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

TL;DR

This work investigates how finite-precision arithmetic and rounding schemes influence direct numerical simulations of turbulence, specifically the ability to accurately capture the PDF of time-velocity changes. It formalizes the dynamic evolution as $\\hat{\\mathbf{u}}^{i+1}=\\mathbf{G}(\\mathbf{F}(\\hat{\\mathbf{u}}^i))$ and compares across FP formats from FP64 to FP8 using CPFloat, perturbing the state or observations with RTN, SR, or uniform additive noise. Key findings show that stochastic rounding and noise reduce bias and stagnation relative to deterministic rounding, with FP16 under SR/Noise reproducing mean-flow trends of higher-precision runs, though second-order statistics and PDFs are more sensitive, particularly in the channel center, while near-wall regions are more robust. The results suggest stochastic rounding as a promising alternative to RTN for DNS and motivate adaptive precision strategies or non-standard numerical formats that better capture turbulence statistics, enabling lower-cost simulations without sacrificing essential flow features.

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.

Figures (2)

• Figure 1: First- (left) and second-order (right) statistical moments of velocity $\tilde{\mathbf{U}}$ (top) and $\hat{\mathbf{U}}$ (bottom). The solid, dashed and dot-dashed lines correspond to the streamwise ($u$), wall-normal ($v$), and spanwise ($w$) velocity components, respectively. Note that only isotropic components of the Reynolds stress tensor are plotted (right).
• Figure 2: 2D PDFs of the streamwise velocity component $u$ and $\Delta u = u^i -u^{i-1}$ normalized by the mean velocity at three different wall-normal locations $y=\{0.0175, 0.14, 0.92\}$. The three different locations are shown for $\tilde{\mathbf{U}}$ (top) in FP32 (left) and FP16 (right) together with the results from $\hat{\mathbf{U}}$ (bottom) with FP16 RTN (left) and FP16 SR (right)