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DOE OSTI2023

Quantification of Physical and Numerical Mixing in a Coastal Ocean Model Using Salinity Variance Budgets

Numerical mixing, the spurious mixing primarily generated by the discretization of advection, is often significant in estuarine and coastal models due to sharp, energetic fronts. In this study, we compare on- and offline estimates of numerical mixing in a submesoscale-resolving realistic simulation of the ocean state over the Texas-Louisiana continental shelf. While offline estimates of numerical mixing differ from online estimates, offline methods may be the only analysis available. This study offers insight into the differences between the on- and offline methods. We use two methods to estimate numerical mixing offline, based on salinity squared $s^2$ and volume-mean salinity variance $s^{\prime^2}$. Numerical mixing estimated from the $s^{\prime^2}$ budget is generally within 60\% of the magnitude for the online method but captures the temporal variability well. However, the $s^2$ budget compares poorly due to larger truncation errors associated with the tendency and advection terms, which can be reduced by increasing the model output frequency. We also investigate the effects of horizontal resolution on numerical mixing using a two-way nested grid. The volume-integrated numerical mixing constitutes 57\% of the bulk physical mixing -- the mixing prescribed by the turbulence closure scheme -- in the coarse model and may exceed the physical mixing by half an order of magnitude. We find that numerical mixing is reduced by 35\% on average in the nested model, less than expected based on scaling of the numerical mixing for an upwind advection scheme, likely due to new dynamical processes that emerge in the nested simulation.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Numerical Mixing Suppresses Submesoscale Baroclinic Instabilities Over Sloping Bathymetry

Abstract The impacts of spurious numerical salinity mixing on the larger‐scale flow and tracer fields are characterized using idealized simulations. The idealized model is motivated by realistic simulations of the Texas‐Louisiana shelf and features oscillatory near‐inertial wind forcing. can exceed the physical mixing from the turbulence closure in frontal zones and within the mixed layer. This suggests that simulated mixing processes in frontal zones are driven largely by . Near‐inertial alongshore wind stress amplitude is varied to identify a base case that maximizes the ratio of to in simulations with no prescribed horizontal mixing. We then test the sensitivity of the base case with three tracer advection schemes (MPDATA, U3HC4, and HSIMT) and conduct ensemble runs with perturbed bathymetry. Instability growth is evaluated using the volume‐integrated eddy kinetic energy and available potential energy . While all schemes have similar total mixing, the HSIMT simulations have over double the volume‐integrated and 20% less relative to other schemes, which suppresses the release of and reduces the by roughly 25%. This results in reduced isohaline variability and steeper isopycnals, evidence that enhanced suppresses instability growth. Differences in and between the MPDATA and U3HC4 simulations are marginal. However, the U3HC4 simulations have 25% more . Experiments with variable horizontal viscosity and diffusivity coefficients show that small amounts of prescribed horizontal mixing improve the representation of the ocean state for all advection schemes by reducing the and increasing the .

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Numerical Diffusion (Mixing) of Material in Numerical Simulations of Hydrodynamics [Slides]

It is often assumed that a material interface between two materials is spread over a few numerical cells in numerical simulations for hydrodynamics. Also, we have the impression that higher order methods introduce less numerical diffusion (mixing) of material. As we know one of the purposes of adaptive mesh refinement (AMR) is to resolve interfaces between materials, but we would like to know how effective AMR is to reduce numerical diffusion of material. We will present our investigation about numerical diffusion (mixing) of material in xRage. The result of the investigation indicates that the assumptions mentioned above are not always valid. In this talk, we will also demonstrate the effectiveness of numerical techniques to reduce numerical diffusion of material, including contact discontinuity steepening, isotropic interface steepening, max interface steepening, material interface reconstruct.

36 MATERIALS SCIENCE↗

DOE OSTI2026

Mixed-precision numerics in scientific applications: survey and perspectives

The explosive demand for artificial intelligence (AI) workloads has led to a significant increase in silicon area dedicated to lower-precision computations on recent high-performance computing hardware designs. However, mixed-precision capabilities, which can achieve performance improvements of up to 8x compared to double-precision in extreme compute-intensive workloads, remain largely untapped in most scientific applications. A growing number of efforts have shown that mixed-precision algorithmic innovations can deliver superior performance without sacrificing accuracy. These developments should prompt computational scientists to seriously consider whether their scientific modeling and simulation applications could benefit from the acceleration offered by new hardware and mixed-precision algorithms. In this survey, we (1) review progress across diverse scientific domains—fluid dynamics, weather and climate, quantum chemistry, and computational genomics—that have begun adopting mixed-precision strategies; (2) examine state-of-the-art algorithmic techniques such as iterative refinement, splitting and emulation schemes, and adaptive precision solvers; (3) assess their implications for accuracy, performance, and resource utilization; and (4) survey the emerging software ecosystem that enables mixed-precision methods at scale. We conclude with perspectives and recommendations on cross-cutting opportunities, domain-specific challenges, and the role of co-design between application scientists, numerical analysts, and computer scientists. Collectively, this survey underscores that mixed-precision numerics can reshape computational science by aligning algorithms with the evolving landscape of hardware capabilities.

