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:

At Mark 22:

lrsave was removed from the interface

Description

nag_pde_1d_parab_dae_fd (d03ph) integrates the system of parabolic-elliptic equations and coupled ODEs

\sum_{j = 1}^{npde} P_{i, j} \frac{\partial U_{j}}{\partial t} + Q_{i} = x^{- m} \frac{\partial}{\partial x} (x^{m} R_{i}), i = 1, 2, \dots, npde, a \leq x \leq b, t \geq t_{0},

(1)

F_{i} (t, V, \dot{V}, ξ, U^{*}, U_{x}^{*}, R^{*}, U_{t}^{*}, U_{x t}^{*}) = 0, i = 1, 2, \dots, ncode,

(2)

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

In (1),

P_{i, j}

and

R_{i}

depend on

x

t

U

U_{x}

and

V

;

Q_{i}

depends on

x

t

U

U_{x}

V

and linearly on

\dot{V}

. The vector

U

is the set of PDE solution values

U (x, t) = {[U_{1} (x, t), \dots, U_{npde} (x, t)]}^{T},

and the vector

U_{x}

is the partial derivative with respect to

x

. The vector

V

is the set of ODE solution values

V (t) = {[V_{1} (t), \dots, V_{ncode} (t)]}^{T},

and

\dot{V}

denotes its derivative with respect to time.

In (2),

ξ

represents a vector of

n_{ξ}

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_{x}^{*}

R^{*}

U_{t}^{*}

and

U_{x t}^{*}

are the functions

U

U_{x}

R

U_{t}

and

U_{x t}

evaluated at these coupling points. Each

F_{i}

may only depend linearly on time derivatives. Hence the equation (2) may be written more precisely as

F = G - A \dot{V} - B (\begin{matrix} U_{t}^{*} \\ U_{x t}^{*} \end{matrix}),

(3)

where

F = {[F_{1}, \dots, F_{ncode}]}^{T}

G

is a vector of length ncode,

A

is an ncode by ncode matrix,

B

is an ncode by

(n_{ξ} \times npde)

matrix and the entries in

G

A

and

B

may depend on

t

ξ

U^{*}

U_{x}^{*}

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 Arguments for the specification of odedef.)

The integration in time is from

t_{0}

t_{out}

, over the space interval

a \leq x \leq b

, where

a = x_{1}

and

b = x_{npts}

are the leftmost and rightmost points of a user-defined mesh

x_{1}, x_{2}, \dots, x_{npts}

. The coordinate system in space is defined by the values of

m

;

m = 0

for Cartesian coordinates,

m = 1

for cylindrical polar coordinates and

m = 2

for spherical polar coordinates.

The PDE system which is defined by the functions

P_{i, j}

Q_{i}

and

R_{i}

must be specified in pdedef.

The initial values of the functions

U (x, t)

and

V (t)

must be given at

t = t_{0}

The functions

R_{i}

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

β_{i} (x, t) R_{i} (x, t, U, U_{x}, V) = γ_{i} (x, t, U, U_{x}, V, \dot{V}), i = 1, 2, \dots, npde,

(4)

where

x = a

x = b

The boundary conditions must be specified in bndary. The function

γ_{i}

may depend linearly on

\dot{V}

The problem is subject to the following restrictions:

(i)	In (1), ${\dot{V}}_{j} (t)$ , for $j = 1, 2, \dots, ncode$ , may only appear linearly in the functions $Q_{i}$ , for $i = 1, 2, \dots, npde$ , with a similar restriction for $γ$ ;
(ii)	$P_{i, j}$ and the flux $R_{i}$ must not depend on any time derivatives;
(iii)	$t_{0} < t_{out}$ , so that integration is in the forward direction;
(iv)	the evaluation of the terms $P_{i, j}$ , $Q_{i}$ and $R_{i}$ is done approximately at the mid-points of the mesh $x (i)$ , for $i = 1, 2, \dots, npts$ , 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 $x_{1}, x_{2}, \dots, x_{npts}$ ;
(v)	at least one of the functions $P_{i, j}$ must be nonzero so that there is a time derivative present in the PDE problem;
(vi)	if $m > 0$ and $x_{1} = 0.0$ , 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 $β_{i} = 1.0$ and $γ_{i} = 0.0$ . See also Further Comments below.

