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DOE OSTI2025

On the Effectiveness of Neural Operators at Zero-Shot Weather Downscaling [SWR-25-20]

Code repository for the experiments performed in the paper: On the Effectiveness of Neural Operators at Zero-Shot Weather Downscaling (https://doi.org/10.1017/eds.2025.11) Overall, our work investigates the zero-shot downscaling potential of neural operators. To summarize, our contributions are: 1. We provide a comparative analysis based on two challenging weather downscaling problems, between various neural operator and non-neural-operator methods with large upsampling factors (e.g., 8x and 15x) and fine grid resolutions (e.g., 2 km × 2 km wind speed). 2. We examine whether neural operator layers provide unique advantages when testing downscaling models on upsampling factors higher than those seen during training, i.e., zero-shot downscaling. Our results instead show the surprising success of an approach that combines a powerful transformer-based model with a parameter-free interpolation step at zero-shot weather downscaling. 3. We find that this Swin-Transformer-based approach mostly outperforms all neural operator models in terms of average error metrics, whereas an enhanced super-resolution generative adversarial network (ESRGAN)-based approach is better than most models in capturing the physics of the system, and suggests their use in future work as strong baselines. However, these approaches still do not capture variations at smaller spatial scales well, including the physical characteristics of turbulence in the HR data. This suggests a potential for improvement in transformer or GAN-based methods and neural-operator-based methods for zero-shot weather downscaling.

Sinha, Saumya [National Renewable Energy Laborator↗

DOE OSTI2020

Laminar-to-turbulent flame transition and cycle-to-cycle variations in large eddy simulation of spark-ignition engines

This paper investigates the effect of laminar-to-turbulent flame transition modeling on the prediction of cycle-to-cycle variations (CCVs) in large eddy simulation (LES) of spark-ignition (SI) engines. A laminar-to-turbulent flame transition model that describes the non-equilibrium sub-filter flame speed evolution during an early stage of flame kernel growth is developed. In the present model, the flame transition is characterized by the flame kernel size at which the flame transition ends, defined here as the flame transition scale. The proposed model captures the effects that variations in a turbulent flow field have on the evolution of early-stage burning rates, through variations in the flame transition scale. The proposed flame transition model is combined with the front propagation formulation (FPF) method and a spark-ignition model to predict CCVs in a gasoline direct injection SI engine. It is found that multi-cycle LES with the proposed flame transition model reproduces experimentally-observed CCVs satisfactorily. When the transition model is not considered or when variations in the transition process are neglected, CCVs are significantly under-predicted for the case considered here. These results indicate the importance of modeling the laminar-to-turbulent flame transition and the effect of turbulence on the transition process, when predicting CCVs, under certain engine conditions. The LES results are also used to analyze sources for variations in the flame transition. It is found, for the present engine case, that the most important source is the cycle-to-cycle variation in the turbulence dissipation rate, which is used to measure the strength of turbulence in the proposed model, near a spark plug. The large-scale velocity field and the variations of the laminar flame speed due to the mixture composition and thermal stratification are also found to be important factors to contribute to the variations in the flame transition.

42 ENGINEERING↗

DOE OSTI2020

Interface capturing simulations of bubble population effects in PWR subchannels

As the computational power of high-performance computing (HPC) facilities grows, so too does the feasibility of using first principle based simulation to study turbulent two-phase flows within complex pressurized water reactor (PWR) geometries. Direct numerical simulation (DNS), integrated with an interface capturing method, allows for the collection of high-fidelity numerical data using advanced analysis techniques. The research presented here employs the massively parallel, finite-element based, unstructured mesh code, PHASTA, to simulate a set of two-phase bubbly flows through PWR subchannel geometries including auxiliary structures (spacer grids and mixing vanes). The main objective of the presented work is to analyze bubble dynamics and turbulence interactions at varying bubble concentrations to support the development of advanced two-phase flow closure models. Turbulent two-phase flows in PWR subchannels were simulated at hydraulic Reynolds numbers of 81,000 with bubble concentrations of 3%–15% by gas volume fraction (768–3928 resolved bubbles, respectively) and compared against a 1% void fraction case (262 bubbles) that had been previously simulated. The finite element mesh utilized for the study at higher bubble concentrations was composed of 1.55 billion elements, compared to the previous study which employed 1.11 billion elements, ensuring all turbulence scales and individual bubbles within the flow are fully resolved. For each case, the resolved initial bubble size was 0.65 mm in diameter (resolved with 25 grid points across the diameter). The simulations were analyzed to find flow features such as the mean velocity profile, bubble relative velocity and the effect of the bubbles on the turbulent conditions.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2025

