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Stencil computations are widely used in the scientific simulation domain, and their performance is critical to the overall efficiency of many large-scale numerical applications. Many optimization techniques, most of them varying strategies of tiling and parallelization, exist to systematically enhance the efficiency of stencil computations. However, the effective- ness of these optimizations vary significantly depending on the wide range of properties demonstrated by the different stencils. This paper studies several well-known optimization strategies for stencils and presents a new approach to effectively guide the composition of these optimizations, by modeling their interactions with four domain-level proper- ties of stencils: spatial dimensionality, temporal order, order of accuracy, and directional dependence. When using our prediction model to guide optimizations for five real-world stencil applications, we were able to identify optimization strategies that outperformed two highly optimized stencil libraries by an average of 2.4x. 

The special computational challenges of simulating 3-D hydrodynamics in deep stellar interiors are discussed, and numerical algorithmic responses described. Results of recent simulations carried out at scale on the NSF's Blue Waters machine at the University of Illinois are presented, with a special focus on the computational challenges they address. Prospects for future work using GPU-accelerated nodes such as those on the DoE's new Summit machine at Oak Ridge National Laboratory are described, with a focus on numerical algorithmic accommodations that we believe will be necessary.