Search NASA⌕ Search

SEARCH · Search NASA

Results for “Time step adaptivity”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 271 records · Page 15
All sourcesNASA NTRSDOE OSTI
NASA NTRS2000

Tactile Sensing for Dexterous Robotic Hands

Robotic systems will be used as precursors to human exploration to explore the solar system and expand our knowledge of planetary surfaces. Robotic systems will also be used to build habitats and infrastructure required for human presence in space and on other planetary surfaces . Such robots will require a high level of intelligence and automation. The ability to flexibly manipulate their physical environment is one characteristic that makes humans so effective at these building and exploring tasks . The development of a generic autonomous grasp ing capability will greatly enhance the efficiency and ability of robotics to build, maintain and explore. To tele-operate a robot over vast distances of space, with long communication delays, has proven to be troublesome. Having an autonomous grasping capability that can react in real-time to disturbances or adapt to generic objects, without operator intervention, will reduce the probability of mishandled tools and samples and reduce the number of re-grasp attempts due to dropping. One aspect that separates humans from machines is a rich sensor set. We have the ability to feel objects and respond to forces and textures. The development of touch or tactile sensors for use on a robot that emulates human skin and nerves is the basis for this discussion. We will discuss the use of new piezo-electric and resistive materials that have emerged on the market with the intention of developing a touch sensitive sensor. With viable tacti le sensors we will be one step closer to developing an autonomous grasping capability.

Martin, Toby B.↗
DOE OSTI2025

A Scale‐Adaptive Urban Hydrologic Framework: Incorporating Network‐Level Storm Drainage Pipes Representation

Abstract Below‐ground urban stormwater networks (BUSNs) significantly influence urban flood dynamics, yet their representation at the watershed or larger scales remains challenging. We introduce a scalable urban hydrologic framework that centers on a novel network‐level BUSN representation, balancing the needs for physical basis, parameter parsimony, and computational efficiency. Our framework conceptualizes an urban watershed into four interacting zones: hillslopes (natural), storm‐sewersheds (urban), a sub‐network channel (tributaries), and a main channel. We develop an innovative Graph Theory‐based algorithm to derive network‐level BUSN parameters from publicly available datasets, enabling efficient, scalable parameterization. We demonstrate this framework's applicability at nine representative watersheds in the Houston metropolitan region, USA, with urban imperviousness ranging from 0% to 64% and drainage areas ranging from 24 to 302 . Our model achieves satisfying computational efficiency, completing hourly time step simulations for 18 years in less than 5 sec per watershed on a standard PC. Validation against observed daily streamflow confirms that the model can capture small‐to‐large flood peaks and seasonal and annual water balance over these watersheds. Comparisons with the National Water Model show better performance in predicting flood peaks and overall water balance, underscoring the promises of our new framework for urban hydrologic modeling at large scales. Furthermore, analysis reveals nonlinear relationships between BUSNs' designed capacities and flood reduction effects. Our approach bridges the gap between detailed hydraulic and large‐scale hydrologic models, providing a valuable tool for urban flood prediction and management across broader spatial and temporal scales.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1993

Additional support for the TDK/MABL computer program

An advanced version of the Two-Dimensional Kinetics (TDK) computer program was developed under contract and released to the propulsion community in early 1989. Exposure of the code to this community indicated a need for improvements in certain areas. In particular, the TDK code needed to be adapted to the special requirements imposed by the Space Transportation Main Engine (STME) development program. This engine utilizes injection of the gas generator exhaust into the primary nozzle by means of a set of slots. The subsequent mixing of this secondary stream with the primary stream with finite rate chemical reaction can have a major impact on the engine performance and the thermal protection of the nozzle wall. In attempting to calculate this reacting boundary layer problem, the Mass Addition Boundary Layer (MABL) module of TDK was found to be deficient in several respects. For example, when finite rate chemistry was used to determine gas properties, (MABL-K option) the program run times became excessive because extremely small step sizes were required to maintain numerical stability. A robust solution algorithm was required so that the MABL-K option could be viable as a rocket propulsion industry design tool. Solving this problem was a primary goal of the phase 1 work effort.

