A second-order parareal algorithm for fractional PDEs

Abstract

We are concerned with using the parareal (parallel-in-time) algorithm for large scale ODEs system U ' ( t ) + A U ( t ) + d A α U ( t ) = F ( t ) arising frequently in semi-discretizing time-dependent PDEs with spatial fractional operators, where d 0 is a constant, α ? ( 0 , 1 ) and A is 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 Δt and large temporal mesh size ΔT. The two mesh sizes satisfy Δ T = J Δ 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 understood that the speedup of the parareal algorithm, namely E , satisfies E = O ( c log ? ( 1 / ? ) ) , where c : = T unit f / T unit g and ? 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 ? can be a satisfactory quantity. Particularly, for F = 2nd-order DIRK (diagonally implicit Runge-Kutta), it holds ? ? 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 A and d A α . 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 ? smaller: ? ? 1 5 ! Numerical results are provided to support our conclusions.