Matrix-Free Finite Volume Kernels on a Dataflow Architecture

Abstract: Fast and accurate numerical simulations are crucial for designing large-scale geological carbon storage projects ensuring safe long-term CO2 containment as a climate change mitigation strategy. These simulations involve solving numerous large and complex linear systems arising from the implicit Finite Volume (FV) discretization of PDEs governing subsurface fluid flow. Compounded with highly detailed geomodels, solving linear systems is computationally and memory expensive, and accounts for the majority of the simulation time. Modern memory hierarchies are insufficient to meet the latency and bandwidth needs of large-scale numerical simulations. Therefore, exploring algorithms that can leverage alternative and balanced paradigms, such as dataflow and in-memory computing is crucial. This work introduces a matrix-free algorithm to solve FV-based linear systems using a dataflow architecture to significantly minimize memory latency and bandwidth bottlenecks. Our implementation achieves two orders of magnitude speedup compared to a GPGPU-based reference implementation, and up to 1.2 PFlops on a single dataflow device.