d03ra is the AD Library version of the primal routine d03raf. Based (in the C++ interface) on overload resolution, d03ra can be used for primal, tangent and adjoint evaluation. It supports tangents and adjoints of first order.

2 Specification

Fortran Interface

Subroutine d03ra_AD_f (

ad_handle, npde, ts, tout, dt, xmin, xmax, ymin, ymax, nx, ny, tols, tolt, pdedef, bndary, pdeiv, monitr, opti, optr, rwk, lenrwk, iwk, leniwk, lwk, lenlwk, itrace, ind, iuser, ruser, lwsav, iwsav, rwsav, ifail)

Integer, Intent (In)	::	npde, nx, ny, opti(4), lenrwk, leniwk, lenlwk, itrace
Integer, Intent (Inout)	::	iwk(leniwk), ind, iuser(*), iwsav(20), ifail
ADTYPE, Intent (In)	::	tout, xmin, xmax, ymin, ymax, tols, tolt, optr(3,npde)
ADTYPE, Intent (Inout)	::	ts, dt(3), rwk(lenrwk), ruser(*), rwsav(25)
Logical, Intent (Inout)	::	lwsav(5)
Logical, Intent (Out)	::	lwk(lenlwk)
Type (c_ptr), Intent (Inout)	::	ad_handle
External	::	pdedef, bndary, pdeiv, monitr

Corresponding to the overloaded C++ function, the Fortran interface provides five routines with names reflecting the type used for active real arguments. The actual subroutine and type names are formed by replacing AD and ADTYPE in the above as follows:

when ADTYPE is	Real(kind=nag_wp)	then AD is p0w
when ADTYPE is	Type(nagad_a1w_w_rtype)	then AD is a1w
when ADTYPE is	Type(nagad_t1w_w_rtype)	then AD is t1w

C++ Header Interface

#include <dco.hpp>
#include <nagad.h>
namespace nag {
namespace ad {

void d03ra (

void *&ad_handle, const Integer &npde, ADTYPE &ts, const ADTYPE &tout, ADTYPE dt[], const ADTYPE &xmin, const ADTYPE &xmax, const ADTYPE &ymin, const ADTYPE &ymax, const Integer &nx, const Integer &ny, const ADTYPE &tols, const ADTYPE &tolt,
void (NAG_CALL pdedef)(void *&ad_handle, const Integer &npts, const Integer &npde, const ADTYPE &t, const ADTYPE x[], const ADTYPE y[], const ADTYPE u[], const ADTYPE ut[], const ADTYPE ux[], const ADTYPE uy[], const ADTYPE uxx[], const ADTYPE uxy[], const ADTYPE uyy[], ADTYPE res[], Integer iuser[], ADTYPE ruser[]),
void (NAG_CALL bndary)(void *&ad_handle, const Integer &npts, const Integer &npde, const ADTYPE &t, const ADTYPE x[], const ADTYPE y[], const ADTYPE u[], const ADTYPE ut[], const ADTYPE ux[], const ADTYPE uy[], const Integer &nbpts, const Integer lbnd[], ADTYPE res[], Integer iuser[], ADTYPE ruser[]),
void (NAG_CALL pdeiv)(void *&ad_handle, const Integer &npts, const Integer &npde, const ADTYPE &t, const ADTYPE x[], const ADTYPE y[], ADTYPE u[], Integer iuser[], ADTYPE ruser[]),
void (NAG_CALL monitr)(void *&ad_handle, const Integer &npde, const ADTYPE &t, const ADTYPE &dt, const ADTYPE &dtnew, const logical &tlast, const Integer &nlev, const Integer ngpts[], const ADTYPE xpts[], const ADTYPE ypts[], const Integer lsol[], const ADTYPE sol[], Integer &ierr, Integer iuser[], ADTYPE ruser[]),
const Integer opti[], const ADTYPE optr[], ADTYPE rwk[], const Integer &lenrwk, Integer iwk[], const Integer &leniwk, logical lwk[], const Integer &lenlwk, const Integer &itrace, Integer &ind, const Integer &liuser, Integer iuser[], const Integer &lruser, ADTYPE ruser[], logical lwsav[], Integer iwsav[], ADTYPE rwsav[], Integer &ifail)

}
}

The function is overloaded on ADTYPE which represents the type of active arguments. ADTYPE may be any of the following types:
double,
dco::ga1s<double>::type,
dco::gt1s<double>::type

Note: this function can be used with AD tools other than dco/c++. For details, please contact NAG.

