A parallel-in-time algorithm for high-order BDF methods for diffusion and subdiffusion equations

Abstract

In this paper, we propose a parallel-in-time algorithm for approximately solving parabolic equations. In particular, we apply the k-step backward differentiation formula and then develop an iterative solver by using the waveform relaxation technique. Each resulting iteration represents a periodic-like system, which could be further solved in parallel by using the diagonalization technique. The convergence of the waveform relaxation iteration is theoretically examined by using the generating function method. The argument could be further applied to the time-fractional subdiffusion equation, whose discretization shares common properties of the standard BDF methods due to the nonlocality of the fractional differential operator. Illustrative numerical results are presented to complement the theoretical analysis.