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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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DOE OSTI2020

Recent Development of Frequency Estimation Methods for Future Smart Grid

The frequency estimated by the Phasor Measurement Unit (PMU) is a critical index of power system status and supports many smart grid applications. The future smart grid features high penetration of renewables and more fast-moving power electronics inverters but raises challenges to the reliable frequency estimation. This article presents three methods to address these challenges. First, an enhanced zero-crossing algorithm was developed to track the fast-changing frequency in system dynamics. Second, we propose a technology that can tolerate the system transient and suppress the outliers. Third, an algorithm was developed to export high time-resolution frequency estimations with minimum computational effort. All of the proposed methods are realized in hardware and compared with classical frequency estimation methods. The testing results indicate that the proposed methods have excellent performance. They can be used in future PMUs and provide reliable and high time resolution data for smart grid applications.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2020

Modeling the Interaction Between Energy Efficiency and Demand Response on Regional Grid Scales: Preprint

With increasing penetration of intermittent renewable generation at grid and distributed scales, flexible building loads can provide significant system value and support the evolving needs of the grid. The growing value of load flexibility may complicate the traditional separation between energy efficiency (EE) and demand response (DR). EE measures may compete in some cases with a building’s DR capabilities but complement one another in other cases. EE can also increase or decrease the need for DR at the system level and change the availability of DR to meet system needs. In this study we present a bottom-up approach to modeling interactive effects between EE and DR in buildings within two regions of the US electricity grid. From a library of building simulation models for different buildings and climates, we synthesize system-level demand profiles and the impacts of potential future EE portfolios. Coupling the underlying building models with a database of DR-enabling technologies, we then compute the quantity of DR that can be delivered in each scenario. The results show that EE and DR interactions are largely driven by the timing of EE savings that are measure-specific and the coincidence with system peak demand that is region-specific. We also find that perspective of the impacts matters – for instance that some EE measures reduce the system need for DR but also reduce the DR potential. Our results imply that utility EE and DR programs developed without considering interactive effects may lead to increased grid-management challenges over the long term.

buildings↗
DOE OSTI2016

System and method for design and optimization of grid connected photovoltaic power plant with multiple photovoltaic module technologies

A system and method of using one or more DC-DC/DC-AC converters and/or alternative devices allows strings of multiple module technologies to coexist within the same PV power plant. A computing (optimization) framework estimates the percentage allocation of PV power plant capacity to selected PV module technologies. The framework and its supporting components considers irradiation, temperature, spectral profiles, cost and other practical constraints to achieve the lowest levelized cost of electricity, maximum output and minimum system cost. The system and method can function using any device enabling distributed maximum power point tracking at the module, string or combiner level.

14 SOLAR ENERGY↗
DOE OSTI2022

Hierarchical Model-Free Transactive Control of Building Loads to Support Grid Services

Residential buildings consume 4.4 quads of electricity annually, approximately 37% of the total electricity consumption in the United States. This represents a vast resource that can be used for demand management and other ancillary services. This project aims to develop a robust, scalable hierarchical transactional control mechanism incorporating elements of model-free control (MFC) and game theory to harness buildings to provide ancillary services to the grid. This approach is being taken to address the challenges of incorporating traditional transactional control schemes into existing buildings. The challenges include small individual building sizes requiring aggregation of many buildings, unpredictable energy usage that makes model identification difficult, and satisfying the sensitive occupant comfort constraints. In the proposed approach, by separating the control mechanism into two layers above and below the load aggregator, MFC can be used below the aggregator to modulate flexible building loads in response to pricing signals with guaranteed performance. This allows the burden of identifying an accurate model of the system to be shifted to the above-aggregator layer, where fluctuations in individual building usage have less impact on predicted building system behavior. Game theory concepts can then be used to determine pricing curves and control signals among regional aggregators. Managing this control in a game-theoretic approach will allow us to build in financial incentives that increase customer engagement. Additionally, the usage of MFC necessitates less burdensome computational and communication requirements, thus, it is easily deployable on small, embedded devices. In a broader sense, developing a strategy capable of effectively incorporating residential and small commercial buildings will allow greater throughput of existing and emerging grid services in addition to future transactive energy grid management methods. Using MFC within a hierarchical control architecture will allow the shifting of existing forecasting challenges to an aggregate level, where dynamics are slower and more predictable. This will enable a smooth interface between the grid services requests of utilities and the reliable control required by participating buildings. MFC, which supports distributed control architecture, permits a scalable solution that can be deployed to neighborhood-size systems as well as individual buildings. This project focuses on three objectives: (1) developing the mathematical framework, algorithm toolkit, and software toolset of the two-layer transactive control testbed; (2) developing a scalable solution for application over many residential and small-size commercial buildings with sparse distributed communication; and (3) field testing and implementation on hardware of the control strategies developed in the previous two objectives. The research and development activities are focused and designed to be impactful within the relevant 2025 targets timeframe. An open-source control framework for exploiting variability and dispatchability of building loads will be delivered as the outcome of the project. This capability enables greater participation of loads in electricity markets and ancillary services that are both useful for the utility and financially beneficial for building owners.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2023

