Parallel High-Order Anisotropic Block-Based Adaptive Mesh Refinement Finite-Volume Scheme

A novel parallel, high-order, anisotropic, block-based, adaptive mesh re nement (AMR), nite-volume scheme is proposed and developed herein for the numerical solution of physically complex ow problems having disparate spatial and temporal scales, and with strong anisotropic features. A block-based AMR approach is used which permits highly e cient and scalable implementations on parallel computing architectures and the use of multiblock, bodytted computational grids for the treatment of complex geometries. Rather than adopting a more usual isotropic approach to the re nement of the grid blocks, the proposed approach uses a binary hierarchical tree data structure that allows for anisotropic re nement of the grid blocks in each of the coordinate directions in an independent fashion. This allows for the more e cient and accurate treatment of narrow layers, discontinuities, and/or shocks in the solutions which occur, for example, in the thin boundaries and mixing layers of high-Reynolds-number viscous ows and in the regions of strong non-linear wave interactions of high-speed compressible ows with shocks. The anisotropic AMR approach is combined with a computationally e cient, high-order, central, essentially non-oscillatory (CENO), cell-centered, upwind, nite-volume scheme and an e cient parallel, implicit and explicit scheme for the solution of general systems of partial di erential equations governing both steady and time-varying problems. The CENO upwind scheme makes use of Riemann-solver based ux functions and a solution smoothness indicator to provide robust, accurate, and monotonic treatment of shocks and under-resolved solution content. The nite-volume scheme is applied to the solution of both a model system, the advectiondi usion equation, as well as the Euler equations governing compressible, inviscid, gaseous ows in two space dimensions. The potential of the parallel adaptive scheme for dealing with ows having disparate scales is clearly demonstrated.