Parallel solution of Partial Differential Equations on Binarized Octrees

Significant savings in computer memory and computational turnaround time can be realized through adaptive mesh refinement (AMR) in fluid dynamics simulations. Tree-based implementations of the AMR technique offer many advantages, but their computer implementation can be quite involved. A binarized octree is a recent pointerless implementation of an octree that relies solely on bitwise representations of the elements of an octree where a red-black tree is then used to insert and remove elements to the octree with a guaranteed worst-case performance of O(log N). The strict adherence to the bitwise representation of an octree also enables deep levels of mesh adaptations as there is no need to convert the bitwise representation to an integer. Furthermore, neighborhood information need not be stored as it is inherent in the bitwise representation. Here, we extend a binarized octree-based AMR technique for the parallel solution of partial differential equations on distributed memory platforms. We develop non-blocking collective communication utilities for intra-processor and inter-processor information exchange and discuss various alternatives to interpolations at the coarse-fine interface. We solve the 3D Poisson equation to verify the accuracy of the overall method.