The algebraic-differential equation system which is defined by the functions

F_{i}

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

U_{i}

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 \times 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:

npde \geq 1

2: $m$ – int64int32nag_int scalar

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

2

3: $ts$ – double scalar

The initial value of the independent variable

t

Constraint:

ts < tout

4: $tout$ – double scalar

The final value of

t

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

P_{i, j}

Q_{i}

and

R_{i}

which define the system of PDEs. The functions may depend on

x

t

U

U_{x}

and

V

Q_{i}

may depend linearly on

\dot{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 $t$ .
3: $x$ – double scalar: The current value of the space variable $x$ .
4: $u (npde)$ – double array: $u (i)$ contains the value of the component $U_{i} (x, t)$ , for $i = 1, 2, \dots, npde$ .
5: $ux (npde)$ – double array: $ux (i)$ contains the value of the component $\frac{\partial U_{i} (x, t)}{\partial x}$ , for $i = 1, 2, \dots, npde$ .
6: $ncode$ – int64int32nag_int scalar: The number of coupled ODEs in the system.
7: $v (ncode)$ – double array: If $ncode > 0$ , $v (i)$ contains the value of the component $V_{i} (t)$ , for $i = 1, 2, \dots, ncode$ .
8: $vdot (ncode)$ – double array: If $ncode > 0$ , $vdot (i)$ contains the value of component ${\dot{V}}_{i} (t)$ , for $i = 1, 2, \dots, ncode$ .
Note: ${\dot{V}}_{i} (t)$ , for $i = 1, 2, \dots, ncode$ , may only appear linearly in $Q_{j}$ , for $j = 1, 2, \dots, npde$ .
9: $ires$ – int64int32nag_int scalar: Set to $- 1 or 1$ .
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)

must be set to the value of

P_{i, j} (x, t, U, U_{x}, V)

, for

i = 1, 2, \dots, npde

and

j = 1, 2, \dots, npde

2: $q (npde)$ – double array

q (i)

must be set to the value of

Q_{i} (x, t, U, U_{x}, V, \dot{V})

, for

i = 1, 2, \dots, npde

3: $r (npde)$ – double array

r (i)

must be set to the value of

R_{i} (x, t, U, U_{x}, V)

, for

i = 1, 2, \dots, 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:

$ires = 2$: 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 $ires = 3$ , then nag_pde_1d_parab_dae_fd (d03ph) returns to the calling function with the error indicator set to $ifail = 4$ .

5: $user$ – Any MATLAB object

6: $bndary$ – function handle or string containing name of m-file

bndary must evaluate the functions

β_{i}

and

γ_{i}

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

t

3: $u (npde)$ – double array

u (i)

contains the value of the component

U_{i} (x, t)

at the boundary specified by ibnd, for

i = 1, 2, \dots, npde

4: $ux (npde)$ – double array

ux (i)

contains the value of the component

\frac{\partial U_{i} (x, t)}{\partial x}

at the boundary specified by ibnd, for

i = 1, 2, \dots, npde

5: $ncode$ – int64int32nag_int scalar

The number of coupled ODEs in the system.

6: $v (ncode)$ – double array

ncode > 0

v (i)

contains the value of the component

V_{i} (t)

, for

i = 1, 2, \dots, ncode

7: $vdot (ncode)$ – double array

ncode > 0

vdot (i)

contains the value of component

{\dot{V}}_{i} (t)

, for

i = 1, 2, \dots, ncode

Note:

{\dot{V}}_{i} (t)

, for

i = 1, 2, \dots, ncode

, may only appear linearly in

Q_{j}

, for

j = 1, 2, \dots, npde

8: $ibnd$ – int64int32nag_int scalar

Specifies which boundary conditions are to be evaluated.

$ibnd = 0$: bndary must set up the coefficients of the left-hand boundary, $x = a$ .
$ibnd \neq 0$: bndary must set up the coefficients of the right-hand boundary, $x = b$ .

9: $ires$ – int64int32nag_int scalar

Set to

- 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)

must be set to the value of

β_{i} (x, t)

at the boundary specified by ibnd, for

i = 1, 2, \dots, npde

2: $gamma (npde)$ – double array

gamma (i)

must be set to the value of

γ_{i} (x, t, U, U_{x}, V, \dot{V})

at the boundary specified by ibnd, for

i = 1, 2, \dots, 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:

$ires = 2$: 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 $ires = 3$ , then nag_pde_1d_parab_dae_fd (d03ph) returns to the calling function with the error indicator set to $ifail = 4$ .

