On the implementation of flux limiters in algebraic frameworks

TL;DR

This paper introduces a novel algebraic formulation of flux limiters using minimal kernels, enhancing portability and efficiency in high-performance computing environments for shock capturing in hyperbolic PDEs.

Contribution

It reformulates flux limiters with algebraic data structures and kernels, simplifying deployment on modern supercomputers and enabling better parallelization.

Findings

New algebraic flux limiter formulation based on algebraic topology and graph theory.

Implementation using only sparse matrix-vector multiplication and binary operators.

Demonstrated effectiveness on a canonical advection problem.

Abstract

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.