On the implementation of flux limiters in algebraic frameworks

The use of flux limiters is widespread within the scientific computing community to capture shock discontinuities 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 programming 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 limiters. 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. (C) 2021 The Author(s). Published by Elsevier B.V.

Automated summary

Flux limiters are widely used in scientific computing to capture shock discontinuities and are crucial for the temporal integration of high-speed aerodynamics, multiphase flows, and hyperbolic equations. However, the emergence of new computing architectures and hybrid supercomputer systems present a challenge for portability, especially for legacy codes that need to adapt their kernels to complex parallel programming paradigms. This study proposes a new formulation of flux limiters based on algebraic data structures and kernels, which simplifies their deployment on state-of-the-art supercomputers by reducing the number of computing kernels. The proposed formulation demonstrates its effectiveness in massively parallel computing systems and is applied to a canonical advection problem.