4: $user$ – Any MATLAB object

7: $u (neqn)$ – double array

The initial values of the dependent variables defined as follows:

$u (npde \times (j - 1) + i)$ contain $U_{i} (x_{j}, t_{0})$ , for $i = 1, 2, \dots, npde$ and $j = 1, 2, \dots, npts$ , and
$u (npts \times npde + i)$ contain $V_{i} (t_{0})$ , for $i = 1, 2, \dots, ncode$ .

8: $x (npts)$ – double array

The mesh points in the space direction.

x (1)

must specify the left-hand boundary,

a

, and

x (npts)

must specify the right-hand boundary,

b

Constraint:

x (1) < x (2) < \dots < x (npts)

9: $ncode$ – int64int32nag_int scalar

The number of coupled ODE components.

Constraint:

ncode \geq 0

10: $odedef$ – function handle or string containing name of m-file

odedef must evaluate the functions

F

, which define the system of ODEs, as given in (3).

If you wish to compute the solution of a system of PDEs only (

ncode = 0

), odedef must be the string nag_pde_1d_parab_remesh_fd_dummy_odedef (d53pck).

[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

t

3: $ncode$ – int64int32nag_int scalar

The number of coupled ODEs in the system.

4: $v (ncode)$ – double array

ncode > 0

v (i)

contains the value of the component

V_{i} (t)

, for

i = 1, 2, \dots, ncode

5: $vdot (ncode)$ – double array

ncode > 0

vdot (i)

contains the value of component

{\dot{V}}_{i} (t)

, for

i = 1, 2, \dots, ncode

6: $nxi$ – int64int32nag_int scalar

The number of ODE/PDE coupling points.

7: $xi (nxi)$ – double array

nxi > 0

xi (i)

contains the ODE/PDE coupling points,

ξ_{i}

, for

i = 1, 2, \dots, nxi

8: $ucp (npde :)$ – double array

The second dimension of the array ucp must be at least

\max (1, nxi)

nxi > 0

ucp (i, j)

contains the value of

U_{i} (x, t)

at the coupling point

x = ξ_{j}

, for

i = 1, 2, \dots, npde

and

j = 1, 2, \dots, nxi

9: $ucpx (npde :)$ – double array

The second dimension of the array ucpx must be at least

\max (1, nxi)

nxi > 0

ucpx (i, j)

contains the value of

\frac{\partial U_{i} (x, t)}{\partial x}

at the coupling point

x = ξ_{j}

, for

i = 1, 2, \dots, npde

and

j = 1, 2, \dots, nxi

10: $rcp (npde :)$ – double array

The second dimension of the array rcp must be at least

\max (1, nxi)

rcp (i, j)

contains the value of the flux

R_{i}

at the coupling point

x = ξ_{j}

, for

i = 1, 2, \dots, npde

and

j = 1, 2, \dots, nxi

11: $ucpt (npde :)$ – double array

The second dimension of the array ucpt must be at least

\max (1, nxi)

nxi > 0

ucpt (i, j)

contains the value of

\frac{\partial U_{i}}{\partial t}

at the coupling point

x = ξ_{j}

, for

i = 1, 2, \dots, npde

and

j = 1, 2, \dots, nxi

12: $ucptx (npde :)$ – double array

The second dimension of the array ucptx must be at least

\max (1, nxi)

ucptx (i, j)

contains the value of

\frac{\partial^{2} U_{i}}{\partial x \partial t}

at the coupling point

x = ξ_{j}

, for

i = 1, 2, \dots, npde

and

j = 1, 2, \dots, nxi

13: $ires$ – int64int32nag_int scalar

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.

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)

must contain the

i

th component of

F

, for

i = 1, 2, \dots, ncode

, where

F

is defined as

F = G - A \dot{V} - B (\begin{matrix} U_{t}^{*} \\ U_{x t}^{*} \end{matrix}),

(5)

F = - A \dot{V} - B (\begin{matrix} U_{t}^{*} \\ U_{x t}^{*} \end{matrix}) .

(6)

The definition of

F

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:

$ires = 2$: 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 $ires = 3$ , then nag_pde_1d_parab_dae_fd (d03ph) returns to the calling function with the error indicator set to $ifail = 4$ .

