A Parallel Adaptive Recursive Solver For ODE Two-Point Boundary Value Problems

In this paper, we present a parallel adaptive recursive solver( pars) for linear variable coefficient ordinary differential equation two-point boundary value problems(BVP). Following the work by Leslie Greengard and June Y ub Lee [12], our numerical solver utilizes a well-conditioned integral equation reformulation, a spectral discretization scheme and a divide-and-conquer strategy to combine the local spectral solutions efficiently to form the global solution. For a prescribed accuracy requirement, an error analysis based adaptive refinement strategy is designed.The algorithm is black-box, parallelized, and achievable for arbitrary high-order accuracy. The solver is programmed mostly in C++ on multi-core computers using the Cilk multithreaded runtime system, which is recursively parallelized to solve the original problems. This paper summarizes the mathematical foundations of the algorithm and its parallel algorithm theory, and presents several numerical examples. As far as we know, our algorithm is the most efficient and numerically stable parallel algorithm for solving ODE two point boundary value problems.One important application of the developed algorithm is to solve higher dimensional time dependent partial differential equations: when these equations are discretized using the “method of lines transpose” in time and “alternating direction implicit” or “locally one-dimension” type techniques, the fundamental building block required is the efficient solution of two point boundary value ordinary differential equation solvers. This is how the work in this paper is motivated.