Nickerson, G. R.↗
NASA NTRS1990

A transient response analysis of the space shuttle vehicle during liftoff

A proposed transient response method is formulated for the liftoff analysis of the space shuttle vehicles. It uses a power series approximation with unknown coefficients for the interface forces between the space shuttle and mobile launch platform. This allows the equation of motion of the two structures to be solved separately with the unknown coefficients at the end of each step. These coefficients are obtained by enforcing the interface compatibility conditions between the two structures. Once the unknown coefficients are determined, the total response is computed for that time step. The method is validated by a numerical example of a cantilevered beam and by the liftoff analysis of the space shuttle vehicles. The proposed method is compared to an iterative transient response analysis method used by Martin Marietta for their space shuttle liftoff analysis. It is shown that the proposed method uses less computer time than the iterative method and does not require as small a time step for integration. The space shuttle vehicle model is reduced using two different types of component mode synthesis (CMS) methods, the Lanczos method and the Craig and Bampton CMS method. By varying the cutoff frequency in the Craig and Bampton method it was shown that the space shuttle interface loads can be computed with reasonable accuracy. Both the Lanczos CMS method and Craig and Bampton CMS method give similar results. A substantial amount of computer time is saved using the Lanczos CMS method over that of the Craig and Bampton method. However, when trying to compute a large number of Lanczos vectors, input/output computer time increased and increased the overall computer time. The application of several liftoff release mechanisms that can be adapted to the proposed method are discussed.

Brunty, J. A.↗
NASA NTRS2002

An End-to-End Architecture for Science Goal Driven Observing

New observatories will have greater on-board storage capacity and on-board processing capabilities. The new bottleneck will be download capacity. The cost of downlink time and limitations of bandwidth will end the era where all exposure data is downloaded and all data processing is performed on the ground. In addition, observing campaigns involving inherently variable targets will need scheduling flexibility to focus observing time and data download on exposures that are scientifically interesting. The ability to quickly recognize and react to such events by re-prioritizing the observing schedule will be an essential characteristic for maximizing scientific returns. It will also be a step towards increasing spacecraft autonomy, a major goal of NASA's strategic plan. The science goal monitoring (SGM) system is a proof-of-concept effort to address these challenges. We are developing an interactive distributed system that will use on-board processing and storage combined with event-driven interfaces with ground-based processing and operations, to enable fast re-prioritization of observing schedules, and to minimize time spent on non-optimized observations. SGM is initially aimed towards time-tagged observing modes used frequently in spectroscopic studies of varying targets. In particular, the SGM is collaborating with the proposed MIDEX-class mission Kronos team. The variable targets that Kronos seeks to study make an adaptive system such as SGM particularly valuable for achieving mission goals. However, the architecture and interfaces will also be designed for easy adaptability to other observing platforms, including ground-based systems and to work with different scheduling and pipeline processing systems. This talk will focus on our strategy for developing SGM and the technical challenges that we have encountered. We will discuss the SGM architecture as it applies to the Kronos mission and explain how it is scalable to other missions.

Jones, Jeremy↗
NASA NTRS2018

Land Surface Reflectances from Geostationary Sensors

GEONEX is a processing pipeline that produces a suite of satellite land surface products using data streams from the latest geostationary (GEO) sensors including the GOES016/ABI and the Himawari-8/AHI. The suite, created collaboratively by scientists from NASA and NOAA, includes top-of-atmosphere (TOA) reflectances, land surface reflectances (LSRs), vegetation indices, LAI/fPAR, and other downstream products. As a key component of the GEONEX product processing, we have adapted the Multi-Angle Implementation of Atmospheric Correction (MAIAC) algorithm to produce LSRs from the TOA data. Because the algorithm depends on building "stacks" of images, we first run internal geo-registration checks to ensure geo-spatial accuracy and consistency of the input (L1B) data before transferring them from the geostationary projection into a tile system in geographic grids. Scan-time is inferred from metadata and applied to calculate the sun-sensor angles for each grid cell. The MAIAC algorithm is run to detect clouds/shadows, estimate aerosol optical thickness (AOT), perform atmospheric corrections, and generate LSRs. We have processed 18-months (from 2016/04 onward) of AHI data over East Asia and Oceania at a 10-minute time step and 10-months (from 2018/01 onward) of ABI data over North and South Americas at a 15-minute time step. As a verification measure, we compare the GEONEX (AHI/ABI) surface reflectances with the standard MODIS products (MOD09GA) and the MODIS MAIAC products over pixels that have similar sun-view geometries. The results indicate general linear relationships between GEONEX and corresponding MODIS LSRs. In particular, the RMSEs between GEONEX and MOD09 data are comparable to those between MOD09 and MODIS MAIAC products, suggesting that the uncertainties of GEONEX LSRs fall into an acceptable range. However, direct comparisons of LSRs over pixels with different sun-view angles are not as straightforward and require more modeling efforts to correct the directional effects. Evaluation of such angular influences on the downstream products (e.g., vegetation indices) is also under investigation.