3: $user$ – Any MATLAB object

11: $xi (nxi)$ – double array

nxi > 0

xi (i)

, for

i = 1, 2, \dots, nxi

, must be set to the ODE/PDE coupling points.

Constraint:

x (1) \leq xi (1) < xi (2) < \dots < xi (nxi) \leq x (npts)

12: $rtol (:)$ – double array

The dimension of the array rtol must be at least

1

itol = 1

2

and at least

neqn

itol = 3

4

The relative local error tolerance.

Constraint:

rtol (i) \geq 0.0

for all relevant

i

13: $atol (:)$ – double array

The dimension of the array atol must be at least

1

itol = 1

3

and at least

neqn

itol = 2

4

The absolute local error tolerance.

Constraint:

atol (i) \geq 0.0

for all relevant

i

Note: corresponding elements of rtol and atol cannot both be

0.0

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

‖e_{i} / w_{i}‖ < 1.0

, where

w_{i}

is defined as follows:

itol	rtol	atol	$w_{i}$
1	scalar	scalar	$rtol (1) \times \|U_{i}\| + atol (1)$
2	scalar	vector	$rtol (1) \times \|U_{i}\| + atol (i)$
3	vector	scalar	$rtol (i) \times \|U_{i}\| + atol (1)$
4	vector	vector	$rtol (i) \times \|U_{i}\| + atol (i)$

In the above,

e_{i}

denotes the estimated local error for the

i

th component of the coupled PDE/ODE system in time,

u (i)

, for

i = 1, 2, \dots, neqn

The choice of norm used is defined by the argument norm_p.

Constraint:

1 \leq itol \leq 4

15: $norm_p$ – string (length ≥ 1)

The type of norm to be used.

$norm_p ='M'$: Maximum norm.
$norm_p ='A'$: Averaged $L_{2}$ norm.

u_{norm}

denotes the norm of the vector u of length neqn, then for the averaged

L_{2}

norm

u_{norm} = \sqrt{\frac{1}{neqn} \sum_{i = 1}^{neqn} {(u (i) / w_{i})}^{2}},

while for the maximum norm

u_{norm} = \max_{i} |u (i) / w_{i}| .

See the description of itol for the formulation of the weight vector

w

Constraint:

norm_p ='M'

'A'

16: $laopt$ – string (length ≥ 1)

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'

'S'

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

ncode = 0

17: $algopt (30)$ – 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)

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 $algopt (1) = 1.0$ .

algopt (1) = 2.0

, then

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, then 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 $P_{i, j} = 0.0$ , for $j = 1, 2, \dots, npde$ , for some $i$ or when there is no ${\dot{V}}_{i} (t)$ dependence in the coupled ODE system. If $algopt (4) = 1.0$ , then the Petzold test is used. If $algopt (4) = 2.0$ , then the Petzold test is not used. The default value is $algopt (4) = 1.0$ .

algopt (1) = 1.0

, then

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. $0.51 \leq algopt (5) \leq 0.99$ . 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 $algopt (7) = 1.0$ , then switching is allowed and if $algopt (7) = 2.0$ , then switching is not allowed. The default value is $algopt (7) = 1.0$ .
$algopt (11)$: Specifies a point in the time direction, $t_{crit}$ , beyond which integration must not be attempted. The use of $t_{crit}$ is described under the argument itask. If $algopt (1) \neq 0.0$ , a value of $0.0$ for $algopt (11)$ , say, should be specified even if itask subsequently specifies that $t_{crit}$ 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 $algopt (14) = 0.0$ , then 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 $algopt (15) = 0.0$ , then 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 $U$ , $U_{t}$ , $V$ and $\dot{V}$ . 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 functions 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 $algopt (30) = 0.0001$ .

18: $rsave (lrsave)$ – double array

lrsave, the dimension of the array, must satisfy the constraint

laopt ='F'

lrsave \geq neqn \times neqn + neqn + nwkres + lenode

laopt ='B'

lrsave \geq (3 \times mlu + 1) \times neqn + nwkres + lenode

laopt ='S'

lrsave \geq 4 \times neqn + 11 \times neqn / 2 + 1 + nwkres + lenode

Note: when

laopt ='S'

, the value of lrsave may be too small when supplied to the integrator. An estimate of the minimum size of lrsave is printed on the current error message unit if

itrace > 0

and the function returns with

ifail = 15

ind = 0

, rsave need not be set on entry.

ind = 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

ind = 0

, isave need not be set on entry.

ind = 1

, isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular:

$isave (1)$: Contains the number of steps taken in time.
$isave (2)$: Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.
$isave (3)$: Contains the number of Jacobian evaluations performed by the time integrator.
$isave (4)$: Contains the order of the last backward differentiation formula method used.
$isave (5)$: Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the $L U$ decomposition of the Jacobian matrix.