Applying a Compact Porous Media Model to Numerically Derive Resistance Coefficients for Lattice Structures

Additive Manufacturing allows for exploring various geometries to achieve specific engineering criteria. Lattices are one geometry with unique properties, including being periodically repeating structures which allow flow through them to be represented as a porous media according to Darcy-Forchheimer equations. These equation’s coefficients are generally experimentally derived, but this work demonstrates the ability to numerically derive them with CFD. Simulations were performed using three-dimensional stead state Reynolds-averaged Navier-Stokes with a k-ω Shear Stress Transport turbulence model using Ansys Fluent. Three lattice geometries were investigated and drag coefficients were derived. The method was validated against externally published data for similar geometries demonstrating strong agreement, and grid convergence for all simulations was calculated with a Grid Convergence Index method. Wall roughness is demonstrated to have a non-negligible impact on results and roughness values are considered for the primary focus Octahedral geometry where both smooth wall and rough wall coefficients were derived. The porosity coefficients for the Octahedral geometry at 1.0 [m/s] were found to be 2.89×10 6 and 2.90×10 6 [1/(Pa*m*s)] for the permeability coefficients, 6.37×10 1 and 5.44×10 1 [m 2 /kg] for the inertial resistance coefficients, and with a max pressure drop of 5116.7 [Pa] and 4429.5 [Pa] for the smooth walls and rough walls, respectively. The derived numerical method enables rapid exploration and optimization of new lattice designs for diverse engineering applications.

42 ENGINEERING↗

DOE OSTI2022

The coexistence of the streaming instability and the vertical shear instability in protoplanetary disks

The streaming instability is a promising mechanism to induce the formation of planetesimals. Nonetheless, this process has been found in previous studies to require either a dust-to-gas surface density ratio or a dust size that is enhanced compared to observed values. Employing two-dimensional global simulations of protoplanetary disks, we show that the vertical shear instability and the streaming instability in concert can cause dust concentration that is sufficient for planetesimal formation for lower surface density ratios and smaller dust sizes than the streaming instability in isolation, and in particular under conditions that are consistent with observational constraints. This is because dust overdensities forming in pressure bumps induced by the vertical shear instability act as seeds for the streaming instability and are enhanced by it. While our two-dimensional model does not include self-gravity, we find that strong dust clumping and the formation (and dissolution) of gravitationally unstable overdensities can be robustly inferred from the evolution of the maximum or the mean dust-to-gas volume density ratio. The vertical shear instability puffs up the dust layer to an average mid-plane dust-to-gas density ratio that is significantly below unity. We therefore find that reaching a mid-plane density ratio of one is not necessary to trigger planetesimal formation via the streaming instability when it acts in unison with the vertical shear instability.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2021

Ultrasonic oscillatory two-phase flow in microchannels

Experimental and numerical investigations are performed to provide an assessment of the transport behavior of an ultrasonic oscillatory two-phase flow in a microchannel. The work is inspired by the flow observed in an innovative ultrasonic fabric drying device using a piezoelectric bimorph transducer with microchannels, where a water-air two-phase flow is transported by harmonically oscillating microchannels. The flow exhibits highly unsteady behavior as the water and air interact with each other during the vibration cycles, making it significantly different from the well-studied steady flow in microchannels. Here, the computational fluid dynamics (CFD) modeling is realized by combing the turbulence Reynolds-averaged Navier-Stokes (RANS) k – ω model with the phase-field method to resolve the dynamics of the two-phase flow. The numerical results are qualitatively validated by the experiment. Through parametric studies, we specifically examined the effects of vibration conditions (i.e., frequency and amplitude), microchannel taper angle, and wall surface contact angle (i.e., wettability) on the flow rate through the microchannel. The results will advance the potential applications where oscillatory or general unsteady microchannel two-phase flows may be present.

