A second-order parareal algorithm for fractional PDEs

We are concerned with using the parareal (parallel-in-time) algorithm for large scale ODEs system U'(t) + AU(t) + dA(alpha)U(t) = F(t) arising frequently in semi-discretizing time-dependent PDEs with spatial fractional operators, where d > 0 is a constant, a is an element of (0, 1) and Ais a spare and symmetric positive definite (SPD) matrix. The parareal algorithm is iterative and is characterized by two propagators F and G, which are respectively associated with small temporal mesh size Delta t tand large temporal mesh size Delta T. The two mesh sizes satisfy Delta T = J Delta t with J >= 2 being an integer, which is called mesh ratio. Let T-unit(f) and T-unit(g) be respectively the computational cost of the two propagators for moving forward one time step. Then, it is well understoodthat the speedup of the parareal algorithm, namely E, satisfies E = O(clog(1/rho)), where c := T-unit(f)/T-unit(g) and rho is the convergence factor. A larger E corresponds a more efficient parareal solver. For G = Backward-Euler and some choices of F, previous studies show that rho can be a satisfactory quantity. Particularly, for F = 2nd-order DIRK (diagonally implicit Runge-Kutta), it holds rho approximate to 1/3 for any choice of the mesh ratio J. In this paper, we continue to consider F = 2nd-order DIRK, but with a new choice for G, the IMEX (implicit-explicit) Euler method, where the 'implicit' and ` explicit' computation is respectively associated with Aand dA(alpha). Compared to the widely used Backward-Euler method, this choice on the one hand increases c (this point is apparent), and interestingly on the other hand it can also make the convergence factor rho smaller: rho approximate to 1/5! Numerical results are provided to support our conclusions. (C) 2015 Elsevier Inc. Allrightsreserved.