Analysis of the Parareal Algorithm Applied to Hyperbolic Problems Using Characteristics

The parareal algorithm is a time domain decomposition algorithm for the time parallel approximation of solutions of evolution problems. It can be interpreted as a multiple shooting method for initial value problems with a particular choice of the approximate Jacobian on a coarse grid. The method can give significant speedup for non-linear systems of ordinary differential equations and discretized diffusive partial differential equations, but has been reported to be less effective for hyperbolic problems. We prove in this paper a convergence result for the advection equation using the technique of characteristics. Our analysis also reveals limitations of the method when applied to the second order wave equation. Key words: Time parallel time integration methods, multiple shooting for initial value problems, parareal algorithm, hyperbolic problems, advection equation, second order wave equation. AMS subject classifications: 65R20, 45L05, 65L20