42 ENGINEERING↗

DOE OSTI2023

Analysis techniques for blob properties from gas puff imaging data

Filamentary structures, also known as blobs, are a prominent feature of turbulence and transport at the edge of magnetically confined plasmas. They cause cross-field particle and energy transport and are, therefore, of interest in tokamak physics and, more generally, nuclear fusion research. Several experimental techniques have been developed to study their properties. Among these, measurements are routinely performed with stationary probes, passive imaging, and, in more recent years, Gas Puff Imaging (GPI). In this work, we present different analysis techniques developed and used on 2D data from the suite of GPI diagnostics in the Tokamak à Configuration Variable, featuring different temporal and spatial resolutions. Although specifically developed to be used on GPI data, these techniques can be employed to analyze 2D turbulence data presenting intermittent, coherent structures. We focus on size, velocity, and appearance frequency evaluation with, among other methods, conditional averaging sampling, individual structure tracking, and a recently developed machine learning algorithm. We describe in detail the implementation of these techniques, compare them against each other, and comment on the scenarios to which these techniques are best applied and on the requirements that the data must fulfill in order to yield meaningful results.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2020

Special Issue: To Commemorate the Legacy of Malcolm J. Andrews—Scientist, Educator, and Leader

This special issue celebrates the scientific legacy of Dr. Malcolm J. Andrews, former Editor of the ASME Journal of Fluids Engineering (2010–2018). Malcolm devoted his career to the development and applications of computational, experimental, and analytical tools to investigate and solve diverse, complex fluid dynamics, and heat transfer problems. Malcolm’s most notable research accomplishments include: (i) contributions to innovative experiments, simulations, and theory to understand buoyancydriven turbulent mixing due to Rayleigh–Taylor instability; (ii) development of the multiphase particle-in-cell method for understanding fragmentation in spray jets, coal-water slurry, and other dense particulate flows; (iii) mixing in nano- and biomaterials in applications ranging from protein glues to carbon nanotube composites; (iv) determining dispersion in proteins to improve therapeutic drug delivery via micro-encapsulation; (v) innovative methods for storing information using nanopinned dipole magnetic vortices in superconducting materials; and (vi) modeling and design modifications for shell-and-tube heat exchangers, as well as heat transfer in rotating systems such as turbine blades. The results of his research have been published in a long series of trend-setting journal and conference papers, book chapters, and patents.

42 ENGINEERING↗

DOE OSTI2020

Development and Validation of SAM Multi-dimensional Flow Model for Thermal Mixing and Stratification Modeling