Graphics processing units↗

DOE OSTI2021

Mixed Precision Numerical Linear Algebra (Final Report)

The objective of this subcontract was to identify opportunities for the use of mixed precision within iterative solvers and to explore these opportunities both theoretically and experimentally. All quarterly milestones were achieved. We summarize the achievements each quarter in sections below.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Mixed Precision Numerical Linear Algebra (Final Report)

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Mixed Precision Numerical Linear Algebra (Final Report)

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Mixed Precision Numerical Linear Algebra (Final Report)

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Improved Prediction of Cold-Air Pools in the Weather Research and Forecasting Model Using a Truly Horizontal Diffusion Scheme for Potential Temperature

The terrain-following vertical coordinate system used by many atmospheric models, including the Weather Research and Forecasting (WRF) Model, is prone to errors in regions of complex terrain. These errors stem, in part, from the calculation of horizontal gradients within the diffusion term of the momentum or scalar evolution equations. In WRF, such gradients can be calculated along coordinate surfaces, or using metric terms that help account for grid skewness. However, neither of these options ensures a truly horizontal gradient calculation, especially if a grid cell is skewed enough that the heights of the neighboring grid points used in the calculation fall outside the vertical range of the cell. In this work, an improved scheme that uses Taylor series approximations to vertically interpolate variables to the level necessary for a truly horizontal gradient calculation is implemented in WRF for the diffusion of potential temperature. The scheme is validated using an atmosphere-at-rest configuration, in which spurious flows develop only as a result of numerical errors and can thus be used as a proxy for model performance. Following validation, the method is applied to the simulation of cold-air pools (CAPs), which occur in regions of complex terrain and are characterized by strong near-surface temperature gradients. Using the truly horizontal scheme, idealized simulations demonstrate reduced numerical mixing in a quiescent CAP, and a realistic case study in the Columbia River basin shows a reduction in positive wind speed bias by up to roughly 20% compared to observations from the Second Wind Forecast Improvement Project.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

A survey of numerical linear algebra methods utilizing mixed-precision arithmetic

The efficient utilization of mixed-precision numerical linear algebra algorithms can offer attractive acceleration to scientific computing applications. Especially with the hardware integration of low-precision special-function units designed for machine learning applications, the traditional numerical algorithms community urgently needs to reconsider the floating point formats used in the distinct operations to efficiently leverage the available compute power. In this study, we provide a comprehensive survey of mixed-precision numerical linear algebra routines, including the underlying concepts, theoretical background, and experimental results for both dense and sparse linear algebra problems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Conditional Point Sampling: A Monte Carlo Method for Radiation Transport in Stochastic Media.

Current methods for stochastic media transport are either computationally expensive or, by nature, approximate. Moreover, none of the well-developed, benchmarked approximate methods can compute the variance caused by the stochastic mixing, a quantity especially important to safety calculations. Therefore, we derive and apply a new conditional probability function (CPF) for use in the recently developed stochastic media transport algorithm Conditional Point Sampling (CoPS), which 1) leverages the full intra-particle memory of CoPS to yield errorless computation of stochastic media outputs in 1D, binary, Markovian-mixed media, and 2) leverages the full inter-particle memory of CoPS and the recently developed Embedded Variance Deconvolution method to yield computation of the variance in transport outputs caused by stochastic material mixing. Numerical results demonstrate errorless stochastic media transport as compared to reference benchmark solutions with the new CPF for this class of stochastic mixing as well as the ability to compute the variance caused by the stochastic mixing via CoPS. Using previously derived, non-errorless CPFs, CoPS is further found to be more accurate than the atomic mix approximation, Chord Length Sampling (CLS), and most of memory-enhanced versions of CLS surveyed. In addition, we study the compounding behavior of CPF error as a function of cohort size (where a cohort is a group of histories that share intra-particle memory) and recommend that small cohorts be used when computing the variance in transport outputs caused by stochastic mixing.