A Reversed Impedance-Based Stability Criterion for IBR Grids

The existing impedance-based stability criterion is effective for analyzing local control interactions; however, it is difficult to scale the existing criterion to analyze wide-area control interactions among numerous IBRs through a complex power system network. The scaled version of the existing criterion requires the impedance response of each IBR in the system as well as of the network looking from all the IBRs. It is quite challenging to obtain all these impedance responses because of the computational effort and the requirement of separately scanning the impedance of the network and the IBRs. We propose a reversed criterion for the impedance-based stability analysis to address these problems. In contrast to the existing criterion, the reversed criterion analyzes the stability of a power system when an IBR is disconnected from the system. The reversed criterion estimates the impact of an IBR on the frequency and damping of power system oscillation modes using the impedance scans of only the IBR and the grid at its terminal. It can be sequentially applied at different IBRs to evaluate their impact on the power system stability. In addition to scalability, the reversed criterion gives flexibility to focus only on a few selected IBRs, depending on their rating, the magnitude of oscillations observed at their terminals, and the vendor support available for implementing stabilizing control system updates. The reversed criterion is demonstrated on a 14-bus power system with 100% IBRs.

control interactions↗
DOE OSTI2025

Explainable AI for Multivariate Time Series Pattern Exploration: Latent Space Visual Analytics With Temporal Fusion Transformer and Variational Autoencoders in Power Grid Event Diagnosis

Detecting and analyzing complex patterns in multivariate time-series data is crucial for decision-making in urban and environmental system operations. However, challenges arise from the high dimensionality, intricate complexity, and interconnected nature of complex patterns, which hinder the understanding of their underlying physical processes. Existing AI methods often face limitations in interpretability, computational efficiency, and scalability, reducing their applicability in real-world scenarios. This paper proposes a novel visual analytics framework that integrates two generative AI models, Temporal Fusion Transformer (TFT) and Variational Autoencoders (VAEs), to reduce complex patterns into lower-dimensional latent spaces and visualize them in 2D using dimensionality reduction techniques such as PCA, t-SNE, and UMAP with DBSCAN. These visualizations, presented through coordinated and interactive views and tailored glyphs, enable intuitive exploration of complex multivariate temporal patterns, identifying patterns’ similarities and uncover their potential correlations for a better interpretability of the AI outputs. The framework is demonstrated through a case study on power grid signal data, where it identifies multi-label grid event signatures, including faults and anomalies with diverse root causes. Additionally, novel metrics and visualizations are introduced to validate the models and assess the performance, efficiency, and consistency of latent maps generated by VAE, which have been utilized in prior studies for latent space cartography and used as a benchmark in this study, and the emerging TFT architecture under various configurations. These analyses provide actionable insights for model parameter tuning and reliability improvements. Comparative results highlight that TFT achieves shorter run times and superior scalability to diverse time-series data shapes compared to VAE. This work advances fault diagnosis in multivariate time series, fostering explainable AI to support critical system operations.

