Runge-Kutta Software for the
         Parallel Solution of Boundary Value ODEs

        P.H. Muir, Saint Mary's University,
   R.N. Pancer, University of Toronto at Scarborough,
       K.R. Jackson, University of Toronto

In this paper we describe the development of parallel software for
the numerical solution of boundary value ordinary differential
equations (BVODEs). The software, implemented on two shared memory,
parallel architectures, is based on a modification of the MIRKDC
package, which employs discrete and continuous mono-implicit
Runge-Kutta schemes within a defect control algorithm.  The primary
computational costs are associated with the almost block diagonal
(ABD) linear systems representing the Newton matrices arising from
the iterative solution of the nonlinear algebraic systems which result
from the discretization of the ODEs. The most significant modification
featured in the parallel version of the code is the replacement of the
sequential ABD linear system software, COLROW, which employs alternating
row and column elimination, with new parallel ABD linear system software,
RSCALE, which is based on a recently developed parallel block eigenvalue
rescaling algorithm. Other modifications are associated with the
parallelization of the setup of the ABD systems, the setup of the
approximate solution interpolants, and the estimation of the defect.
The numerical results show that nearly optimal speedups can be obtained,
and that substantial speedups in overall solution time are achieved,
compared with the sequential version of MIRKDC.