Geostationary satellite; Remote Sensing; Atmospher↗
DOE OSTI2025

Chemical timescale effects on detonation convergence

Numerical simulations of detonation-containing flows have emerged as crucial tools for designing next- generation power and propulsion devices. As these tools mature, it is important for the combustion community to properly understand and isolate grid resolution effects when simulating detonations. To this end, the objective of this work is to provide a comprehensive analysis of the numerical convergence of unsteady detonation simulations, with focus on isolating the impacts of chemical timescale modifications on convergence characteristics in the context of operator splitting. With the aid of an AMReX-based adaptive mesh refinement flow solver-which enables resolutions up to ($\mathcal{O}$ (1000) cells-per-induction length-the convergence analysis is conducted using two kinetics configurations: (1) the simplified three-step Arrhenius-based model mechanism of Short and Quirk (1997), where chemical timescales in the detonation are modified by adjusting activation energies in the initiation and branching reactions, and (2) a detailed hydrogen- air mechanism, where the chemical timescales are adjusted by varying the ambient pressure. The convergence of unsteady self-sustained detonations in one-dimensional channels is then analyzed with reference to steady-state theoretical baseline solutions using these mechanisms. The goal of the analysis is to provide a detailed comparison of the effects of grid resolution on both macroscopic (peak pressures and wave speeds) and microscopic (wave structure) quantities of interest, drawing connections between the deviations from steady-state baselines and minimum chemical timescales. In particular, chemical timescale reductions were found to have minimal impact on the convergence of macroscopic properties. Furthermore, analyses of microscopic convergence trends, particularly in the reaction front location, revealed a key insight: maintaining the induction time while eliminating prohibitive chemical timescales through mechanism simplifications and combustion modeling can significantly enhance detonation convergence properties. Ultimately, this work uncovers resolution-dependent unsteady detonation convergence regimes and highlights the important role played by not only the chemical timescales, but also the ratio between the chemical timescale and induction time on the numerical convergence of the detonation wave structure.

Adaptive mesh refinement↗
NASA NTRS2016

On-orbit RNA Purification and qRT-PCR Capabilities of the WetLab-2 System

NASA Ames Research Center’s WetLab-2 system brings new capabilities to the ISS for researchers. The system can lyse cells and extract RNA on-orbit from different sample types ranging from microbial cultures to animal tissues. Our purification method has the advantage of using non-toxic chemicals and does not require alcohols or other organics. The purified RNA can then either be stabilized for return to Earth or can be used to conduct on-orbit quantitative Reverse Transcriptase PCR (qRT-PCR) analysis without the need for sample return. qRT-PCR reactions are performed by dispensing the RNA into reaction tubes that contain all lyophilized reagents needed to perform the analysis. The system uses a Cepheid® SmartCycler that allows for multiplexing of assays, this can be used to normalize for RNA concentration and integrity and to study multiple genes of interest in each tube. There are a total of 16 independent PCR modules each capable of detecting up to four fluorescent channels. The WetLab- 2 system can downlink data from the ISS to the ground after a completed run and uplink new thermal cycling programs. The ability to purify and stabilize RNA on-orbit can eliminate the confounding effects of reentry stresses and shock acting on live cells and organisms or the concern of RNA degradation of some samples. It also has the benefit of minimizing the needed downmass. Conducting qRT-PCR and generating results on-orbit is also an important step towards utilizing the ISS as a National Laboratory facility. Specifically, the ability to get on-orbit data will provide investigators with the opportunity to adjust experimental parameters in real time without the need for sample return and re-flight. On orbit gene expression analysis can also provide benchmarking prior to sample return. The system can also be used for analysis of air, surface, water, and clinical samples to monitor environmental pathogens and crew health. The validation flight of the WetLab-2 system using E. coli bacteria and mouse liver is scheduled to launch on SpaceX-8 this spring. Pending operations, the preliminary results from the validation flight will be presented. To support the needs of future researchers, we are adapting our system to purify RNA from two additional sample types: fibrous tissue such as muscle and mammalian adherent cells grown on alginate beads. Progress of this work will also be presented. The WetLab-2 Project is supported by the Research Integration Office in the ISS Program.