Explainable AI↗
DOE OSTI2015

Strategy and gaps for modeling, simulation, and control of hybrid systems

The purpose of this report is to establish a strategy for modeling and simulation of candidate hybrid energy systems. Modeling and simulation is necessary to design, evaluate, and optimize the system technical and economic performance. Accordingly, this report first establishes the simulation requirements to analysis candidate hybrid systems. Simulation fidelity levels are established based on the temporal scale, real and synthetic data availability or needs, solution accuracy, and output parameters needed to evaluate case-specific figures of merit. Accordingly, the associated computational and co-simulation resources needed are established; including physical models when needed, code assembly and integrated solutions platforms, mathematical solvers, and data processing. This report first attempts to describe the figures of merit, systems requirements, and constraints that are necessary and sufficient to characterize the grid and hybrid systems behavior and market interactions. Loss of Load Probability (LOLP) and effective cost of Effective Cost of Energy (ECE), as opposed to the standard Levelized Cost of Electricty (LCOE), are introduced as technical and economical indices for integrated energy system evaluations. Financial assessment methods are subsequently introduced for evaluation of non-traditional, hybrid energy systems. Algorithms for coupled and iterative evaluation of the technical and economic performance are subsequently discussed. This report further defines modeling objectives, computational tools, solution approaches, and real-time data collection and processing (in some cases using real test units) that will be required to model, co-simulate, and optimize; (a) an energy system components (e.g., power generation unit, chemical process, electricity management unit), (b) system domains (e.g., thermal, electrical or chemical energy generation, conversion, and transport), and (c) systems control modules. Co-simulation of complex, tightly coupled, dynamic energy systems requires multiple simulation tools, potentially developed in several programming languages and resolved on separate time scales. Whereas further investigation and development of hybrid concepts will provide a more complete understanding of the joint computational and physical modeling needs, this report highlights areas in which co-simulation capabilities are warranted. The current development status, quality assurance, availability and maintainability of simulation tools that are currently available for hybrid systems modeling is presented. Existing gaps in the modeling and simulation toolsets and development needs are subsequently discussed. This effort will feed into a broader Roadmap activity for designing, developing, and demonstrating hybrid energy systems.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2016

Benchmark of Spacer Grid Models used in Nuclear Power Plants

The fuel bundles of both PWR and BWR reactors require spacers to maintain the integrity and spacing of the slender fuel rods. These spacers can provide benefits such as increased mixing and turbulence, which increases heat distribution, leading to better Departure from Nucleate Boiling (DNB) conditions in PWRs. They also enhance the critical power ratios by increasing the thickness of the liquid layer in BWRs. However, the spacers also cause a pressure drop which must be engineered into the performance and operation. The flow within a rod bundle involves a number of complicated factors, such as cross flow, (de)entrainment of liquid layer, bubble or droplet break up/join. This requires specialized 3 dimensional computational fluid dynamics for the complete flow, however, most of the newly developed models have been accomplished by industry, so they are proprietary and in addition require extensive programming and cost to implement. In the public domain, a number of 1 dimensional equations have been developed that could benefit in initial design and decision making, if the equations are valid in the new and different applications the models are used. In 2002, an OECD/NRC BFBT benchmark workshop determined the variables used in the development of models, to predict flow within nuclear fuel bundles, had not been updated with technological advances to replace general theory and characteristics. If those properties could be identified and modeled, they could be applied to a wider range of conditions. This study utilizes NUPEC BFBT tests' measured data, on an average spacer over a range of pressure and flows, which was determined to be the most detailed measured data available. The focus is to describe the mathematical models of spacer grids and validate them with experimental results. This study consists of compiling and comparing available and established single phase equations and models computations. While these models may not have the desired degree of accuracy for final decisions, they may be accurate enough for the initial phases of design. No conclusion on a universal equation that fits all boundary conditions accurately or detailed effect of properties was determined. However, by using the BFBT experimental data, boundary conditions where the spacer grid models can be useful were identified. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2026

