Fine Grid Research Articles

Two‐Grid Finite Element Method for the Time‐Fractional Allen–Cahn Equation With the Logarithmic Potential

ABSTRACTIn this paper, we propose a two‐grid finite element method for solving the time‐fractional Allen–Cahn equation with the logarithmic potential. Firstly, with the L1 method to approximate Caputo fractional derivative, we solve the fully discrete time‐fractional Allen–Cahn equation on a coarse grid with mesh size and time step size . Then, we solve the linearized system with the nonlinear term replaced by the value of the first step on a fine grid with mesh size and the same time step size . We obtain the energy stability of the two‐grid finite element method and the optimal order of convergence of the two‐grid finite element method in the L2 norm when the mesh size satisfies . The theoretical results are confirmed by arithmetic examples, which indicate that the two‐grid finite element method can keep the same convergence rate and save the CPU time.

Methods and techniques used to produce electricity forecasts on Enedis’ distribution network at a finer grid than the HV/MV substation

Methods and techniques used to produce electricity forecasts on Enedis’ distribution network at a finer grid than the HV/MV substation

Loss of bimolecular reactions in reaction-diffusion master equations is consistent with diffusion limited reaction kinetics in the mean field limit.

We show that the resolution-dependent loss of bimolecular reactions in spatiotemporal Reaction-Diffusion Master Equations (RDMEs) is in agreement with the mean-field Collins-Kimball (C-K) theory of diffusion-limited reaction kinetics. The RDME is a spatial generalization of the chemical master equation, which enables studying stochastic reaction dynamics in spatially heterogeneous systems. It uses a regular Cartesian grid to partition space into locally well-mixed reaction compartments and treats diffusion as a jump reaction between neighboring grid cells. As the chance for reactants to be in the same grid cell decreases for smaller cell widths, the RDME loses bimolecular reactions in finer grids. We show that for a single homo-bimolecular reaction, the mesh spacing can be interpreted as the reaction radius of a well-mixed C-K rate. Then, the bimolecular reaction loss is consistent with diffusion-limited kinetics in the mean-field steady state. In this interpretation, the constant in a bimolecular reaction propensity is no longer the macroscopic reaction rate but the rate of the ballistic C-K step. For the same grid resolution, different diffusion models in RDME, such as those based on finite differences and Gaussian jumps, represent different reaction radii.

APPROXIMATION AND SMOOTHING OF A FUNCTION BASED ON GODUNOV REGULARIZATION

A new approach to function approximation is presented, based on S.K. Godunov’s ideas on the regularization of ill-conditioned systems. The proposed method allows for determining function values at nodes of a finer grid from data on a coarser grid while ensuring control over the smoothness of the resulting function. Convergence and smoothness estimates are substantiated, and results from computational experiments illustrate the effectiveness of the proposed method.

Causes of Differences in Depiction of Convective Lines in 1-km and 3-km CM1 Simulations

Abstract Previous research has shown that 3-km horizontal grid spacing simulations depicting clusters of cells often change to showing linear structures when grid spacing is refined to 1-km. This increase in linear structures at finer horizontal grid spacings may be due simply to the resolving of stronger vertical motion along the leading edge of the MCS cold pool resulting in more continuous zones of convection in higher resolution runs. However, prior work has suggested that the cold pools themselves are stronger with finer grid spacing, enhancing lift to grow linear morphologies faster. In the present study, Cloud Model 1 was used to simulate an array of MCSs with varying wind profiles and a constant thermodynamic profile (Weisman-Klemp analytic sounding) at both 1- and 3-km horizontal grid spacings and with 50 and 100 vertical levels. A line of seven randomly-spaced warm bubbles was used to initiate convection. In 1-km Δx simulations, gravity waves dominated in initiating new convection for growth into lines, and the ascent associated with them was much greater than in 3-km runs. Upscale growth into lines in 3-km Δx simulations was driven more by ascent caused by the collision of convective cold pools.

