Robustness and uncertainty of direct numerical simulation under the influence of rounding and noise

TL;DR

This study investigates how reduced numerical precision and noise affect the accuracy and robustness of direct numerical simulations of turbulence, highlighting the benefits of stochastic rounding over deterministic methods.

Contribution

It introduces an experimental methodology to assess the impact of precision and noise on DNS and compares different rounding schemes and noise effects.

Findings

Stochastic rounding impacts results less than deterministic rounding.

Precision is crucial in regions with small velocity changes.

Noise has a lesser effect than deterministic rounding in turbulence simulations.

Abstract

Numerical precision in large-scale scientific computations has become an emerging topic due to recent developments in computer hardware. Lower floating point precision offers the potential for significant performance improvements, but the uncertainty added from reducing the numerical precision is a major obstacle for it to reach prevalence in high-fidelity simulations of turbulence. In the present work, the impact of reducing the numerical precision under different rounding schemes is investigated and compared to the presence of white noise in the simulation data to obtain statistical averages of different quantities in the flow. To investigate how this impacts the simulation, an experimental methodology to assess the impact of these sources of uncertainty is proposed, in which each realization $u^i$ at time $t_i$ is perturbed, either by constraining the flow to a coarser discretization of the phase space (corresponding to low precision formats rounded with deterministic and stochastic rounding) or by perturbing the flow with white noise with a uniform distribution. The purpose of this approach is to assess the limiting factors for precision, and how robust a direct numerical simulation (DNS) is to noise and numerical precision. Our results indicate that for low-Re turbulent channel flow, stochastic rounding and noise impacts the results significantly less than deterministic rounding, indicating potential benefits of stochastic rounding over conventional round-to-nearest. We find that to capture the probability density function of the velocity change in time, the floating point precision is especially important in regions with small relative velocity changes and low turbulence intensity, but less important in regions with large velocity gradients and variations such as in the near-wall region.