Three rapidly convergent parareal solvers with application to time‐dependent PDEs with fractional Laplacian

Abstract

Time‐dependent PDEs with fractional Laplacian ( − Δ)α play a fundamental role in many fields and approximating ( − Δ)α usually leads to ODEs' system like u′(t) + Au(t) = g(t) with A = Qα, where is a sparse symmetric positive definite matrix and α > 0 denotes the fractional order. The parareal algorithm is an ideal solver for this kind of problems, which is iterative and is characterized by two propagators and . The propagators and are respectively associated with large step size ΔT and small step size Δt, where ΔT = JΔt and J⩾2 is an integer. If we fix the ‐propagator to the Implicit‐Euler method and choose for some proper Runge–Kutta (RK) methods, such as the second‐order and third‐order singly diagonally implicit RK methods, previous studies show that the convergence factors of the corresponding parareal solvers can satisfy and , where σ(A) is the spectrum of the matrix A. In this paper, we show that by choosing these two RK methods as the ‐propagator, the convergence factors can reach , provided the one‐stage complex Rosenbrock method is used as the ‐propagator. If we choose for both and , the complex Rosenbrock method, we show that the convergence factor of the resulting parareal solver can also reach . Numerical results are given to support our theoretical conclusions. Copyright © 2017 John Wiley & Sons, Ltd.