CFD Code Parallelization on GPU and the Code Portability

AbstractGoal of this paper is to develop a fully functional parallel Computational Fluid Dynamics (CFD) code that is optimized to run on a single Graphics Processing Unit (GPU). This is achieved by writing the code in FORTRAN and OpenACC (Open Accelerators), providing them with an easily portable, platform independent code. Existing CFD code is significantly modified to allow for parallel asynchronous execution. Also, due to strong recursive dependencies in Tridiagonal Matrix Algorithm (TDMA) solver, it is replaced with Jacobi, which provides fast execution in environments with large number of parallel cores. In this research a computer code for simulation of 2D flow of water through the axisymmetric channel is used as a base for development. The parallel code is executed on GPU, single, and multicore Central Processing Unit (CPU), and the execution times are compared between platforms. Even though that Jacobi solver performs worse on single core computers, compared to its Gauss‐seidel counterpart, it is used to provide a baseline for comparison. In this work, it is shown that computation on finer grids takes less time on GPU than on CPU. The computation time increase with the number of cells in grid on GPU should follow the observed linear trend until the GPUs physical limitations are reached depending on memory size and core count.

Evaluation of the Mountain Hydroclimate across the Western United States in Dynamically Downscaled Climate Models

Abstract This study evaluates the ability of 15 CMIP6 global climate models (GCMs), dynamically downscaled to a 9-km grid, to effectively simulate the observed regional hydroclimate across the complex terrain of the western United States. The evaluation focuses on orographic precipitation, surface temperature, and snow water equivalent (SWE), evaluated over a 33-yr period (1981–2014) using gridded gauge- and station-based datasets and a snowpack reanalysis product. Additional comparisons are made against two ERA5-driven climate reconstructions: one at 9-km resolution, with the same physical choices, and one at 4-km resolution. The latter better captures the terrain and orographic processes and uses different physics. Model performance is evaluated in four geographic regions, and mountains are contrasted against the surrounding plains. The evaluation is challenged by the fact that gridded observational estimates of climate parameters in complex terrain have a poorly quantified uncertainty related to measurement and/or representativeness issues. The ensemble mean of the downscaled GCMs overestimates cold-season precipitation, its orographic enhancement, and peak SWE in the mountains. Its diurnal temperature cycle and its mountain–plain temperature contrast are suppressed, compared to observations: its temperature estimates err toward the mean on both sides of both distributions. But the same applies to the identically downscaled (9 km) ERA5, indicating that these biases are due to model physics, not a misrepresentation of the climate system. The 4-km ERA5-based reconstruction has smaller biases in terms of precipitation and temperature, especially over mountains, but underestimates SWE. Model performance differs between mountains and plains, and the differences vary by region. Significance Statement In the western United States, much of the precipitation contributing to streamflow falls over mountains. In addition, the mountain snowpack seasonally stores water for consumptive use in the warm season. Amid growing concerns about changing water availability associated with the changing hydrometeorological patterns in a warming climate, it is essential to evaluate how well climate models capture precipitation and the snowpack across the headwater regions. Dynamically downscaled CMIP6 models, on average, capture the historical climate quite well, but uncertainty remains about how well they capture key climate parameters over the mountains. This uncertainty is due, in part, to the paucity of measurements in the mountains. Here, we demonstrate that dynamic downscaling to finer grid resolutions reduces this uncertainty.

Nighttime light imagery or mobile phone footprints: Which better reflects urban socio-economics at the grid level? A case study in the Pearl River Delta, China

Traditional socioeconomic censuses rely on manual statistical surveys at the administrative division level, incurring significant costs while also facing the issue of data fabrication. The lack of information at the fine-scale spatial level limits more accurate policy formulation at the local and global levels. Nighttime lights have been proven to reflect human activities and estimate socio-economic indicators. Meanwhile, with the widespread use of smart devices, mobile phone data recorded as sensor data also provide various information about human footprints. This research elucidates the revealing ability of mobile phone footprints (MOB) and nighttime lights (NTL) to estimate various socio-economic indicators at a fine grid scale, establishing them as valuable proxies for understanding complex urban patterns. A comparative analysis within the Pearl River Delta (PRD), China demonstrates MOB's superior capacity in accurately reflecting socio-economic indicators such as population density and gross domestic product (GDP) distribution, effectively mitigating the oversaturation shortcomings of NTL in reflecting socioeconomic conditions. Especially in urban built-up areas, MOB and NTL data synergistically provide a refined depiction of socio-economic conditions, with MOB elucidating urban structure and density, and NTL closely associated with the service sector's footprint. The insights of the study highlight the value of integrating MOB and NTL data to refine the accuracy of socioeconomic indicators, which could be instrumental in the creation of nuanced urban planning and policy interventions. Such data-driven approaches promise to more effectively address socioeconomic inequalities and support sustainable urban development initiatives.