Designing resilient IoT and Edge Computing with federated tinyML

The rapid growth of the Internet of Things (IoT) and Edge Computing (EC) has brought significant conveniences to modern society but has also greatly expanded the cyber attack surfaces, particularly as these technologies are being increasingly integrated into critical systems such as power grids, healthcare, and smart homes. Here, to improve IoT/EC’s cybersecurity posture, we leveraged Artificial Intelligence (AI) and Machine Learning (ML) by employing tinyML to monitor voluminous IoT data for cyber threats while addressing devices’ resource constraints, and utilizing Federated Learning (FL) to share local detection knowledge across the system while preserving privacy. Building on our three-layer architecture combining tinyML and FL to enhance autonomous cyber attack detection, this paper demonstrated that the architecture improves detection accuracy, reduces resource consumption, and enables lightweight, secure IoT device monitoring. These results were validated using the public N-BaIoT dataset as well as real IoT network traffic data collected under multiple attack scenarios from our testbeds. Additionally, we introduced an enhanced FL methodology with a novel preprocessing stage, including federated feature selection and global preprocessor construction, to address IoT/EC data heterogeneity. We developed a physical IoT testbed for attack simulations and data collection, implemented a tinyML-powered detector for realistic model validation, and also built a virtual testbed for scalable evaluations of FL models across diverse network environments.

Cognitive cyber↗
DOE OSTI2023

Comparative Study of Nonlinear Black-Box Modeling for Power Electronics Converters

With the increasing penetration level of renewable sources and power electronics loads in modern power systems, accurate and computationally efficient models are needed. Black-box model (BBM) could be a useful method in such systems. However, not very extensive research efforts have been made for power electronics BBM so far, and existing works mostly focus on steady-state operation, neglecting the important transient behaviors such as load transients, voltage transients, and faults. This paper presents a comparative study of three commonly used nonlinear BBM approaches for transient behaviors of power electronics converters. Comparison methods are proposed, and the evaluations are conducted under different transients using a grid-connected single-phase photovoltaic inverter. The findings of this study provide valuable references for further feasibility investigations on implementing BBMs in large-scale power electronics-rich power systems.

Qiao, Liang↗
DOE OSTI2018

NO-LESS: Near optimal curtailment strategy selection for net load balancing in micro grids

The integration of renewable energy sources such as solar PVs into micro grids has increased in recent years. However, the intermittent and unpredictable nature of renewable energy generation makes supply uncertain, in turn requiring continuous net load balancing to ensure grid stability. One technique to achieve net load balancing is curtailment, however, selection of the optimal set of curtailment strategies to achieve a targeted curtailment is NP-hard. State-of-the-art curtailment selection techniques approach this problem by either developing computationally expensive optimal curtailment strategies or reduce accuracy to generate sub-optimal solutions quickly. In this work, we develop NO-LESS: a Near OptimaL CurtailmEnt Strategy Selection algorithm. NO-LESS is a novel Fully Polynomial Time Approximation Scheme (FPTAS) which determines bounded (near optimal) curtailment strategies in a small amount of time while simultaneously satisfying practical constraints on strategy switching overhead and curtailment fairness. We perform both theoretical analysis and practical evaluation to show that NO-LESS is a scalable solution for performing net load balancing through curtailment strategy selection in Micro Grids.

14 SOLAR ENERGY↗
DOE OSTI2013

Numerical computation of doppler-broadening in the resonance domain

We have implemented an accurate and fast calculation of the Doppler-broadened kernel PT(E {yields} E') for neutron elastic scattering based on a gas model. An exponential cutoff which accounts for the asymptotic behavior of the error function helps limit the range of integration while eliminating difference effects. This allows for calculating a kernel library for {sup 238}U over a very fine energy grid covering the resonance range in only a few hours in a laptop. We give an example showing the impact of {sup 238}U elastic up-scattering on the values of self shielded cross sections. (authors)

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Preface to the Special Issue on TOUGH Symposium 2015

