Abstract
Parareal is a kind of time parallel numerical methods for time-dependent systems. In this paper, we consider a general linear parabolic PDE, use optimal quadratic spline collocation (QSC) method for the space discretization, and proceed with the parareal technique on the time domain. Meanwhile, deferred correction technique is also used to improve the accuracy during the iterations. In fact, the optimal QSC method is a correction of general QSC method. Along the temporal direction we embed the iterations of deferred correction into parareal to construct a hybrid method, parareal deferred correction (PDC) method. The error estimation is presented and the stability is analyzed. To save computational cost, we find out a simple way to balance the two kinds of iterations as much as possible. We also argue that the hybrid algorithm has better system efficiency and costs less running time. Numerical experiments by multicore computers are attached to exhibit the effectiveness of the hybrid algorithm.
Highlights
The parareal algorithm was firstly proposed by Lions et al [1] in 2001 as a parallel-in-time approach for solving timedependent differential equations, mainly for ordinary differential equations (ODEs) and parabolic equations [2,3,4]
As a purely parallel algorithm, the parareal algorithm has been applied to many practical problems, such as hyperbolic problems [5], fluid mechanics [6], quantum control [7, 8], and optimized control problem [9, 10]
We have proposed a kind of parareal waveform relaxation methods for ODEs [15] and parabolic PDEs [16], as well as parareal deferred correction methods for PDEs [17]
Summary
The parareal algorithm was firstly proposed by Lions et al [1] in 2001 as a parallel-in-time approach for solving timedependent differential equations, mainly for ordinary differential equations (ODEs) and parabolic equations [2,3,4]. The parareal algorithm was initially derived from multiple shooting, time multigrid, and algebraic deferred correction [18, 19] through borrowing the idea of domain decomposition on the time domain It employs two operators on two time decompositions of different step sizes with the name of coarse and fine grids to propagate the values along the time direction. A kind of iterative methods for solving ODEs, is originally proposed in [20] and has been combined with parareal in [11] It makes use of the previously computed solution on the same time subdomain. We apply the idea of correction to the problem of parabolic PDEs. In detail, we employ the optimal QSC method for the spatial variables and PDC method for the temporal variables, which will result in high accuracy approximations with low computational cost.
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