  In this paper, we study the parallel simulation of the magnetohydrodynamic
(MHD) dynamo in a rapidly rotating spherical shell with pseudo-vacuum magnetic
boundary conditions. A second-order finite volume scheme based on a collocated
quasi-uniform cubed-sphere grid is applied to the spatial discretization of the
MHD dynamo equations. To ensure the solenoidal condition of the magnetic field,
we adopt a widely-used approach whereby a pseudo-pressure is introduced into
the induction equation. The temporal integration is split by a second-order
approximate factorization approach, resulting in two linear algebraic systems
both solved by a preconditioned Krylov subspace iterative method. A multi-level
restricted additive Schwarz preconditioner based on domain decomposition and
multigrid method is then designed to improve the efficiency and scalability.
Accurate numerical solutions of two benchmark cases are obtained with our code,
comparable to the existing local method results. Several large-scale tests
performed on the Sunway TaihuLight supercomputer show good strong and weak
scalabilities and a noticeable improvement from the multi-level preconditioner
with up to 10368 processor cores.