The TOUGH Symposium 2015 was held in Berkeley, California, September 28-30, 2015. The TOUGH family of codes, developed at the Energy Geosciences Division of Lawrence Berkeley National Laboratory (LBNL), is a suite of computer programs for the simulation of multiphase and multicomponent fluid and heat flows in porous and fractured media with applications in many geosciences fields, such as geothermal reservoir engineering, nuclear waste disposal, geological carbon sequestration, oil and gas reservoirs, gas hydrate research, vadose zone hydrology and environmental remediation. Since the first release in the 1980s, many modifications and enhancements have been continuously made to TOUGH and its various descendants (iTOUGH2, TOUGH+, TOUGH-MP, TOUGHREACT, TOUGH+HYDRATE, TMVOC...), at LBNL and elsewhere. Today, these codes are used worldwide in academia, government organizations and private companies in problems involving coupled hydrological, thermal, biogeochemical and geomechanical processes. The Symposia, organized every 2-3 years, bring together developers and users for an open exchange on recent code enhancements and applications. In 2015, the Symposium was attended by one hundred participants, representing thirty-four nationalities. This Special Issue in Computers & Geosciences gathers extended versions of selected Symposium proceedings related to (i) recent enhancements to the TOUGH family of codes and (ii) coupled flow and geomechanics processes modeling. In this special issue, a paper by Jung et al. provides a new base version of TOUGH, TOUGH3. This latest version addresses the computational robustness and efficiency needed for the increasing complexity of the simulated processes and the growing size of model domains. Moreover, it consolidates the serial and parallel versions into a single code, hence, inconsistencies between different versions are less likely to occur, and code maintenance and development become more efficient. Finsterle et al. present a review of the capabilities of iTOUGH2 and illustrate the workflow of an integrated simulation-optimization framework. Liu et al. present an approach to reduce the computational cost of performing Bayesian inverse modeling and demonstrate their approach using a synthetic experiment simulated with TOUGH2. Regarding the mesh generation process, the paper by Doughty presents a generalization of the grid-generation algorithm embedded in TOUGH2 (MESHMAKER) to create one-column grids with fractal dimension, useful for representing flow through fractures or highly heterogeneous media with fractal geometry, and also for inverse methods. Sentís and Gable present a module to interface the open source code Lagrit with TOUGH2, that assists in the processes of mesh generation and model setup, and that can be particularly useful for complex geometries and large models. Bonduà et al. present a pre-processing tool (VORO2MESH) that features a full 3D Voronoi approach for structured and unstructured grids, and an enhanced post-processor (TOUGH2Viewer) for the visualization of 3D Voronoi grids. The literature review shows that coupled fluid flow and geomechanics has gained increasing attention in the last years in many geosciences fields. In this context, several codes of the TOUGH family have been successfully coupled with geomechanics software. The paper by Rutqvist presents a review of some of these TOUGH-based geomechanics models, of which the TOUGH-FLAC simulator – presented for the first time in the TOUGH Symposium 2003 – is well-known worldwide and has been used in a large variety of applications, encompassing the application domains of the TOUGH codes. Blanco-Martín et al. present an extension of TOUGH-FLAC to the finite strain framework, motivated by the existence of large strains associated with rock salt deformation in the context of heat-generating nuclear waste disposal. Following a paper published in the Special Issue of the TOUGH Symposium 2012 in this journal, Kim et al. present the current status of the TOUGH-RSBN simulator for coupled hydro-mechanical modeling of hydraulic fracturing using a discrete fracture network approach, and present some validation and demonstration cases. Rinaldi and Nespoli present TOUGH-SEED, an injection-induced seismicity modeling approach using TOUGH2 and a geomechanics stochastic model, and apply it to the Basel EGS project. Finally, Rinaldi et al. use the PEST protocol implemented in iTOUGH2, combined with TOUGH-FLAC as the forward model, to perform inverse modeling for the In Salah CO2 demonstration site in Algeria. Using pore pressures and InSAR displacements as observations, this approach allows for a multi-physics joint inversion and takes advantage of the inverse capabilities of iTOUGH2 (parameter estimation, sensitivity analysis, uncertainty estimation …), applied to coupled fluid flow and geomechanics. On behalf of the members of the TOUGH Symposium 2015 Organizing Committee, I hope that this collection of papers will provide the readers of Computers & Geosciences with a recent view of the capabilities of the TOUGH family of codes, as well as with current modeling research domains in the geosciences. Last but not least, I would like to express my gratitude to all the reviewers who kindly contributed to this Special Issue, as well as the Journal staff who have made this publication possible.