20: $itask$ – int64int32nag_int scalar

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 $t = t_{crit}$ where $t_{crit}$ is described under the argument algopt.
$itask = 5$: Take one step in the time direction and return, without passing $t_{crit}$ , where $t_{crit}$ is described under the argument algopt.

Constraint:

itask = 1

2

3

4

5

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

0

1

2

3

$itrace = - 1$: No output is generated.
$itrace = 0$: Only warning messages from the PDE solver are printed on the current error message unit (see nag_file_set_unit_error (x04aa)).
$itrace > 0$: 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.

itrace < - 1

, then

- 1

is assumed and similarly if

itrace > 3

, then

3

is assumed.

The advisory messages are given in greater detail as itrace increases. You are advised to set

itrace = 0

, unless you are experienced with Sub-chapter D02M–N.

22: $ind$ – int64int32nag_int scalar

Indicates whether this is a continuation call or a new integration.

$ind = 0$: Starts or restarts the integration in time.
$ind = 1$: Continues the integration after an earlier exit from the function. In this case, only the arguments tout and ifail should be reset between calls to nag_pde_1d_parab_dae_fd (d03ph).

Constraint:

ind = 0

1

23: $cwsav (10)$ – cell array of strings

24: $lwsav (100)$ – logical array

25: $iwsav (505)$ – int64int32nag_int array

26: $rwsav (1100)$ – 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]

Constraint:

npts \geq 3

2: $nxi$ – int64int32nag_int scalar

Default: the dimension of the array xi.

The number of ODE/PDE coupling points.

Constraints:

if $ncode = 0$ , $nxi = 0$ ;
if $ncode > 0$ , $nxi \geq 0$ .

3: $neqn$ – int64int32nag_int scalar

Default: the dimension of the array u.

The number of ODEs in the time direction.

Constraint:

neqn = npde \times npts + ncode

4: $lisave$ – int64int32nag_int scalar

Default: the dimension of the array isave.

The dimension of the array isave. its size depends on the type of matrix algebra selected:

if $laopt ='F'$ , $lisave \geq 24$ ;
if $laopt ='B'$ , $lisave \geq neqn + 24$ ;
if $laopt ='S'$ , $lisave \geq 25 \times neqn + 24$ .

Note: when using the sparse option, the value of lisave may be too small when supplied to the integrator. An estimate of the minimum size of lisave is printed on the current error message unit if

itrace > 0

and the function returns with

ifail = 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.

Output Parameters

1: $ts$ – double scalar

The value of

t

corresponding to the solution values in u. Normally

ts = tout

2: $u (neqn)$ – double array

The computed solution

U_{i} (x_{j}, t)

, for

i = 1, 2, \dots, npde

and

j = 1, 2, \dots, npts

, and

V_{k} (t)

, for

k = 1, 2, \dots, ncode

, evaluated at

t = ts

3: $rsave (lrsave)$ – double array

ind = 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

ind = 1

, isave must be unchanged from the previous call to the function because it contains required information about the iteration. In particular:

$isave (1)$: Contains the number of steps taken in time.
$isave (2)$: Contains the number of residual evaluations of the resulting ODE system used. One such evaluation involves computing the PDE functions at all the mesh points, as well as one evaluation of the functions in the boundary conditions.
$isave (3)$: Contains the number of Jacobian evaluations performed by the time integrator.
$isave (4)$: Contains the order of the last backward differentiation formula method used.
$isave (5)$: Contains the number of Newton iterations performed by the time integrator. Each iteration involves an ODE residual evaluation followed by a back-substitution using the $L U$ decomposition of the Jacobian matrix.

5: $ind$ – int64int32nag_int scalar

ind = 1

6: $user$ – Any MATLAB object

7: $cwsav (10)$ – cell array of strings

8: $lwsav (100)$ – logical array

9: $iwsav (505)$ – int64int32nag_int array

10: $rwsav (1100)$ – double array

11: $ifail$ – int64int32nag_int scalar

ifail = 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.

