Three-Dimensional MHD on Cubed-Sphere Grids: Parallel Solution-Adaptive Simulation Framework

An accurate, effcient and scalable cubed-sphere grid framework is described for simulation of magnetohydrodynamic (MHD) space-physics flows in domains between two concentric spheres. The unique feature of the proposed formulation compared to existing cubed-sphere codes lies in the design of a cubed-sphere framework that is based on a genuine and consistent multi-block implementation, leading to flux calculations, adaptivity, implicit solves, and parallelism that are fully transparent to the boundaries between the six grid root blocks that correspond to the six sectors of the cubed-sphere grid. Crucial elements of the proposed approach that facilitate this flexible design are: an unstructured connectivity of the six root blocks of the grid, multi-dimensional k-exact reconstruction that automatically takes into account information from neighbouring cells, and adaptive division of the six root blocks into smaller blocks of varying resolution that are all treated exactly equally for ghost cell information transfers, flux calculations, adaptivity, implicit solves and parallel distribution. The approach requires significant initial investment in developing a general and sophisticated adaptive multi-block implementation, with the added complexity of unstructured root-block connectivity, but once this infrastructure is in place, a simulation framework that is uniformly accurate and easily scalable can be developed naturally, since blocks that are adjacent to sector boundaries or sector corners are not treated specially in any way. The general design principles of the adaptive multi-block approach are described and, in particular, how they are used in the implementation of the cubed-sphere framework. The finite-volume discretization, parallelization, and implicit solves are also described. The adaptive mesh refinement (AMR) algorithm uses an upwind spatial discretization procedure in conjunction with limited linear solution reconstruction and Riemann-solver based flux functions to solve the governing equations on multi-block hexahedral mesh. A flexible block-based hierarchical data structure is used to facilitate automatic solution-directed mesh adaptation according to physics-based refinement criteria. The parallel implicit approach is based on a matrix-free inexact Newton method that solves the system of discretized nonlinear equations and a preconditioned generalized minimal residual (GMRES) method that is used at each step of the Newton algorithm. An additive Schwarz global preconditioner in conjunction with local block-fill incomplete lower-upper (BFILU) type preconditioners provides robustness and improves convergence efficiency of the iterative method. The Schwarz preconditioning and block-based data structure readily allow efficient and scalable parallel implementations of the implicit AMR approach on distributed-memory multi-processor architectures. Numerical test problems, including grid convergence studies and realistic global modelling of solar wind conditions, are discussed to demonstrate the accuracy and efficiency of the proposed solution procedure. ∗Postdoctoral Fellow, Email: [email protected], Member AIAA. †Associate Professor, Email: [email protected]. ‡PhD Candidate, Email: [email protected]. §Professor, Email: [email protected], Senior Member AIAA.