Self-adaptive turbulence eddy simulation of a supersonic cavity-ramp flow

High Reynolds number supersonic wall-bounded turbulent flow is common in the aviation industry. However, accurately predicting this flow remains a long-standing challenge for turbulence modeling, particularly in separation cases. This study presents a numerical investigation of a Mach 2.92 turbulent cavity-ramp flow employing a new hybrid turbulence modeling approach, denoted as the self-adaptive turbulence eddy simulation (SATES) method. The results are compared with experimental data, as well as with outcomes from the conventional Reynolds-averaged Navier–Stokes (RANS) method and the previous delayed detached eddy simulation (DDES). The SATES model demonstrates a satisfactory prediction of time-averaged and fluctuating quantities in the free shear layer, redeveloping boundary layer, and ramp, showing better accuracy than the RANS results. Notably, the SATES results from the medium grid exhibit similarities to the DDES results from the extremely fine grid which is about 26 times compared with the SATES mesh. Furthermore, the significant features of the supersonic reattached flow, including turbulent properties, vortex structures, and unsteadiness characteristics, are analysed in detail, including the Reynolds stresses anisotropy invariant maps, transport process of vorticity, and frequency spectra. This work confirms that accurate numerical predictions of high Reynolds number supersonic separated and reattached flows are achievable using the SATES method with a relatively coarse mesh maintaining an affordable computational cost.

A consistent, volume preserving, and adaptive mesh refinement-based framework for modeling non-isothermal gas–liquid–solid flows with phase change

This work expands on our recently introduced low Mach enthalpy method (Thirumalaisamy and Bhalla 2023) for simulating the melting and solidification of a phase change material (PCM) alongside (or without) an ambient gas phase. The method captures PCM’s volume change (shrinkage or expansion) by accounting for density change-induced flows. We present several improvements to the original work. First, we introduce consistent time integration schemes for the mass, momentum, and enthalpy equations, which enhance the method stability. Demonstrating the effectiveness of this scheme, we show that a system free of external forces and heat sources can conserve its initial mass, momentum, enthalpy, and phase composition. This allows the system to transition from a non-isothermal, non-equilibrium, phase-changing state to an isothermal, equilibrium state without exhibiting unrealistic behavior. Furthermore, we show that the low Mach enthalpy method accurately simulates thermocapillary flows without introducing spurious phase changes. To reduce computational costs, we solve the governing equations on adaptively refined grids. We investigate two cell tagging/untagging criteria and find that a gradient-based approach is more effective. This approach ensures that the moving thin mushy region is always captured at fine grid levels, even when it temporarily falls within a subgrid level. We propose an analytical model to validate advanced computational fluid dynamics (CFD) codes used to simulate metal manufacturing processes (welding, 3D printing). These processes involve a heat source (like a laser) melting metal or its alloy in the presence of an ambient (inert) gas. Traditionally, studies relied on artificially manipulating material properties to match complex experiments for validation purposes. Leveraging the analytical solution to the Stefan problem with a density jump, this model offers a straightforward approach to validating multiphysics simulations involving heat sources and phase change phenomena in three-phase flows. Lastly, we demonstrate the practical utility of the method in modeling porosity defects (gas bubble trapping) during metal solidification. A field extension technique is used to accurately apply surface tension forces in a three-phase flow situation. This is where part of the bubble surface is trapped within the (moving) solidification front.

Optimization of heat transfer in a double lid-driven cavity with isoperimetric heated blocks using GFEM

This article is concerned with the examination of flow dynamics and heat transfer characteristics in a 1:4 double lid driven cavity in presence of isoperimetric heated blocks of various shapes. The focus is to identify the optimal shape that enhances the heat transfer in a tall cavity. The parametric settings are chosen in such a way that all the convection regimes including natural, forced and mixed convection could be generated. This cavity has lids positioned at the top and bottom, moving in opposite directions along the x-axis. The physical system is represented as a set of coupled partial differential equations incorporating the rheological properties of the power-law fluids (PL). The governing equations in conjunction with various non-dimensional physical parameters are simulated via Galerkin’s Finite Element Method (GFEM) on a very fine hybrid grid. The study includes the computation of the Kinetic Energy and Average Nusselt number to determine the optimal shape. It is concluded that the circular block is superior to the other two in terms of heat transmission efficiency.