$ifail = 1$

On entry,	$tout - ts$ is too small,
or	$itask \neq 1$ , $2$ , $3$ , $4$ or $5$ ,
or	$m \neq 0$ , $1$ or $2$ ,
or	at least one of the coupling points defined in array xi is outside the interval [ $x (1), x (npts)$ ],
or	$m > 0$ and $x (1) < 0.0$ ,
or	$npts < 3$ ,
or	$npde < 1$ ,
or	$norm_p \neq'A'$ or $'M'$ ,
or	$laopt \neq'F'$ , $'B'$ or $'S'$ ,
or	$itol \neq 1$ , $2$ , $3$ or $4$ ,
or	$ind \neq 0$ or $1$ ,
or	mesh points $x (i)$ are badly ordered,
or	lrsave is too small,
or	lisave is too small,
or	ncode and nxi are incorrectly defined,
or	$neqn \neq npde \times npts + ncode$ ,
or	either an element of rtol or $atol < 0.0$ ,
or	all the elements of rtol and atol are zero.

W $ifail = 2$: The underlying ODE solver cannot make any further progress, with the values of atol and rtol, across the integration range from the current point $t = ts$ . The components of u contain the computed values at the current point $t = ts$ .

W $ifail = 3$: In the underlying ODE solver, there were repeated error test failures on an attempted step, before completing the requested task, but the integration was successful as far as $t = ts$ . The problem may have a singularity, or the error requirement may be inappropriate.

$ifail = 4$: In setting up the ODE system, the internal initialization function was unable to initialize the derivative of the ODE system. This could be due to the fact that ires was repeatedly set to $3$ in at least pdedef, bndary or odedef, when the residual in the underlying ODE solver was being evaluated.

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

W $ifail = 6$: When evaluating the residual in solving the ODE system, ires was set to $2$ in at least pdedef, bndary or odedef. Integration was successful as far as $t = ts$ .

$ifail = 7$: The values of atol and rtol are so small that the function is unable to start the integration in time.

$ifail = 8$: In one of pdedef, bndary or odedef, ires was set to an invalid value.

$ifail = 9$ (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 arguments and array dimensions. Setting $itrace = 1$ may provide more information. If the problem persists, contact NAG.

W $ifail = 10$: The required task has been completed, but it is estimated that a small change in atol and rtol is unlikely to produce any change in the computed solution. (Only applies when you are not operating in one step mode, that is when $itask \neq 2$ or $5$ .)

$ifail = 11$: An error occurred during Jacobian formulation of the ODE system (a more detailed error description may be directed to the current error message unit). If using the sparse matrix algebra option, the values of $algopt (29)$ and $algopt (30)$ may be inappropriate.

$ifail = 12$: In solving the ODE system, the maximum number of steps specified in $algopt (15)$ have been taken.

W $ifail = 13$: Some error weights $w_{i}$ 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 $i$ th) which has become zero. The integration was successful as far as $t = ts$ .

$ifail = 14$: The flux function $R_{i}$ was detected as depending on time derivatives, which is not permissible.

$ifail = 15$: When using the sparse option, the value of lisave or lrsave was not sufficient (more detailed information may be directed to the current error message unit).

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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 arguments atol and rtol.

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 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

This example provides a simple coupled system of one PDE and one ODE.

\begin{matrix} {(V_{1})}^{2} \frac{\partial U_{1}}{\partial t} - x V_{1} {\dot{V}}_{1} \frac{\partial U_{1}}{\partial x} = \frac{\partial^{2} y U_{1}}{\partial x^{2}} \\ {\dot{V}}_{1} = V_{1} U_{1} + \frac{\partial U_{1}}{\partial x} + 1 + t, \end{matrix}

for

t \in [10^{- 4}, 0.1 \times 2^{i}]

;

i = 1, 2, \dots, 5

;

x \in [0, 1]

The left boundary condition at

x = 0

\frac{\partial U_{1}}{\partial x} = - V_{1} \exp t .

The right boundary condition at

x = 1

\frac{\partial U_{1}}{\partial x} = - V_{1} {\dot{V}}_{1} .

