A parallelintime algorithm for highorder BDF methods for diffusion and subdiffusion equations
Abstract
In this paper, we propose a parallelintime 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 periodiclike 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 approach we established in this paper extends the existing argument of singlestep methods in Gander and Wu [Numer. Math., 143 (2019), pp. 489527] to general BDF methods up to order six. The argument could be further applied to the timefractional subdiffusion equation, whose discretization shares common properties of the standard BDF methods, because of the nonlocality of the fractional differential operator. Illustrative numerical results are presented to complement the theoretical analysis.
 Publication:

arXiv eprints
 Pub Date:
 July 2020
 arXiv:
 arXiv:2007.13125
 Bibcode:
 2020arXiv200713125W
 Keywords:

 Mathematics  Numerical Analysis;
 65Y05;
 65M15;
 65M12