All of the popular methods for solving boundary value problems (BVPs) for systems of ordinary differential equations (ODEs) involve the solution of systems of linear algebraic equations. These systems are large when the BVP involves many ODEs, or has difficult behavior. They arise directly when the ODEs are linear. If the ODEs are nonlinear, the algebraic equations of the method are linearized and solved iteratively. It may then be necessary to solve many linear systems. Fortunately, these have a special form called almost block diagonal (ABD). The fact that ABD systems can be solved much more efficiently, both in terms of runtime and storage, than general linear systems is what makes the solution of large or difficult BVPs practical. If the boundary conditions are not separated, the linear equations have a variant form called bordered ABD. These equations can be much more difficult to solve numerically, reflecting the behavior of solutions of the BVP itself.

Muir, Pancer, and Jackson provide an excellent survey of methods for solving BVPs numerically, and, in particular, for the stable numerical solution of ABD systems. Because solving ABD systems is a very important, and perhaps dominant, part of the cost of solving a BVP, it is natural to try to solve them in parallel. The authors survey the various approaches, and implement the RSCALE method of Jackson and Pancer in a new code, PMIRKDC. This is a development of Enright and Muir’s MIRKDC, a quality BVP solver based on mono-implicit Runge-Kutta formulas with defect control. In PMIRKDC, the authors parallelize important computations particular to the underlying method, such as estimation of the defect and evaluation of continuous extensions, but the substantial improvements they obtain by parallel solution of ABD systems can be expected for other kinds of methods for solving BVPs as well.