The initial conditions at

t = 10^{- 4}

are defined by the exact solution:

V_{1} = t,   and U_{1} (x, t) = \exp \{t (1 - x)\} - 1.0, x \in [0, 1],

and the coupling point is at

ξ_{1} = 1.0

Open in the MATLAB editor: d03ph_example

function d03ph_example


fprintf('d03ph example results\n\n');

npde   = int64(1);
m      = int64(0);
ts     = 0.0001;
tout   = 0.2;
x      = [0:0.05:1];
u      = [exp(ts*(1-x))-1,ts];
ncode  = int64(1);
xi     = [1];
rtol   = [0.0001];
atol   = [0.0001];
itol   = int64(1);
normt  = 'A';
laopt  = 'F';
algopt = zeros(30,1);
rsave  = zeros(4000, 1);
isave  = zeros(100, 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);

% Loop over t values
nt = 20;
t = zeros(nt,1);
alpha = (32)^(1/19);
for j = 1:nt
  t(j) = 0.1*alpha^(j-1);
end

for t_i = 1:nt
  tout = t(t_i);
  [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, normt, laopt, algopt, ...
         rsave, isave, itask, itrace, ind, cwsav, lwsav, iwsav, rwsav);

  if mod(t_i+4,6)==0
    fprintf('\nThe solution at t = %7.4f is:\n',ts);
    for j = 0:2
      fprintf('%10s%12s','x','u(x,t)');
    end
    fprintf('\n');
    for i = 1:7
      for j = 0:2
        fprintf('%12.2f%10.4f',x(i+j*7),u(i+j*7));
      end
      fprintf('\n');
    end
  end
  v(:,t_i) = u(1:(end-1));
end

fig1 = figure;
mesh(t,x,v);
xlabel('t');
ylabel('x');
zlabel('u(x,t)');
title('Coupled Parabolic PDE/ODE using Finite-differences and BDF');


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 - t;
  elseif (ires == -1)
    f(1) = vdot(1);
  end

d03ph example results


The solution at t =  0.1200 is:
         x      u(x,t)         x      u(x,t)         x      u(x,t)
        0.00    0.1279        0.35    0.0816        0.70    0.0371
        0.05    0.1212        0.40    0.0751        0.75    0.0309
        0.10    0.1145        0.45    0.0687        0.80    0.0248
        0.15    0.1078        0.50    0.0623        0.85    0.0186
        0.20    0.1012        0.55    0.0559        0.90    0.0126
        0.25    0.0946        0.60    0.0496        0.95    0.0065
        0.30    0.0881        0.65    0.0434        1.00    0.0005

The solution at t =  0.3585 is:
         x      u(x,t)         x      u(x,t)         x      u(x,t)
        0.00    0.4327        0.35    0.2640        0.70    0.1154
        0.05    0.4073        0.40    0.2416        0.75    0.0956
        0.10    0.3823        0.45    0.2196        0.80    0.0762
        0.15    0.3578        0.50    0.1980        0.85    0.0572
        0.20    0.3337        0.55    0.1768        0.90    0.0385
        0.25    0.3101        0.60    0.1560        0.95    0.0201
        0.30    0.2869        0.65    0.1355        1.00    0.0020

The solution at t =  1.0711 is:
         x      u(x,t)         x      u(x,t)         x      u(x,t)
        0.00    1.9236        0.35    1.0146        0.70    0.3902
        0.05    1.7720        0.40    0.9104        0.75    0.3187
        0.10    1.6283        0.45    0.8116        0.80    0.2509
        0.15    1.4920        0.50    0.7180        0.85    0.1868
        0.20    1.3629        0.55    0.6294        0.90    0.1260
        0.25    1.2405        0.60    0.5453        0.95    0.0683
        0.30    1.1245        0.65    0.4657        1.00    0.0138

The solution at t =  3.2000 is:
         x      u(x,t)         x      u(x,t)         x      u(x,t)
        0.00   23.4723        0.35    7.0665        0.70    1.6807
        0.05   19.8823        0.40    5.8860        0.75    1.2948
        0.10   16.8191        0.45    4.8793        0.80    0.9662
        0.15   14.2055        0.50    4.0210        0.85    0.6865
        0.20   11.9756        0.55    3.2891        0.90    0.4485
        0.25   10.0733        0.60    2.6653        0.95    0.2460
        0.30    8.4506        0.65    2.1337        1.00    0.0740

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox