On the implementation of flux limiters in algebraic frameworks

The use of flux limiters is widespread within the scientific computing community to capture shock dis- continuities and are of paramount importance for the temporal integration of high-speed aerodynamics, multiphase flows and hyperbolic equations in general. Meanwhile, the breakthrough of new computing architectures and the hybridization of supercomputer systems pose a huge portability challenge, particularly for legacy codes, since the computing subroutines that form the algorithms, the so-called kernels, must be adapted to various complex parallel program- ming paradigms. From this perspective, the development of innovative implementations relying on a minimalist set of kernels simplifies the deployment of scientific computing software on state-of-the-art supercomputers, while it requires the reformulation of algorithms, such as the aforementioned flux lim- iters. Equipped with basic algebraic topology and graph theory underlying the classical mesh concept, a new flux limiter formulation is presented based on the adoption of algebraic data structures and kernels. As a result, traditional flux limiters are cast into a stream of only two types of computing kernels: sparse matrix-vector multiplication and generalized pointwise binary operators. The newly proposed formulation eases the deployment of such a numerical technique in massively parallel, potentially hybrid, computing systems and is demonstrated for a canonical advection problem. The work of N. V. and X. Á. F. has been supported by the Government of Catalonia, FI AGAUR predoctoral grants 2019FI_B2_ 000104 and 2019FI_B2_00076. N. V., X. Á. F., J. C., A. O. and F. X. T. have been funded by the Spanish Research Agency, ANUMESOL project ENE2017-88697-R. J. C. has also been funded by Spanish Research Agency, GALIFLOW project ENE2015-70662-P. The studies of this work have been carried out using the MareNostrum 4 supercomputer of the Barcelona Supercomput- ing Center, projects IM-2019-3-0026 and IM-2020-1-0006; the TSUBAME3.0 supercomputer of the Global Scientific Information and Computing Center at Tokyo Institute of Technology; the Lomonosov-2 supercomputer of the shared research facilities of HPC computing resources at Lomonosov Moscow State University; the K-60 hybrid cluster of the collective use center of the Keldysh Institute of Applied Mathematics; the JFF cluster of the Heat and Mass Transfer Technological Center at Technical University of Cat- alonia. The authors thankfully acknowledge these institutions for the compute time and technical support.