Parallel Domain Decomposition Methods with Mixed Order Discretization for Fully Implicit Solution of Tracer Transport Problems on the Cubed-Sphere

In this paper, a fully implicit finite volume Eulerian scheme and a corresponding scalable parallel solver are developed for some tracer transport problems on the cubed-sphere. To efficiently solve the large sparse linear system at each time step on parallel computers, we introduce a Schwarz preconditioned Krylov subspace method using two discretizations. More precisely speaking, the higher order method is used for the residual calculation and the lower order method is used for the construction of the preconditioner. The matrices from the two discretizations have similar sparsity pattern and eigenvalue distributions, but the matrix from the lower order method is a lot sparser, as a result, excellent scalability results (in total computing time and the number of iterations) are obtained. Even though Schwarz preconditioner is originally designed for elliptic problems, our experiments indicate clearly that the method scales well for this class of purely hyperbolic problems. In addition, we show numerically that the proposed method is highly scalable in terms of both strong and weak scalabilities on a supercomputer with thousands of processors.