Domain sampling methods for an inverse boundary value problem of the heat equation

Abstract We investigate the reconstruction of an unknown cavity inside a heat conductor with only single boundary measurement, which is a typical non-destructive testing of defects in material science using thermography. Mathematically, it can be formulated as an inverse boundary value problem for the heat equation. Two domain sampling methods, i.e. the range test (RT) and no-response test (NRT), are developed to realize the reconstruction. We first justify the convergence of the RT, which is based on the analytical extension property of solutions to the heat equation. Then we prove the duality of the RT and NRT by looking at the relation between their indicator functions. Consequently, the unknown cavity can be reconstructed by using either the RT or NRT if the solution associated to the single measurement cannot be analytically extended across its boundary. Next, we design two efficient algorithms for the RT and NRT to reconstruct the unknown cavity numerically. Each algorithm consists of two steps. The first step is to determine an approximation of the unknown cavity by implementing the sampling methods in a coarse grid, while the second step is to improve the reconstruction by using a fine grid in the approximate domain. The numerical results suggest that both algorithms can generate some rough information on the cavity with only one boundary measurement. Finally, by using this information as an initial guess for Newton’s method, we improve the accuracy of the mentioned reconstruction results.

Modeling Hydraulic Fracture Entering Stress Barrier: Theory and Practical Recommendations

Numerical modeling of hydraulic fracturing is complicated when a fracture reaches a stress barrier. For high barriers, it may require changing the computational scheme. In view of the strong influence of stress barriers on the final footprint and opening of a hydraulic fracture, for decades, their modeling has been the subject of special investigations. Actually, classical models of propagation within a pay layer with impenetrable boundaries referred to the case of extremely high-stress contrast in neighbor layers. Further improvements tended to account for the fracture growth in these layers by including stress contrasts as input parameters of a model. This tendency resulted in the suggestion and successive enhancements of pseudo-three-dimensional models. All of them have used stress intensity factors (SIFs) to characterize the combined resistance caused by stress contrasts, material strength, and fluid viscosity. Specifically, the SIFs served to formulate the conditions that control the front penetration into a neighbor layer. This key concept presents the background of our research. Despite examples of modeling propagation through barriers, there is no general theory clarifying when and why conventional schemes may become inefficient and how to overcome computational difficulties. This paper presents the theory and practical recommendations following it. We start with the definition of the barrier intensity, which exposes that the barrier strength may change from zero for contrast-free propagation to infinity for channelized propagation. The analysis reveals two types of computational difficulties caused by spatial discretization as follows: (i) general, arising for fine grids and aggravated by a barrier, and (ii) specific, caused entirely by a strong barrier. The asymptotic approach, which avoids spatial discretization, is suggested. It is illustrated by solving benchmark problems for barriers of arbitrary intensity. The analysis distinguishes three typical stages of the fracture penetration into a barrier and provides theoretical values of the Nolte–Smith slope parameter and arrest time as functions of the barrier intensity. Special analysis establishes the accuracy and bounds of the asymptotic approach. It appears that the approach provides physically significant and accurate results for fracture penetration into high, intermediate, and even weak stress barriers. On this basis, simple practical recommendations are given for modeling hydraulic fractures in rocks with stress barriers. The recommendations may be promptly implemented in any program using spatial discretization to model fracture propagation.

A dynamic linearized wall model for turbulent flow simulation: Towards grid convergence in wall-modeled simulations

This paper addresses the common issue of (no-)grid convergence in wall-modeled numerical simulations and proposes a dynamic linearization technique applied to the Spalart-Allmaras wall model to achieve a proper behavior on fine grids and low-friction areas. A theoretical analysis of the numerical error committed on the shear stress balance close to the walls is performed. It shows that the error is due to the inappropriate imposition of too steep wall-normal velocity gradients that cannot be properly accounted for on the typical grids used for wall-modeled simulations. Based on this error quantification, a dedicated wall model linearization technique is proposed, following the approach developed by Tamaki, Harada and Imamura in 2017. In the proposed modified linearization method, the linearization distance is modified and adjusted dynamically. This is done according to the theoretical shear stress error estimate, in order to keep the numerical error below a user-defined threshold. The method is applied to well-referenced test cases of increasing complexity from the Turbulence Modeling Resource. Overall, the proposed wall model clearly exhibits appropriate grid convergence properties and is also able to predict accurately non-equilibrium boundary layers and flow separation using proper grid refinement.

