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

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.