Waveform relaxation methods for second order linear differential equations

A waveform relaxation (WR) method is one of the numerical methods for solving large systems of ordinary differential equations (ODEs) on parallel computers. Picard iteration, which is one of the simplest WR methods, needs an enormous number of iterations to get a suitably accurate answer, when the interval of integration is long or the stiffness ratio of the equation is large. Therefore, overlapping schemes which make some of the elements between the adjacent subsystems overlapped have been proposed to accelerate the convergence. In this paper, the convergence of the overlapping methods for the second order linear ODEs y'' = Qy + g is investigated. Moreover, the method is implemented on a distributed memory parallel computer and the efficiencies and the elapsed times for various overlapping numbers are examined. (author)