Dynamic adaptive and fully unstructured tetrahedral gridding: Application to CO2 sequestration with consideration of full fluid compressibility

Numerical simulations of complex subsurface flow problems and geomechanics will advance enormously by dynamic adaptive gridding. Only a small part of a large domain where sharp changes occur may require fine gridding. In this work, we introduce a methodology to carry out dynamic adaptive gridding for large-scale flow problems. The algorithm allows 2D and 3D unstructured gridding with consideration of full fluid compressibility in single-phase, and two-phase compositional flow. We divide a triangular element into four and a tetrahedron element into eight which creates hanging nodes at each level of refinement. The handing nodes are eliminated by splitting extra elements. As a result, a transition region exists between the fine and coarse grid regions of the domain. We have applied the method to CO2 sequestration in subsurface aquifers. The conditions are selected such that gravity fingers develop from density increase by the dissolution of CO2 in the aqueous phase. The selected examples include large domains where neither systematic studies nor dynamic adaptive gridding have been reported in the past. Results from comparison with uniform gridding reveal a speedup of up to three orders of magnitude in 3D.

Efficient aerodynamic shape optimization using transfer learning based multi-fidelity deep neural network

Computational efficiency and precision pose a classic contradiction in aerodynamic shape optimization. To address this challenge, this study introduces an effective optimization framework based on multi-fidelity fully connected neural network (MFFCN). The framework utilizes transfer learning (TL) to train a multi-fidelity surrogate model that establishes direct mappings between geometric configuration parameters and aerodynamic performance by adaptively capturing linear or nonlinear relationships concealed between high-fidelity (HF) and low-fidelity (LF) information. The HF and LF data are derived from fine and coarse grids, respectively, evaluated using the same computational fluid dynamics (CFD) model. The MFFCN-TL framework is applied to optimize the National Advisory Committee for Aeronautics 0012 (NACA0012) airfoil (12 design variables) and the Office National d'Études et de Recherches Aérospatiales M6 (ONERA M6) wing (50 design variables). Simulation results demonstrate that the NACA0012 airfoil achieves a 69.47% enhancement in lift–drag ratio in 1.069 s, compared to a 12.76% gain over 24.8 h in single-fidelity CFD-based optimization. The ONERA M6 wing achieves a 24.66% reduction in drag coefficient in 694 ms compared to 18.37% over 237.3 h in the CFD model. Statistical results show that the MFFCN-TL framework can reduce optimization cost by more than 90% compared to the single-fidelity CFD-based model. These findings suggest that the MFFCN-TL framework significantly enhances optimization efficiency and provides superior feasible solutions over single-fidelity methods.

A rapid method for composition tracking in hydrogen-blended pipeline using Fourier neural operator

Blending hydrogen into natural gas for transportation is a crucial approach for achieving the widespread utilization of hydrogen. Tracking the concentration of the hydrogen within the pipeline is important for monitoring gas quality and managing pipeline operations. This study develops a rapid computational model to predict the hydrogen and natural gas concentrations within the pipeline during transportation based on the Fourier Neural Operator (FNO), an operator neural network capable of learning the differential operator in the partial differential equation. In the proposed model, the numerical method is employed to generate datasets, with the spline interpolation used to enhance data smoothness. The initial and boundary conditions are taken as the inputs to accommodate varying transportation scenarios. Comparison results indicate that the proposed model can notably reduce the time needed to predict the hydrogen and natural gas concentrations while maintaining prediction accuracy. The accuracy of the proposed model is validated by comparing its calculated results with the analytical solution and the concentrations of hydrogen and natural gas within the pipeline under two transportation scenarios, with relative errors of 0.49%, 0.31%, and 0.45%, respectively. Notably, the trained model demonstrates strong grid invariance, a type of model generalization. Trained on data generated from a coarse grid of 101 × 41 spatial-temporal resolution, the proposed model can accurately predict results on a fine grid of 401 × 81 spatial-temporal resolution with a relative error of only 0.38%. Regarding the prediction efficiency, the proposed model achieves an average 17.7-fold speedup compared to the numerical method. The positive results indicate that the proposed model can serve as a rapid and accurate solver for the composition transport equation.

