We describe a parallel, adaptive, multiblock algorithm for explicit integration of time dependent partial differential equations on two-dimensional Cartesian grids. The grid layout we consider consists of a nested hierarchy of fixed size, non-overlapping, logically Cartesian grids stored as leaves in a quadtree. Dynamic grid refinement and parallel partitioning of the grids is done through the use of the highly scalable quadtree/octree library p4est. Because our concept is multiblock, we are able to easily solve on a variety of geometries including the cubed sphere. In this paper, we pay special attention to providing details of the parallel ghost-filling algorithm needed to ensure that both corner and edge ghost regions around each grid hold valid values. We have implemented this algorithm in the ForestClaw code using single-grid solvers from Clawpack, a software package for solving hyperbolic PDEs using finite volumes methods. We show weak and strong scalability results for scalar advection problems on two-dimensional manifold domains on 1 to 64Ki MPI processes, demonstrating neglible regridding overhead.
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