  In this paper, we propose and test a novel diagonal sweeping domain
decomposition method (DDM) with source transfer for solving the high-frequency
Helmholtz equation in $\mathbb{R}^n$. In the method the computational domain is
partitioned into overlapping checkerboard subdomains for source transfer with
the perfectly matched layer (PML) technique, then a set of diagonal sweeps over
the subdomains are specially designed to solve the system efficiently. The
method improves the additive overlapping DDM (W. Leng and L. Ju, 2019) and the
L-sweeps method (M. Taus, et al., 2019) by employing a more efficient subdomain
solving order. We show that the method achieves the exact solution of the
global PML problem with $2^n$ sweeps in the constant medium case. Although the
sweeping usually implies sequential subdomain solves, the number of sequential
steps required for each sweep in the method is only proportional to the $n$-th
root of the number of subdomains when the domain decomposition is quasi-uniform
with respect to all directions, thus it is very suitable for parallel computing
of the Helmholtz problem with multiple right-hand sides through the pipeline
processing. Extensive numerical experiments in two and three dimensions are
presented to demonstrate the effectiveness and efficiency of the proposed
method.