Thermal mixing and stratification in large pools or enclosures are very important phenomena that are critical to nuclear reactor safety. Because of the wide ranges of time and length scales associated with such phenomena, accurate modeling and simulation of thermal mixing and stratification remain as the key unresolved, challenging problems for reactor transient analyses. In traditional system analysis codes, simplified zero-dimensional (0-D) models are widely used for their high numerical efficiency, but they generally suffer from very limited prediction accuracies or range of applicability. Like traditional system analysis codes, the current version of the SAM code has implemented such simplified 0-D and one-dimensional mixing models. On the other hand, high-resolution Computational Fluid Dynamics (CFD) tools are often used to model complex thermal mixing and stratification phenomena. They are, however, generally numerically expensive, and they require large amounts of computational resources. It is therefore desirable to implement advanced and efficient thermal mixing and stratification modeling capabilities embedded in a system analysis code. This approach will improve the accuracy of reactor safety analyses when thermal mixing and stratification are involved, and also avoid using the large computational resources needed for high-resolution CFD analysis. Currently, with the support of the U.S. DOE Office of Nuclear Energy’s Nuclear Energy Advanced Modeling and Simulation program, an effort has been launched to develop and implement a multi-dimensional flow model in the system analysis code SAM, and demonstrate its applications to model thermal mixing and stratification phenomena in large enclosures. The main outcomes of this research and development activity are summarized in this report, which presents an attempt to include a built-in advanced multi-dimensional flow model in a system analysis code with the focus on overcoming the simulation challenges of thermal mixing and stratification phenomena. In this report, we start with the introduction of existing SAM code capabilities to simulate thermal mixing and stratification phenomena, which is followed by a short summary of the multi-dimensional model implemented in the SAM code, including both the physical model and the Finite Element Method code implementation. In this study, two options were implemented in the SAM code to model turbulent flows: a relatively simple built-in turbulence model and an interface to accept externally computed turbulent viscosities (e.g., from a high- resolution CFD simulation). Code validation studies on this newly added capability were then carried out to compare SAM simulation results with experimental data from the SUPERCAVNA facility, which was designed to study the complex flow recirculation and thermal stratification phenomena relevant to sodium fast reactor designs. In this study, one transient and two steady-state test cases were used for code validation. Different approaches have been used to model the complex turbulence flow fields in the SUPERCAVNA facility. A highly simplified zero-equation turbulence model was first used, but it was determined that it is too simple to capture the complex turbulence flow fields in these test cases. Subsequently, the code validation continued with the use of turbulent viscosity data from high-resolution STAR-CCM+ CFD simulations to improve the accuracy of the results. Using this approach, the SAM simulation results showed very good agreement with both the SUPERCAVNA experimental data and STAR-CCM+ simulation results. In this report, we demonstrate the development, implementation, and successful validation of a multi-dimensional flow model in the SAM code, which aims to improve the simulation accuracy for complex thermal mixing and stratification phenomena. Lessons have also been learned, including that in cases where the flow fields are not well predicted by the zero-equation model, the iinput of turbulent viscosities from an external source can enhance the overall predictive capabilities needed to accurately capture complex thermal-hydraulic phenomena. Therefore, future research will be needed to further improve the code’s capabilities, such as by developing a more efficient and robust approach to capture the turbulence effects in the SAM code.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2020

Analysis of combustion acoustic phenomena in compression–ignition engines using large eddy simulation

As computational capabilities continue to grow, exploring the limits of computational fluid dynamics to capture complex and elusive phenomena, which are otherwise difficult to study by experimental techniques, is one of the main targets for the research community. This paper presents a detailed analysis of the physical processes that lead to combustion noise emissions in internal combustion engines. In particular, diesel combustion in a compression-ignition (CI) engine is studied in order to understand the singular behavior of the in-cylinder flow field responsible for the acoustic emissions. The main objective is, therefore, to improve the understanding of the phenomena involved in CI engine noise using large eddy simulations. Several visualization methods are employed to investigate the connection between combustion behavior and its effects on the pressure field. In addition, proper orthogonal decomposition is used to analyze the modal energy distribution among all the acoustic modes. The results show that the acoustic signature is fundamentally conditioned by the intensity of the premixed combustion rather than by the pressure oscillations generated by turbulent fluctuations in the flame surface established during the diffusion stage.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

Toward Exascale: Overview of Large Eddy Simulations and Direct Numerical Simulations of Nuclear Reactor Flows with the Spectral Element Method in Nek5000

At the beginning of the last decade, Petascale supercomputers (i.e., computers capable of more than 1 petaFLOP) emerged. Now, at the dawn of exascale supercomputing, we provide a review of recent landmark simulations of portions of reactor components with turbulence-resolving techniques that this computational power has made possible. In fact, these simulations have provided invaluable insight into flow dynamics, which is difficult or often impossible to obtain with experiments alone. We focus on simulations performed with the spectral element method, as this method has emerged as a powerful tool to deliver massively parallel calculations at high fidelity by using large eddy simulation or direct numerical simulation. We also limit this paper to constant-property incompressible flow of a Newtonian fluid in the absence of other body or external forces, although the method is by no means limited to this class of flows. We briefly review the fundamentals of the method and the reasons it is compelling for the simulation of nuclear engineering flows. We review in detail a series of Petascale simulations, including the simulations of helical coil steam generators, fuel assemblies, and pebble beds. Even with Petascale computing, however, limitations for nuclear modeling and simulation tools remain. In particular, the size and scope of turbulence-resolving simulations are still limited by computing power and resolution requirements, which scale with the Reynolds number. In the final part of this paper, we discuss the future of the field, including recent advancements in emerging architectures such as GPUbased supercomputers, which are expected to power the next generation of high-performance computers.