3 Description

d03ra is the AD Library version of the primal routine d03raf.

d03raf integrates a system of linear or nonlinear, time-dependent partial differential equations (PDEs) in two space dimensions on a rectangular domain. The method of lines is employed to reduce the PDEs to a system of ordinary differential equations (ODEs) which are solved using a backward differentiation formula (BDF) method. The resulting system of nonlinear equations is solved using a modified Newton method and a Bi-CGSTAB iterative linear solver with ILU preconditioning. Local uniform grid refinement is used to improve the accuracy of the solution. d03raf originates from the VLUGR2 package (see Blom and Verwer (1993) and Blom et al. (1996)). For further information see Section 3 in the documentation for d03raf.

4 References

Adjerid S and Flaherty J E (1988) A local refinement finite element method for two-dimensional parabolic systems SIAM J. Sci. Statist. Comput. 9 792–811

Blom J G, Trompert R A and Verwer J G (1996) Algorithm 758. VLUGR2: A vectorizable adaptive grid solver for PDEs in 2D Trans. Math. Software 22 302–328

Blom J G and Verwer J G (1993) VLUGR2: A vectorized local uniform grid refinement code for PDEs in 2D Report NM-R9306 CWI, Amsterdam

Brown P N, Hindmarsh A C and Petzold L R (1994) Using Krylov methods in the solution of large scale differential-algebraic systems SIAM J. Sci. Statist. Comput. 15 1467–1488

Trompert R A (1993) Local uniform grid refinement and systems of coupled partial differential equations Appl. Numer. Maths 12 331–355

Trompert R A and Verwer J G (1993) Analysis of the implicit Euler local uniform grid refinement method SIAM J. Sci. Comput. 14 259–278

5 Arguments

In addition to the arguments present in the interface of the primal routine, d03ra includes some arguments specific to AD.

A brief summary of the AD specific arguments is given below. For the remainder, links are provided to the corresponding argument from the primal routine. A tooltip popup for all arguments can be found by hovering over the argument name in Section 2 and in this section.

1: ad_handle – Pointer to AD Data Input/Output

On entry: a handle to the AD configuration data object, as created by x10aa.

2: npde – Integer Input

3: ts – ADTYPE Input/Output

4: tout – ADTYPE Input

5: dt( $3$ ) – ADTYPE array Input/Output

6: xmin – ADTYPE Input

7: xmax – ADTYPE Input

8: ymin – ADTYPE Input

9: ymax – ADTYPE Input

10: nx – Integer Input

11: ny – Integer Input

12: tols – ADTYPE Input

13: tolt – ADTYPE Input

14: pdedef – Subroutine External Procedure

The specification of pdedef is:

Fortran Interface

Subroutine pdedef (

ad_handle, npts, npde, t, x, y, u, ut, ux, uy, uxx, uxy, uyy, res, iuser, ruser)

Integer, Intent (In)	::	npts, npde
Integer, Intent (Inout)	::	iuser(*)
ADTYPE, Intent (In)	::	t, x(npts), y(npts), u(npts,npde), ut(npts,npde), ux(npts,npde), uy(npts,npde), uxx(npts,npde), uxy(npts,npde), uyy(npts,npde)
ADTYPE, Intent (Inout)	::	ruser(*)
ADTYPE, Intent (Out)	::	res(npts,npde)
Type (c_ptr), Intent (Inout)	::	ad_handle

C++ Header Interface

void pdedef (

void *&ad_handle, const Integer &npts, const Integer &npde, const ADTYPE &t, const ADTYPE x[], const ADTYPE y[], const ADTYPE u[], const ADTYPE ut[], const ADTYPE ux[], const ADTYPE uy[], const ADTYPE uxx[], const ADTYPE uxy[], const ADTYPE uyy[], ADTYPE res[], Integer iuser[], ADTYPE ruser[])

1: ad_handle – Pointer to AD Data Input/Output: On entry: a handle to the AD configuration data object.
2: npts – Integer Input
3: npde – Integer Input
4: t – ADTYPE Input
5: x – ADTYPE array Input
6: y – ADTYPE array Input
7: u – ADTYPE array Input
8: ut – ADTYPE array Input
9: ux – ADTYPE array Input
10: uy – ADTYPE array Input
11: uxx – ADTYPE array Input
12: uxy – ADTYPE array Input
13: uyy – ADTYPE array Input
14: res – ADTYPE array Output
15: iuser( $*$ ) – Integer array User Workspace
16: ruser( $*$ ) – ADTYPE array User Workspace

15: bndary – Subroutine External Procedure

The specification of bndary is:

Fortran Interface

Subroutine bndary (

ad_handle, npts, npde, t, x, y, u, ut, ux, uy, nbpts, lbnd, res, iuser, ruser)

