Advanced algorithms for computations on block-structured adaptively refined meshes

Block-structured adaptively refined meshes are an efficient means of discretizing a domain characterized by a large spectrum of spatiotemporal scales. Further, they allow the use of simple data structures (multidimensional arrays) which considerably assist the task of using them in conjunction with sophisticated numerical algorithms. In this work, we show how such meshes may be used with high order (i.e. greater than 2(nd) order) discretization to achieve greater accuracies at significantly less computational expense, as compared to conventional second order approaches. Our study explores how these high order discretizations are coupled with high-order interpolations and filters to achieve high order convergence on such meshes. One of the side-effects of using high order discretizations is that one now obtains shallow grid hierarchies, which are easier to load balance. As a part of this work, we introduce the concept of bi-level (grid) partitioning and motivate, via an analytical model, how it holds the potential to significantly reduce load-imbalances while incurring a minimal communication cost.