We analyze in this paper the convergence properties of the parareal algorithm for the symmetric positive definite problem u + Au = g. The coarse propagator G is fixed to the backwardEuler method and three time integrators are chosen for the F -propagator: the trapezoidal rule, the third-order diagonal implicit Runge-Kutta (RK) (DIRK) method, and the fourth-order Gauss RK method. It is well known that the Parareal-Euler algorithm using the backward-Euler method for F and G converges rapidly, but less is known when one uses for F the trapezoidal rule, or the fourthorder Gauss RK method, especially when the mesh ratio J(= ΔT /Δt) is small. We show that for a specified λmax(the maximal eigenvalue of A or its upper bound), there exists some critical J cri such that the parareal solvers derived from these three choices of F converge as fast as Parareal-Euler, provided J ≥ J cri . Precisely, for F the trapezoidal rule and the fourth-order Gauss RK method, J cri depends on ΔT, Δt, and λmax and we present concise formulas to calculate J cri , while for F the third-order DIRK method, J cri = 4, independently of these parameters. Numerical examples with applications in fractional PDEs and uncertainty quantification are presented to support the theoretical predictions.1. Introduction. The parareal algorithm, proposed by Lions, Maday, and Turinici in [18], received considerable attention in recent years, because of its potential for parallel-in-time computation. It permits the computation of the solution later in time, before having fully accurate approximations at earlier times, while the global accuracy of the iterative process after a few iterations is comparable to that given by a sequential numerical method used on a fine discretization in time. Nowadays, this algorithm, as well as some other relevant algorithms [5,7,9,10,11,22,29,21], have been used in many fields by many researchers, such as molecular-dynamics simulations [2], morphological transformation simulations [15], structural (fluid) dynamics simulations [5, 10], optimal control [6, 20, 24, 23], Hamiltonian simulations [8,14], simulations of turbulent plasmas [26,27], fast computation of wave equations [7] and Volterra integral equations [17], etc. For a survey of parallel-in-time algorithms, see [16].The algorithm is defined by two time propagators, namely, F and G, which are associated with the small step size Δt and the big step size ΔT . The two step sizes satisfy ΔT = JΔt, where J ≥ 2 is an integer. Since the backward-Euler method is stable for all possible choices of ΔT and the coarse propagator is often forced to take large step sizes, choosing for G the backward-Euler method makes sense. (We can also consider other implicit methods, but they are all more expensive than the backward-Euler method.) Throughout this paper, the G-propagator is fixed to the