Integer, Intent (In)	::	npts, npde, nbpts, lbnd(nbpts)
Integer, Intent (Inout)	::	iuser(*)
ADTYPE, Intent (In)	::	t, x(npts), y(npts), u(npts,npde), ut(npts,npde), ux(npts,npde), uy(npts,npde)
ADTYPE, Intent (Inout)	::	res(npts,npde), ruser(*)
Type (c_ptr), Intent (Inout)	::	ad_handle

C++ Header Interface

void bndary (

void *&ad_handle, const Integer &npts, const Integer &npde, const ADTYPE &t, const ADTYPE x[], const ADTYPE y[], const ADTYPE u[], const ADTYPE ut[], const ADTYPE ux[], const ADTYPE uy[], const Integer &nbpts, const Integer lbnd[], ADTYPE res[], Integer iuser[], ADTYPE ruser[])

1: ad_handle – Pointer to AD Data Input/Output: On entry: a handle to the AD configuration data object.
2: npts – Integer Input
3: npde – Integer Input
4: t – ADTYPE Input
5: x – ADTYPE array Input
6: y – ADTYPE array Input
7: u – ADTYPE array Input
8: ut – ADTYPE array Input
9: ux – ADTYPE array Input
10: uy – ADTYPE array Input
11: nbpts – Integer Input
12: lbnd – Integer array Input
13: res – ADTYPE array Input/Output
14: iuser( $*$ ) – Integer array User Workspace
15: ruser( $*$ ) – ADTYPE array User Workspace

16: pdeiv – Subroutine External Procedure

The specification of pdeiv is:

Fortran Interface

Subroutine pdeiv (

ad_handle, npts, npde, t, x, y, u, iuser, ruser)

Integer, Intent (In)	::	npts, npde
Integer, Intent (Inout)	::	iuser(*)
ADTYPE, Intent (In)	::	t, x(npts), y(npts)
ADTYPE, Intent (Inout)	::	ruser(*)
ADTYPE, Intent (Out)	::	u(npts,npde)
Type (c_ptr), Intent (Inout)	::	ad_handle

C++ Header Interface

void pdeiv (

void *&ad_handle, const Integer &npts, const Integer &npde, const ADTYPE &t, const ADTYPE x[], const ADTYPE y[], ADTYPE u[], Integer iuser[], ADTYPE ruser[])

1: ad_handle – Pointer to AD Data Input/Output: On entry: a handle to the AD configuration data object.
2: npts – Integer Input
3: npde – Integer Input
4: t – ADTYPE Input
5: x – ADTYPE array Input
6: y – ADTYPE array Input
7: u – ADTYPE array Output
8: iuser( $*$ ) – Integer array User Workspace
9: ruser( $*$ ) – ADTYPE array User Workspace

17: monitr – Subroutine External Procedure

The specification of monitr is:

Fortran Interface

Subroutine monitr (

ad_handle, npde, t, dt, dtnew, tlast, nlev, ngpts, xpts, ypts, lsol, sol, ierr, iuser, ruser)

Integer, Intent (In)	::	npde, nlev, ngpts(nlev), lsol(nlev)
Integer, Intent (Inout)	::	ierr, iuser(*)
ADTYPE, Intent (In)	::	t, dt, dtnew, xpts(), ypts(), sol(*)
ADTYPE, Intent (Inout)	::	ruser(*)
Logical, Intent (In)	::	tlast
Type (c_ptr), Intent (Inout)	::	ad_handle

C++ Header Interface

void monitr (

void *&ad_handle, const Integer &npde, const ADTYPE &t, const ADTYPE &dt, const ADTYPE &dtnew, const logical &tlast, const Integer &nlev, const Integer ngpts[], const ADTYPE xpts[], const ADTYPE ypts[], const Integer lsol[], const ADTYPE sol[], Integer &ierr, Integer iuser[], ADTYPE ruser[])

1: ad_handle – Pointer to AD Data Input/Output: On entry: a handle to the AD configuration data object.
2: npde – Integer Input
3: t – ADTYPE Input
4: dt – ADTYPE Input
5: dtnew – ADTYPE Input
6: tlast – logical Input
7: nlev – Integer Input
8: ngpts – Integer array Input
9: xpts – ADTYPE array Input
10: ypts – ADTYPE array Input
11: lsol – Integer array Input
12: sol – ADTYPE array Input
13: ierr – Integer Input/Output
14: iuser( $*$ ) – Integer array User Workspace
15: ruser( $*$ ) – ADTYPE array User Workspace

18: opti( $4$ ) – Integer array Input

19: optr( $3$ , npde) – ADTYPE array Input

20: rwk(lenrwk) – ADTYPE array Communication Array

21: lenrwk – Integer Input

22: iwk(leniwk) – Integer array Communication Array

23: leniwk – Integer Input

24: lwk(lenlwk) – logical array Workspace

25: lenlwk – Integer Input

26: itrace – Integer Input

27: ind – Integer Input/Output

28: liuser Input

User workspace dimension (C++ only), see x10af to specify the dimension from Fortran.