61 RADIATION PROTECTION AND DOSIMETRY↗

DOE OSTI2022

Uncertainty-aware mixed-variable machine learning for materials design

Abstract Data-driven design shows the promise of accelerating materials discovery but is challenging due to the prohibitive cost of searching the vast design space of chemistry, structure, and synthesis methods. Bayesian optimization (BO) employs uncertainty-aware machine learning models to select promising designs to evaluate, hence reducing the cost. However, BO with mixed numerical and categorical variables, which is of particular interest in materials design, has not been well studied. In this work, we survey frequentist and Bayesian approaches to uncertainty quantification of machine learning with mixed variables. We then conduct a systematic comparative study of their performances in BO using a popular representative model from each group, the random forest-based Lolo model (frequentist) and the latent variable Gaussian process model (Bayesian). We examine the efficacy of the two models in the optimization of mathematical functions, as well as properties of structural and functional materials, where we observe performance differences as related to problem dimensionality and complexity. By investigating the machine learning models’ predictive and uncertainty estimation capabilities, we provide interpretations of the observed performance differences. Our results provide practical guidance on choosing between frequentist and Bayesian uncertainty-aware machine learning models for mixed-variable BO in materials design.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Impact of Advection Schemes on Tracer Interrelationships in Large-Eddy Simulations of Deep Convection

This study investigates the preservation of tracer interrelationships during advection in large-eddy simulations of an idealized deep convective cloud, which is particularly relevant to chemistry, aerosol, and cloud microphysics models. Employing the Cloud Model 1, advection is represented using third-, fifth-, and seventh-order weighted essentially non-oscillatory schemes. As a simplified analogy for cloud hydrometeors and aerosols, several inert passive tracers following linear and nonlinear relationships are initialized after the cloud reaches ~6-km depth. Numerical mixing in the simulated turbulent convective clouds leads to significant deviations from the initial nonlinear relationships between tracers. In these simulations, a considerable fraction of the grid points where the tracers’ nonlinear relationships are altered from advection are classified as unrealistic (e.g., ~13% for the environmental tracers on average), including errors from range-preserving unmixing and overshooting. Errors in the sum of three tracers are also relatively large, ranging between ~1% and 16% for 5% of the grid points in and near the cloud. The magnitude of unrealistic mixing and errors in the sum of three tracers generally increase with the order of accuracy of the advection scheme. These results are consistent across model grid spacings ranging from 50 to 200 m, and across three different flow realizations for each combination of grid spacing and advection scheme tested. Tests employing a previously proposed scalar normalization procedure show substantially reduced errors in the sum of three tracers with a relatively small negative impact on other tracer relationships. This analysis, therefore, suggests efficacy of the normalization procedure when applied to turbulent three-dimensional cloud simulations.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

Energy-efficient, Large-scale Molecular Dynamics Simulations via Hardware- and Algorithm-level Optimization

This work aims to develop a framework for energy-efficient computing that will enable molecular dynamics (MD) simulations of large-scale phenomena with atomic precision and simultaneously remove computational bottlenecks limiting the speed of MD simulations. We seek to implement such an approach through the development of surrogate models for the interatomic force calculation combined with the use of mixed numerical precision formats. For a model system of neutral atoms (only pairwise interactions), significant force calculation efficiency improvements were achieved, without detrimental effects on atomic structures or average energies, using single precision, by developing a surrogate model (deep neural network), and by quantizing this surrogate model. For a model system of charged atoms, the reciprocal-space calculation of electrostatic interactions was identified as the main bottleneck, and the development of a surrogate model should be pursued to achieve an estimated one-order-of-magnitude additional speedup.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

DOE OSTI2022

Influences of an entrainment–mixing parameterization on numerical simulations of cumulus and stratocumulus clouds

Abstract. Different entrainment–mixing processes can occur in clouds; however, a homogeneous mixing mechanism is often implicitly assumed in most commonly used microphysics schemes. Here, we first present a new entrainment–mixing parameterization that uses the grid mean relative humidity without requiring the relative humidity of the entrained air. Then, the parameterization is implemented in a microphysics scheme in a large eddy simulation model, and sensitivity experiments are conducted to compare the new parameterization with the default homogeneous entrainment–mixing parameterization. The results indicate that the new entrainment–mixing parameterization has a larger impact on the number concentration, volume mean radius, and cloud optical depth in the stratocumulus case than in the cumulus case. This is because inhomogeneous and homogeneous mixing mechanisms dominate in the stratocumulus and cumulus cases, respectively, which is mainly due to the larger turbulence dissipation rate in the cumulus case. Because stratocumulus clouds break up during the dissipation stage to form cumulus clouds, the effects of this new entrainment–mixing parameterization during the stratocumulus dissipation stage are between those during the stratocumulus mature stage and the cumulus case. A large aerosol concentration can enhance the effects of this new entrainment–mixing parameterization by decreasing the cloud droplet size and evaporation timescale. The results of this new entrainment–mixing parameterization with grid mean relative humidity are validated by the use of a different entrainment–mixing parameterization that uses parameterized entrained air properties. This study sheds new light on the improvement of entrainment–mixing parameterizations in models.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Numerical investigation of mixing and heat transfer in a 7.9 m JP-8 pool fire

