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.
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:
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.
3Description
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.
4References
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. Software22 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. Maths12 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
5Arguments
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.
On entry: a configuration object that holds information on the differentiation strategy. Details on setting the AD strategy are described in AD handle object in the NAG AD Library Introduction.
pdedef
needs to be callable with the specification listed below. This can be a C++ lambda, a functor or a (static member) function pointer.
If using a lambda, parameters can be captured safely by reference. No copies of the callable are made internally.
bndary
needs to be callable with the specification listed below. This can be a C++ lambda, a functor or a (static member) function pointer.
If using a lambda, parameters can be captured safely by reference. No copies of the callable are made internally.
pdeiv
needs to be callable with the specification listed below. This can be a C++ lambda, a functor or a (static member) function pointer.
If using a lambda, parameters can be captured safely by reference. No copies of the callable are made internally.
monitr
needs to be callable with the specification listed below. This can be a C++ lambda, a functor or a (static member) function pointer.
If using a lambda, parameters can be captured safely by reference. No copies of the callable are made internally.
d03ra preserves all error codes from d03raf and in addition can return:
An unexpected AD error has been triggered by this routine. Please
contact NAG.
See Error Handling in the NAG AD Library Introduction for further information.
The routine was called using a strategy that has not yet been implemented.
See AD Strategies in the NAG AD Library Introduction for further information.
A C++ exception was thrown.
The error message will show the details of the C++ exception text.
Dynamic memory allocation failed for AD.
See Error Handling in the NAG AD Library Introduction for further information.
7Accuracy
Not applicable.
8Parallelism and Performance
d03ra
is not threaded in any implementation.
9Further Comments
None.
10Example
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 is
for and , with initial conditions for , and boundary conditions
The heat release argument , the Damkohler number , the activation energy , the reaction rate , and the diffusion argument .
For small times the temperature gradually increases in a circular region about the origin, and at about
‘ignition’ occurs causing the temperature to suddenly jump from near unity to , 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 and of the prey and the predator respectively are
with
where and , and .
The initial conditions are taken to be simple peaked functions which satisfy the boundary conditions and very nearly satisfy the PDEs:
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.