The Parareal Algorithm

Abstract

A short introduction to the parareal algorithm is made on basis of [17]. Then, for the autonomous differential equation, a theorem for the stability property of the algorithm is derived. The theorem states that G∆T must be strong A-stable with limz→−∞ |R(z)| ≤ 1 , in order for the algorithm to 2 be stable for all n, k. Several test problems involving periodic solutions, semidiscretized parabolic PDE’s, stiffness and nonlinearity are calculated using different implicit Runge-Kutta schemes with different order and stabiliy properties. Using the theta-method, the property limz→−∞ |R(z)| ≤ 1 2 is tested, and the results verifies the theorem. Instabilities in the two different heat-equation test problems are predicted using the theorem. Different formulations of the system, for which the coarse propagator G∆T operates, are tested. Substantial differences is found, and explenations are discussed.