e-06;
     0;
     0.0001];
x = [0;
     0.05;
     0.1;
     0.15;
     0.2;
     0.25;
     0.3;
     0.35;
     0.4;
     0.45;
     0.5;
     0.55;
     0.6;
     0.65;
     0.7;
     0.75;
     0.8;
     0.85;
     0.9;
     0.95;
     1];
ncode = int64(1);
xi = [1];
rtol = [0.0001];
atol = [0.0001];
itol = int64(1);
normtype = 'A';
laopt = 'F';
algopt = [0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0;
     0];
rsave = zeros(994, 1);
isave = zeros(24, 1, 'int64');
itask = int64(1);
itrace = int64(0);
ind = int64(0);
cwsav = {''; ''; ''; ''; ''; ''; ''; ''; ''; ''};
lwsav = false(100, 1);
iwsav = zeros(505, 1, 'int64');
rwsav = zeros(1100, 1);
[tsOut, uOut, rsaveOut, isaveOut, indOut, user, ...
 cwsavOut, lwsavOut, iwsavOut, rwsavOut, ifail] = ...
    d03ph(npde, m, ts, tout, @pdedef, @bndary, u, x, ncode, ...
    @odedef, xi, rtol, atol, itol, normtype, laopt, algopt, ...
    rsave, isave, itask, itrace, ind, cwsav, lwsav, iwsav, rwsav);
 tsOut, uOut, indOut, ifail

function [p, q, r, ires, user] = pdedef(npde, t, x, u, ux, ncode, v, vdot, ires, user)
      p = zeros(npde, npde);
      q = zeros(npde, 1);
      r = zeros(npde, 1);

      p(1,1) = v(1)*v(1);
      r(1) = ux(1);
      q(1) = -x*ux(1)*v(1)*vdot(1);
function [beta, gamma, ires, user] = ...
    bndary(npde, t, u, ux, ncode, v, vdot, ibnd, ires, user)
      beta = zeros(npde, 1);
      gamma = zeros(npde, 1);
      beta(1) = 1.0d0;
      if (ibnd == 0)
         gamma(1) = -v(1)*exp(t);
      else
         gamma(1) = -v(1)*vdot(1);
      end
function [f, ires, user] = ...
    odedef(npde, t, ncode, v, vdot, nxi, xi, ucp, ucpx, rcp, ucpt, ucptx, ires, user)
      f = zeros(ncode,1);
      if (ires == 1)
         f(1) = vdot(1) - v(1)*ucp(1,1) - ucpx(1,1) - 1.0d0 - t;
      elseif (ires == -1)
         f(1) = vdot(1);
      end
 

tsOut =

    0.2000


uOut =

    0.2221
    0.2099
    0.1979
    0.1860
    0.1742
    0.1625
    0.1510
    0.1396
    0.1282
    0.1170
    0.1059
    0.0949
    0.0840
    0.0733
    0.0626
    0.0520
    0.0416
    0.0312
    0.0210
    0.0109
    0.0008
    0.2000


indOut =

                    1


ifail =

                    0



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Chapter Contents
Chapter Introduction
NAG Toolbox

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