Fluid dynamics simulations with the lattice Boltzmann method (LBM) are very memory intensive. Alongside reduction in memory footprint, significant performance benefits can be achieved by using FP32 (single) precision compared to FP64 (double) precision, especially on GPUs. Here we evaluate the possibility to use even FP16 and posit16 (half) precision for storing fluid populations, while still carrying arithmetic operations in FP32. For this, we first show that the commonly occurring number range in the LBM is a lot smaller than the FP16 number range. Based on this observation, we develop customized 16-bit formats—based on a modified IEEE-754 and on a modified posit standard—that are specifically tailored to the needs of the LBM. We then carry out an in-depth characterization of LBM accuracy for six different test systems with increasing complexity: Poiseuille flow, Taylor-Green vortices, Karman vortex streets, lid-driven cavity, a microcapsule in shear flow (utilizing the immersed-boundary method), and, finally, the impact of a raindrop (based on a volume-of-fluid approach). We find that the difference in accuracy between FP64 and FP32 is negligible in almost all cases, and that for a large number of cases even 16-bit is sufficient. Finally, we provide a detailed performance analysis of all precision levels on a large number of hardware microarchitectures and show that significant speedup is achieved with mixed FP32/16-bit.

Learning content in mathematics, such as vector geometry, is still predominantly taught in an abstract manner, as the visualization and interaction of three-dimensional problems are limited with classical forms of teaching such as blackboard lessons or exercise sheets. This research article proposes the use of augmented reality (AR) in mathematics education. The proposed approach aims at easing the learning process related to vector geometry currently taught in senior mathematics classes by using intuitive visualization. The article introduces the concept of AR and presents the didactic foundations and the influence on the learning process based on an extensive literature review. Although studies see great potential in the use of AR for teaching mathematics, the method has so far hardly been used in schools. This can be mainly explained by the technological entry barrier of AR and the lack of simple, robust AR applications, in particular for vector geometry. To fill this gap, the authors developed “cleARmaths”, a developed android application for augmented reality-based teaching in vector geometry that allows widespread use. As a didactical concept, some example exercises sessions with the app are proposed, demonstrating how the app could be used in a mathematics classroom. Finally, the app was evaluated in a mathematics class and the results analyzed in a detailed study. It was found by the teacher and students to be beneficial and amusing, demonstrating the potential for AR in mathematics classes.