computational fluid dynamics↗

DOE OSTI2022

Adaptive clipping‐and‐redistribution algorithms for bounded and conservative high‐order interpolations applied to discontinuous and reactive flows

Abstract A new adaptive clipping‐and‐redistribution method is presented which provides bounds‐preservation for multidimensional interpolation in the context of high‐order finite‐volume discretizations with adaptive mesh refinement (AMR). The underlying finite‐volume method (FVM) for the computational fluid dynamics applications is fourth‐order accurate for smooth solutions and utilizes AMR for computational efficiency in solving multiscale problems involving turbulence and combustion. High‐order interpolation between different AMR levels is required. However, this operation often leads to numerical issues because combustion species must have physical bounds preserved. The present study overcomes two major challenges in the development of the high‐order interpolation method. First, the method needs to be bound‐preserving near extrema or discontinuities to prevent the emergence of unphysical oscillations while maintaining fourth‐order accuracy in smooth flows. Second, the method needs to satisfy the conservation requirement in multiple dimensions, particularly in the context of curvilinear coordinate transformations. Additionally, the method is designed to be localized and computationally inexpensive. The new interpolation scheme is demonstrated by solving reacting flows, which are extremely sensitive to unphysical overshoots in conserved quantities. The test problems are shock‐induced ‐ combustion and a ‐air flame in a practical bluff‐body combustor. Results show the method prevents new extrema near discontinuities while maintaining high‐order accuracy in smooth regions. In particular, the method is extremely beneficial for combustion with stiff chemistry. With the proposed new method, even if flame fronts cross AMR interfaces or new grids are created in the vicinity of the flame, solution stability is retained.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Internship Final Report on the unsupervised learning sensor fusion (ULSF) approach

This paper describes a summer internship project undertaken at Sandia National Labs (SNL), both current status and future work. The project was to explore various machine learning approaches for use on turbulent flow data. Specifically, unsupervised classification of turbulent flow data was explored. First, the usage of models in this field is discussed, and several issues in the common usage of the models are identified. Solutions to these issues are then proposed, in the form of a Bayesian filtering approach which probabilistically incorporates multiple sources of data to improve confidence in a result. Several types of sensors are suggested for this method, the incorporation of which range from semi-supervised learning approaches to fully unsupervised. These approaches are then tested on several turbulent flow cases.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Instability and turbulent relaxation in a stochastic magnetic field

An analysis of instability dynamics in a stochastic magnetic field is presented for the tractable case of the resistive interchange. Externally prescribed static magnetic perturbations convert the eigenmode problem to a stochastic differential equation, which is solved by the method of averaging. The dynamics are rendered multi-scale, due to the size disparity between the test mode and magnetic perturbations. Maintaining quasi-neutrality at all orders requires that small-scale convective cell turbulence be driven by disparate scale interaction. Here, the cells in turn produce turbulent mixing of vorticity and pressure, which is calculated by fluctuation-dissipation type analyses, and are relevant to pump-out phenomena. The development of correlation between the ambient magnetic perturbations and the cells is demonstrated, showing that turbulence will 'lock on' to ambient stochasticity. Magnetic perturbations are shown to produce a magnetic braking effect on vorticity generation at large scale. Detailed testable predictions are presented. The relations of these findings to the results of available simulations and recent experiments are discussed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Quantifying the Impact of Vertical Resolution on the Representation of Marine Boundary Layer Physics for Global-Scale Models

Abstract While GCM horizontal resolution has received the majority of scale improvements in recent years, ample evidence suggests that a model’s vertical resolution exerts a strong control on its ability to accurately simulate the physics of the marine boundary layer. Here we show that, regardless of parameter tuning, the ability of a single-column model (SCM) to simulate the subtropical marine boundary layer improves when its vertical resolution is improved. We introduce a novel objective tuning technique to optimize the parameters of an SCM against profiles of temperature and moisture and their turbulent fluxes, horizontal winds, cloud water, and rainwater from large-eddy simulations (LES). We use this method to identify optimal parameters for simulating marine stratocumulus and shallow cumulus. The novel tuning method utilizes an objective performance metric that accounts for the uncertainty in the LES output, including the covariability between model variables. Optimization is performed independently for different vertical grid spacings and value of time step, ranging from coarse scales often used in current global models (120 m, 180 s) to fine scales often used in parameterization development and large-eddy simulations (10 m, 15 s). Uncertainty-weighted disagreement between the SCM and LES decreases by a factor of ∼5 when vertical grid spacing is improved from 120 to 10 m, with time step reductions being of secondary importance. Model performance is shown to converge at a vertical grid spacing of 20 m, with further refinements to 10 m leading to little further improvement. Significance Statement In successive generations of computer models that simulate Earth’s atmosphere, improvements have been mainly accomplished by reducing the horizontal sizes of discretized grid boxes, while the vertical grid spacing has seen comparatively lesser refinements. Here we advocate for additional attention to be paid to the number of vertical layers in these models, especially in the model layers closest to Earth’s surface where climatologically important marine stratocumulus and shallow cumulus clouds reside. Our experiments show that the ability of a one-dimensional model to represent physical processes important to these clouds is strongly dependent on the model’s vertical grid spacing.

Meteorology & Atmospheric Sciences↗

DOE OSTI2020

CO2 mass transfer enhancement of aqueous amine solvents by particle additives

The present invention relates to methods for improving carbon capture by providing particles within an amine solvent. The particles provide for increased turbulence at the interface between the counter-flowing gas and solvent, which allows for increased amine and carbamate salt diffusion between the liquid film and bulk.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

A review of tidal energy—Resource, feedbacks, and environmental interactions

The ocean contains a variety of renewable energy resources, little of which has been exploited. Here, we review both tidal range and tidal stream energy, with a focus on the resource, feedbacks, and environmental interactions. The review covers a wide range of timescales of relevance to tidal energy, from fortnightly (spring-neap) and semi-diurnal variability, down to array, and device-scale turbulence. When simulating the regional tidal energy resource, and to assess environmental impacts, it is necessary to account for feedbacks between the tidal array and the resource itself. We critically review various methods for simulating energy extraction, from insights gained through theoretical studies of “tidal fences” in idealized channels, to realistic three-dimensional model studies with complex geometry and arrays of turbines represented by momentum sinks and additional turbulence due to the presence of rotors and support structures. We discuss how variability can be reduced by developing multiple (aggregated) sites with a consideration of the enhanced phase diversity offered by exploiting less energetic tidal currents. This leads to future research questions that have not yet been explored in depth at first-generation tidal sites in relatively sheltered channels (e.g., the interaction of waves with currents). Such enhanced understanding of real sea conditions, including the effects of wind and waves, leads to our other identified primary future research direction—reduced uncertainties in turbulence predictions, including the development of realistic models that simulate the interaction between ambient turbulence and the turbulence resulting from multiple wakes, and changes to system-wide hydrodynamics, water quality, and sedimentation.

Environmental impacts↗

DOE OSTI2023

On the use of a multigrid-reduction-in-time algorithm for multiscale convergence of turbulence simulations

Simulations of turbulent flow present challenges in terms of accuracy and affordability on modern highly-parallel computer architectures. A multigrid-reduction-in-time algorithm is used to provide a framework for separately evolving different scales of turbulence and for parallelizing the temporal domain, thereby increasing the concurrency. It is hypothesized that the space–time locality of the small scales of turbulence can be used to circumvent difficulties in applying temporal multigrid to flows dominated by inertial physics. For algorithms that fall well short of spectral accuracy (fourth-order is used in this work) attention must be paid to the accuracy of features on scales transferred between multigrid levels. Numerical experiments were performed using implicit large-eddy simulation. Results from applying the approach to an infinite-Reynolds number Taylor–Green flow and a double-shear flow at a Reynolds number of 11650 provide strong evidence that the approach has merit. The multigrid-reduction-in-time framework can be used to parallelize the temporal domain of a high-Reynolds-number turbulent flow and permit independent convergence of different scales. Establishing this foundation allows for future research in reducing the wall-clock time to solve turbulent flows while retaining the same accuracy as sequential solvers. In conclusion, current performance results from parallelizing the temporal domain are not competitive with those from sequential-in-time methods.

97 MATHEMATICS AND COMPUTING↗