Normalizing Basis Functions: Approximate Stationary Models for Large Spatial Data

ABSTRACTIn geostatistics, traditional spatial models often rely on the Gaussian process (GP) to fit stationary covariances to data. It is well known that this approach becomes computationally infeasible when dealing with large data volumes, necessitating the use of approximate methods. A powerful class of methods approximate the GP as a sum of basis functions with random coefficients. Although this technique offers computational efficiency, it does not inherently guarantee a stationary covariance. To mitigate this issue, the basis functions can be ‘normalized’ to maintain a constant marginal variance, avoiding unwanted artefacts and edge effects. This allows for the fitting of nearly stationary models to large, potentially non‐stationary datasets, providing a rigorous base to extend to more complex problems. Unfortunately, the process of normalizing these basis functions is computationally demanding. To address this, we introduce two fast and accurate algorithms to the normalization step, allowing for efficient prediction on fine grids. The practical value of these algorithms is showcased in the context of a spatial analysis on a large dataset, where significant computational speedups are achieved. While implementation and testing are done specifically within the LatticeKrig framework, these algorithms can be adapted to other basis function methods operating on regular grids.

Fast interactive simulations of cardiac electrical activity in anatomically accurate heart structures by compressing sparse uniform cartesian grids

Background and Objective:Numerical simulations are valuable tools for studying cardiac arrhythmias. Not only do they complement experimental studies, but there is also an increasing expectation for their use in clinical applications to guide patient-specific procedures. However, numerical studies that solve the reaction–diffusion equations describing cardiac electrical activity remain challenging to set up, are time-consuming, and in many cases, are prohibitively computationally expensive for long studies. The computational cost of cardiac simulations of complex models on anatomically accurate structures necessitates parallel computing. Graphics processing units (GPUs), which have thousands of cores, have been introduced as a viable technology for carrying out fast cardiac simulations, sometimes including real-time interactivity. Our main objective is to increase the performance and accuracy of such GPU implementations while conserving computational resources. Methods:In this work, we present a compression algorithm that can be used to conserve GPU memory and improve efficiency by managing the sparsity that is inherent in using Cartesian grids to represent cardiac structures directly obtained from high-resolution MRI and mCT scans. Furthermore, we present a discretization scheme that includes the cross-diagonal terms in the computational cell to increase numerical accuracy, which is especially important for simulating thin tissue sections without the need for costly mesh refinement. Results:Interactive WebGL simulations of atrial/ventricular structures (on PCs, laptops, tablets, and phones) demonstrate the algorithm’s ability to reduce memory demand by an order of magnitude and achieve calculations up to 20x faster. We further showcase its superiority in slender tissues and validate results against experiments performed in live explanted human hearts. Conclusions:In this work, we present a compression algorithm that accelerates electrical activity simulations on realistic anatomies by an order of magnitude (up to 20x), thereby allowing the use of finer grid resolutions while conserving GPU memory. Additionally, improved accuracy is achieved through cross-diagonal terms, which are essential for thin tissues, often found in heart structures such as pectinate muscles and trabeculae, as well as Purkinje fibers. Our method enables interactive simulations with even interactive domain boundary manipulation (unlike finite element/volume methods). Finally, agreement with experiments and ease of mesh import into WebGL paves the way for virtual cohorts and digital twins, aiding arrhythmia analysis and personalized therapies.

ZFP: A compressed array representation for numerical computations

HPC trends favor algorithms and implementations that reduce data motion relative to FLOPS. We investigate the use of lossy compressed data arrays in place of traditional IEEE floating point arrays to store the primary data of calculations. Simulation is fundamentally an exercise in controlled approximation, and error introduced by finite-precision arithmetic (or lossy compression) is just one of several sources of error that need to be managed to ensure sufficient accuracy in a computed result. We describe ZFP, a compressed numerical format designed for in-memory storage of multidimensional arrays, and summarize theoretical results that demonstrate that the error of repeated lossy compression can be bounded and controlled. Furthermore, we establish a relationship between grid resolution and compression-induced errors and show that, contrary to conventional floating point, ZFP reduces finite-difference errors with finer grids. We present example calculations that demonstrate data reduction by 4x or more with negligible impact on solution accuracy. Our results further demonstrate several orders-of-magnitude increase in accuracy using ZFP over IEEE floating point and Posits for the same storage budget.