Numerical Analysis/ DCS/ University of Toronto

The PMIRKDC package

PMIRKDC is a collection of Fortran subroutines for solving boundary value ordinary differential equations (BVODEs) on a parallel shared-memory computer. PMIRKDC is based on the package MIRKDC [1], which employs mono-implicit Runge-Kutta schemes within a defect control algorithm. The primary computational costs involve the treatment of large, almost block diagonal (ABD) linear systems. The most significant feature of PMIRKDC is the replacement of sequential ABD software, COLROW, with new parallel ABD software, RSCALE, based on a parallel block eigenvalue rescaling algorithm. Other modifications involve parallelization of the setup of the ABD systems and solution interpolants, and defect estimation. When run on a sequential computer, all parallel directives in PMIRKDC are ignored (they are treated as comments).

The software contained on this page was used to generate many of the numerical results discussed in [2], in which a modification of the nonlinear problem Swirling Flow III [3] is solved with the PMIRKDC/RSCALE package. Complete download and installation instructions, along with sample input and output, are included below.

Source code

There are 10 files in total:

Parallel MirkDC subroutines
- pmirkdc (22 subroutines)
Swirling Flow III (SWF3) subroutines (solution uses parameter continuation on SWF3's epsilon)
- swf_iii (6 subroutines + MAIN)
Timing statistics collection subroutines
- LAstats (for ABD system factorizations and solves)
- nonLAstats (for residual and Jacobian evaluation, defect estimation)
- TOTALstats
ABD system reformatting subroutines
Parallel processor set-up subroutines
- M_SET_PROCS
- init_procs

Installation

PMIRKDC must be linked with RSCALE, LAPACK and the level-3 BLAS. Follow these steps to install on a sequential computer:

Download the 10 files listed above and the 16 files listed on the RSCALE page to a common directory.
Compile with f77 -c -u *.f
Link and load with f77 *.o -LYourLibrary -llapack -lblas

Execution / input and output

The executable a.out reads from standard (keyboard) input and writes to standard (terminal screen) output. Input/output may be redirected from/to files with a.out < input.txt > output.txt

a.out expects the input described below. Further details on these input values may be found in swf_iii and pmirkdc.

order of MIRK scheme (integer)
defect tolerance (double precision)
number of subintervals in initial mesh (integer)
SWF3's epsilon (double precision)
SWF3's disk coordinates (double precision)
number of partitions (integer)
output control (integer; controls verbosity of PMIRKDC's diagnostics - see pmirkdc for further details)
selection criterion for hybrid ABD system solver kernel (integer; ignored in this implementation - just set to 0)
parameter continuation strategy nconts (integer; see below)

The SWF3 problem is solved using a parameter continuation strategy on epsilon. The input value nconts is interpreted as follows:

if (nconts = 1 .or. nconts = 0) parameter continuation is not used; i.e., we attempt to solve the problem for the given value of epsilon with a single call to PMIRKDC
if (iabs(nconts) > 1) parameter continuation is used; the next line of input contains nconts values for epsilon, in descending order of magnitude, with the last value the desired epsilon for the final solution, and with sign(nconts) specifying how statistics will be recorded:
- if (nconts > 0) statistics are accumulated over all nconts continuation problems
- if (nconts < 0) statistics are recorded for the last continuation problem only

For example, the data given in input.txt specifies an (epsilon = .0001) SWF3 problem on the interval [-1, 1]. The problem is to be solved using a 4th-order MIRK scheme, starting with an initial uniform mesh of 10 subintervals. Five continuation iterations will be invoked, for epsilon = .002, .001, .0004, .0002, and finally .0001. For each continuation problem, the computed solution must satisfy a defect tolerance of 1.d-07. All parallelized code segments are to be partitioned into 2 slices and executed concurrently on 2 processors. Timing statistics will be reported for the final continuation iteration only.

When a.out is run with input.txt, the result is output.txt. The output is more or less self-explanatory. Three tables are generated showing a breakdown of (1) the monitored non-(Linear Algebra) time, (2) the monitored Linear Algebra time, and (3) the overall statistics. (``Linear Algebra'', in this context, refers to the factorization and solution of the ABD linear systems, and ``non-(Linear Algebra)'' refers to the residual and Jacobian evaluations, and defect estimation.) The total time ``not monitored'' shown in the overall statistics refers to the cumulative time taken by the non-parallelized segments of PMIRKDC. It is the least significant time when a problem is solved on one processor, but clearly becomes increasingly significant as more processors are used and the parallelized segments of the code begin to speed up.

References

W.H. Enright and P.H. Muir, Runge-Kutta software with defect control for boundary value ODEs, SIAM J. Sci. Comput., 17 (1996) pp. 479-497.
P.H. Muir, R.N. Pancer and K.R. Jackson, PMIRKDC: a parallel mono-implicit Runge-Kutta code with defect control for boundary value ODEs, Parallel Comput., 29/6 (2003) pp. 711-741.
U.M. Ascher, R.M.M. Mattheij and R.D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Classics in Applied Mathematics Series, SIAM, Philadelphia (1995).

Last modified by Richard Pancer, 13 November 2004.
Comments, complaints and reports of broken links to pancer@utsc.utoronto.ca.