The response of objects engulfed in, and adjacent to, large-scale pool fires is of interest in accident and safety assessments. In this study, Fuego, a low-Mach turbulent reacting flow code was used to study conjugate heat transfer in a 7.9 m diameter JP-8 pool fire. Simulations were designed to replicate past experimental measurements (Blanchat et al., 2006) of incident heat flux to three cylindrical calorimeters in and around the pool fire. Two turbulent combustion models were compared directly - the eddy dissipation concept and a more recently developed unsteady flamelet model. First and second order spatial and temporal discretization schemes were also compared to assess the performance of low-dissipation numerical operators. Heat flux predictions to the transportation size calorimeter outside the fire were within experimental uncertainties. Inside the fire, experimental measurements were higher than predicted values and may have been a consequence of soot deposition and augmented participating media radiation from soot and fuel vapor. Simulation predictions improved in cases where turbulent kinetic energy and mixing were more resolved. This work, and others referenced herein, suggest that spatial resolution on the order of 0.5–1.0 cm may be required to fully resolve fluid instabilities, vortex production between fire plumes and crosswind, soot production, and fuel-air mixing. This presents a substantial computational challenge for safety assessments of engulfed objects in fully turbulent pool fires.

Conjugate heat transfer↗

DOE OSTI2025

Convergence of Cloud Droplet Spectral Relative Dispersion During Entrainment‐Mixing Based on Particle‐Resolved Direct Numerical Simulations

Entrainment-mixing processes critically impact cloud microphysical properties, but their effects on the relative dispersion (d) of cloud droplet size distributions (CDSDs) remain elusive. A direct numerical simulation model is initialized with different CDSDs to fill the gap. These results show that d decreases for broad CDSDs and increases for narrow ones, ultimately converging to approximately 0.5 regardless of initial CDSDs during the evaporation-dominated mixing stage. The supersaturation fluctuation and the shape of CDSDs jointly influence the convergence behavior of d. Further sensitivity tests show that the initial microphysical/dynamical/thermodynamical conditions exert negligible effects on the final converged value of d but affect the convergence rate (k). The k generally increases with increasing droplet number concentration and dissipation rate, and increases with decreasing liquid water content, relative humidity of entrained air, and mixing fraction of cloudy air. A conceptual model with two timescales is proposed; k and the timescales are negatively correlated, meaning that slow mixing and/or evaporation process results in slow convergence of d. In conclusion, this finding provides an important reference for improving understanding and parameterization of d during the entrainment-mixing processes.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2020

Relationships between Immersion Freezing and Crystal Habit for Arctic Mixed-Phase Clouds—A Numerical Study

Abstract The number concentration of ice particles in Arctic mixed-phase clouds is a major controlling factor of cloud lifetime. The relationships between ice nucleation mode and ice crystal habit development are not yet constrained by observations. This study uses a habit-predicting microphysical scheme within a 3D large-eddy simulation model to evaluate the relationship between immersion freezing and ice habit in a simulated Arctic mixed-phase cloud case. Three immersion freezing parameterizations are considered: a volume-dependent freezing scheme (VF), a parameterization limited to activated droplets (C-AC), and a parameterization limited to coarse aerosol particles (C-CM). Both C-AC and C-CM are based on classical nucleation theory. The freezing rate with VF is found to be greater in downdraft regions than in updraft regions due to the downdraft having a higher number concentration of large droplets. The C-AC cases show active freezing of small droplets near cloud top, whereas in the C-CM cases, mainly the 8–32- μ m-sized droplets freeze in updraft regions near the cloud base. Because the initial crystal size is assumed to affect the axis ratio of hexagonal plates, the VF cases produce crystals with larger axis ratios, resulting in smaller mode radii than the C-AC cases. In all cases, irregular polycrystals dominate near cloud top and a band-like structure develops within the cloud, which qualitatively agrees with previous observations. In the VF and C-CM cases, unactivated large droplets arising from coarse-mode aerosol particles contributed significantly to the freezing rate, producing an important influence on crystal habit.

54 ENVIRONMENTAL SCIENCES↗