99 GENERAL AND MISCELLANEOUS↗
DOE OSTI2016

Vlasov simulations of electron-ion collision effects on damping of electron plasma waves

Collisional effects can play an essential role in the dynamics of plasma waves by setting a minimum damping rate and by interfering with wave-particle resonances. Kinetic simulations of the effects of electron-ion pitch angle scattering on Electron Plasma Waves (EPWs) are presented here. In particular, the effects of such collisions on the frequency and damping of small-amplitude EPWs for a range of collision rates and wave phase velocities are computed and compared with theory. Both the Vlasov simulations and linear kinetic theory find the direct contribution of electron-ion collisions to wave damping significantly reduced from that obtained through linearized fluid theory. To our knowledge, this simple result has not been published before. Simulations have been carried out using a grid-based (Vlasov) approach, based on a high-order conservative finite difference method for discretizing the Fokker-Planck equation describing the evolution of the electron distribution function. Details of the implementation of the collision operator within this framework are presented. Such a grid-based approach, which is not subject to numerical noise, is of particular interest for the accurate measurements of the wave damping rates.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Artificial Intelligence-Enhanced CMIP6 Climate Projections Across the Conterminous United States

This dataset comprises high-resolution climate projections at 1/24 degree grid (~4km) over the conterminous United States (CONUS) based on ten Global Climate Models (GCMs) that are part of the Coupled Models Intercomparison Project phase 6 (CMIP6). The CMIP6 GCMs are downscaled using two artificial intelligence (AI) techniques, primarily based on the computer vision approach called super-resolution. We train two separate networks: super-resolution convolutional neural network (SRCNN) and super-resolution generative adversarial network (SRGAN). The networks are trained using Daymet observations, originally available at a 1 km resolution. For training purposes, the Daymet data is interpolated to 1/24 degree (~4km), 0.25 degree and 1 degree, which serve as high, intermediate and low-resolution inputs respectively. For each of the SRCNN and SRGAN network, we use a two-step resolution enhancement, the first step generates 4x refinement from 1 degree to 0.25 degree and the second step generates 6x refinement from 0.25 degree to 1/24 degree (~4km). We downscale daily scale precipitation, maximum temperature and minimum temperature for the six CMIP6 GCMs for 1980 to 2019 in the historical period and 2020 to 2059 in the near-term future under the shared socioeconomic pathway 585 and 245 (SSP585 and SSP245) emission scenarios. We also perform double bias-correction with Daymet observations using a quantile mapping approach, first for GCMs prior to making predictions at 1 degree grid and second after making final predictions at ~4km.

13 HYDRO ENERGY↗
DOE OSTI2022

Development of lean, efficient, and fast physics-framed deep-learning-based proxy models for subsurface carbon storage

In this work, we present deep-learning-based surrogate models for CCUS developed with four different algorithms and a physics-framed two-phase flow problem involving displacement of water by CO 2 . The deep-learning models were trained using 3D datasets describing the pressure plume, CO 2 saturation plume, and water extraction rate generated by numerical simulation. The hyperparameters defining the architecture of the neural networks were optimized to determine the slimmest network size and training parameters that give the most efficient performance at the least training cost. To develop a robust model that closely mimics the governing physical laws, the discretized form of the two-phase fluid transport equation was used to formulate the supervised deep-learning task. The algorithms investigated in this study predicted the data to above 95% accuracy, with the multi-layer perceptron model demonstrating the best performance by balancing training speed, prediction time, and prediction accuracy with lean network capacity. Furthermore, the surrogate models simultaneously predict reservoir pressure and CO 2 saturation in every grid block, including the surface well extraction rate and bottomhole pressure, at all simulation times for a given static model realization in just a few seconds on a standard desktop computer. A key outcome of this study is that limits can be placed on network design parameters to avoid over designing neural networks, with associated efficiencies in training and prediction times. This is very useful because large volumes of data may be generated in CCUS projects and over-design of neural network architectures imposes penalties that are antithetical to the goal of near-real time forecasting.

58 GEOSCIENCES↗
DOE OSTI2013

An implicit logarithmic finite-difference technique for two dimensional coupled viscous Burgers’ equation

This article describes a new implicit finite-difference method: an implicit logarithmic finite-difference method (I-LFDM), for the numerical solution of two dimensional time-dependent coupled viscous Burgers’ equation on the uniform grid points. As the Burgers’ equation is nonlinear, the proposed technique leads to a system of nonlinear systems, which is solved by Newton's iterative method at each time step. Computed solutions are compared with the analytical solutions and those already available in the literature and it is clearly shown that the results obtained using the method is precise and reliable for solving Burgers’ equation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

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