29: iuser( $*$ ) – Integer array User Workspace

User workspace.

30: lruser Input

User workspace dimension (C++ only), see x10ae to specify the dimension from Fortran.

31: ruser( $*$ ) – ADTYPE array User Workspace

User workspace.

32: lwsav( $5$ ) – logical array Communication Array

Communication Array.

33: iwsav( $20$ ) – Integer array Communication Array

Communication Array.

34: rwsav( $25$ ) – ADTYPE array Communication Array

Communication Array.

35: ifail – Integer Input/Output

6 Error Indicators and Warnings

d03ra preserves all error codes from d03raf and in addition can return:

$ifail = - 89$: An unexpected AD error has been triggered by this routine. Please contact NAG.
See Section 4.8.2 in the NAG AD Library Introduction for further information.

$ifail = - 199$: The routine was called using a mode that has not yet been implemented.

$ifail = - 443$: On entry: ad_handle is nullptr.
This check is only made if the overloaded C++ interface is used with arguments not of type double.

$ifail = - 444$: A C++ exception was thrown.
The error message will show the details of the C++ exception text.

$ifail = - 899$: Dynamic memory allocation failed for AD.
See Section 4.8.1 in the NAG AD Library Introduction for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

d03ra is not threaded in any implementation.

9 Further Comments

None.

10 Example

The following examples are variants of the example for d03raf, modified to demonstrate calling the NAG AD Library.

Description of the primal example.

For this routine two examples are presented, with a main program and two example problems given in Example 1 (EX1) and Example 2 (EX2).

Example 1 (EX1)

This example stems from combustion theory and is a model for a single, one-step reaction of a mixture of two chemicals (see Adjerid and Flaherty (1988)). The PDE for the temperature of the mixture

u

\frac{\partial u}{\partial t} = d (\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}) + D (1 + α - u) \exp (- \frac{δ}{u})

for

0 \leq x, y \leq 1

and

t \geq 0

, with initial conditions

u (x, y, 0) = 1

for

0 \leq x, y \leq 1

, and boundary conditions

u_{x} (0, y, t) = 0, u (1, y, t) = 1 for ​ 0 \leq y \leq 1,

u_{y} (x, 0, t) = 0, u (x, 1, t) = 1 for ​ 0 \leq x \leq 1 .

The heat release argument

α = 1

, the Damkohler number

D = R \exp (δ) / (α δ)

, the activation energy

δ = 20

, the reaction rate

R = 5

, and the diffusion argument

d = 0.1

For small times the temperature gradually increases in a circular region about the origin, and at about

t = 0.24

‘ignition’ occurs causing the temperature to suddenly jump from near unity to

1 + α

, and a reaction front forms and propagates outwards, becoming steeper. Thus during the solution, just one grid level is used up to the ignition point, then two levels, and then three as the reaction front steepens.

Example 2 (EX2)

This example is taken from a multispecies food web model, in which predator-prey relationships in a spatial domain are simulated (see Brown et al. (1994)). In this example there is just one species each of prey and predator, and the two PDEs for the concentrations

c_{1}

and

c_{2}

of the prey and the predator respectively are

\frac{\partial c_{1}}{\partial t} = c_{1} (b_{1} + a_{11} c_{1} + a_{12} c_{2}) + d_{1} (\frac{\partial^{2} c_{1}}{\partial x^{2}} + \frac{\partial^{2} c_{1}}{\partial y^{2}}),

0 = c_{2} (b_{2} + a_{21} c_{1} + a_{22} c_{2}) + d_{2} (\frac{\partial^{2} c_{2}}{\partial x^{2}} + \frac{\partial^{2} c_{2}}{\partial y^{2}}),

with

\begin{matrix} a_{11} = a_{22} = −1, \\ a_{12} = - 0.5 \times 10^{−6}, and \\ a_{21} = 10^{4}, and \\ b_{1} = 1 + α x y + β \sin (4 π x) \sin (4 π y), \end{matrix}

where

α = 50

and

β = 300

, and

b_{2} = - b_{1}

The initial conditions are taken to be simple peaked functions which satisfy the boundary conditions and very nearly satisfy the PDEs:

c_{1} = 10 + {(16 x (1 - x) y (1 - y))}^{2},

c_{2} = b_{2} + a_{21} c_{1},

and the boundary conditions are of Neumann type, i.e., zero normal derivatives everywhere.

During the solution a number of peaks and troughs develop across the domain, and so the number of levels required increases with time. Since the solution varies rapidly in space across the whole of the domain, refinement at intermediate levels tends to occur at all points of the domain.