Exploiting repeated matrix block structures for more efficient CFD on modern supercomputers

TL;DR

This paper presents a new method for CFD that exploits repeated matrix block structures to transform sparse matrix-vector operations into more efficient matrix-matrix operations, significantly improving performance on supercomputers.

Contribution

It introduces a novel approach to increase arithmetic intensity in CFD by leveraging repeated matrix block structures and a mesh-refinement strategy to accelerate convergence.

Findings

Speed-ups of over 50% in test cases.

Effective transformation of SpMV to SpMM for better compute utilization.

Validation on turbulent flow, convection, and airfoil simulations.

Abstract

Computational Fluid Dynamics (CFD) simulations are often constrained by the memory-bound nature of sparse matrix-vector operations, which eventually limits performance on modern high-performance computing (HPC) systems. This work introduces a novel approach to increase arithmetic intensity in CFD by leveraging repeated matrix block structures. The method transforms the conventional sparse matrix-vector product (SpMV) into a sparse matrix-matrix product (SpMM), enabling simultaneous processing of multiple right-hand sides. This shifts the computation towards a more compute-bound regime by reusing matrix coefficients. Additionally, an inline mesh-refinement strategy is proposed: simulations initially run on a coarse mesh to establish a statistically steady flow, then refine to the target mesh. This reduces the wall-clock time to reach transition, leading to faster convergence with equivalent computational cost. The methodology is evaluated using theoretical performance bounds and validated through three test cases: a turbulent channel flow, Rayleigh-B\'{e}nard convection, and an industrial airfoil simulation. Results demonstrate substantial speed-ups - from modest improvements in basic configurations to over 50% in the mesh-refinement setup - highlighting the benefits of integrating SpMM across all CFD operators, including divergence, gradient, and Laplacian.