An efficient parareal algorithm for a class of time-dependent problems with fractional Laplacian

Time-dependent diffusion equations with fractional Laplacian have received considerable attention in recent years, for which numerical methods play an important role because a simple and analytic solution is often unavailable. We analyze in this paper a parareal algorithm for this kind of problem, which realizes parallel-in-time computation. The algorithm is iterative and uses the 3rd-order SDIRK (singly diagonally implicit Runge-Kutta) method with a small step-size Δt as the F-propagator and the implicit-explicit Euler method with a large step-size ΔT as the G-propagator. The two step-sizes satisfy ΔT/Δt=J with J ≥ 2 being an integer. Using the implicit-explicit Euler method as the G-propagator potentially improves the parallel efficiency, but complicates the convergence analysis. By employing some technical analysis, we provide a sharp estimate of the convergence rate, which is independent of the mesh ratio J and the distribution of the eigenvalues of the coefficient matrix. An extension of the results to problems with time-periodic conditions is also given. Several numerical experiments are carried out to verify the theoretical results.