Parra, Macarena↗
DOE OSTI2026

Captan+X Data Converter Integration

Fermi National Accelerator Laboratory's CAPTAN (Compact And Programmable daTa Acquisition Node) series provides a flexible hardware platform for data acquisition across a range of experiments and facilities. The latest iteration, CAPTAN+X, is built around a Kintex-7 FPGA supporting four FPGA Mezzanine Card (FMC) connections. As part of a broader laboratory effort to bring facility systems under a Model-Based Systems Engineering (MBSE) framework, CAPTAN+X is one of several systems slated to be incorporated into this modeling environment in the near term. A necessary step toward that goal is incorporating the platform's core functionality, which centers on integration with the LXD31K4 FMC, a data converter module combining dual AD9652 analog-to-digital converters and dual AD9142A digital-to-analog converters. Achieving compatibility required resolving pin-mapping conflicts between the LXD31K4's High Pin Count connector and the CAPTAN+X's available pin types, adapting a Board Support Project originally written for an UltraScale-class evaluation board to the Kintex-7 architecture, replacing incompatible primitives, restructuring clock distribution, and manually configuring chip initialization in place of an unsupported soft-processor-based approach. Functional verification of the ADC and DAC channels, followed by closed-loop testing combining both converters with real-time filtering, confirmed correct operation of the integrated system. These results establish a working hardware and firmware baseline for the CAPTAN+X platform, positioning it for future inclusion in the laboratory's growing MBSE modeling effort.

Espinoza, David [Illinois U., Urbana (main)]↗
DOE OSTI2024

A Methodology for determining optimal milling parameters for low-taper microtensile sample production

Femtosecond lasers are beginning to see an increase in interest for industrial and high throughput microsample milling applications. Despite this, much of the literature regarding removal rate studies for ultrashort pulse laser milling continues to focus on single-pulse or similar experiments that have a very small time on target. The material volume removal rates of femtosecond milling can vary wildly depending on parameters like laser scan speed, hatch spacing, z-axis step size, and pulses on the material. Currently, there is no established methodology for determining the optimal values of these parameters for micromilling applications. This methodology was developed in this study. The methodology consists of a pulse study, a repeat study, and a z-step study that will return the milling parameters that resulted in the largest volume removal rate for the laser used for this research. The methodology was created for the laser system used in this study, and was tested on 316 Stainless Steel, but was developed in such a way that it may be adapted to any material, and with some tweaking of variables might be used in other laser systems. The laser system used in this study utilizes a novel positioning stage that introduces 6 degrees of freedom to the translation of laser samples, being able to tilt and move in all three axes. Using both the developed methodology and the unique capabilities of the laser microtensile samples were produced with taper angles of <1° in 316 Stainless Steel.

Barker, Zachary Wayne↗
DOE OSTI2026

CAPTAN+X Data Converter Integration

Fermi National Accelerator Laboratory's CAPTAN (Compact And Programmable daTa Acquisition Node) series provides a flexible hardware platform for data acquisition across a range of experiments and facilities. The latest iteration, CAPTAN+X, is built around a Kintex-7 FPGA supporting four FPGA Mezzanine Card (FMC) connections. As part of a broader laboratory effort to bring facility systems under a Model-Based Systems Engineering (MBSE) framework, CAPTAN+X is one of several systems slated to be incorporated into this modeling environment in the near term. A necessary step toward that goal is incorporating the platform's core functionality, which centers on integration with the LXD31K4 FMC, a data converter module combining dual AD9652 analog-to-digital converters and dual AD9142A digital-to-analog converters. Achieving compatibility required resolving pin-mapping conflicts between the LXD31K4's High Pin Count connector and the CAPTAN+X's available pin types, adapting a Board Support Project originally written for an UltraScale-class evaluation board to the Kintex-7 architecture, replacing incompatible primitives, restructuring clock distribution, and manually configuring chip initialization in place of an unsupported soft-processor-based approach. Functional verification of the ADC and DAC channels, followed by closed-loop testing combining both converters with real-time filtering, confirmed correct operation of the integrated system. These results establish a working hardware and firmware baseline for the CAPTAN+X platform, positioning it for future inclusion in the laboratory's growing MBSE modeling effort.

Espinoza, David [Illinois U., Urbana (main)]↗
NASA NTRS2003

Enhanced Multi-Modal Access to Planetary Exploration

Tomorrow's Interplanetary Network (IPN) will evolve from JPL's Deep-Space Network (DSN) and provide key capabilities to future investigators, such as simplified acquisition of higher-quality science at remote sites and enriched access to these sites. These capabilities could also be used to foster public interest, e.g., by making it possible for students to explore these environments personally, eventually perhaps interacting with a virtual world whose models could be populated by data obtained continuously from the IPN. Our paper looks at JPL's approach to making this evolution happen, starting from improved communications. Evolving space protocols (e.g., today's CCSDS proximity and file-transfer protocols) will provide the underpinning of such communications in the next decades, just as today's rich web was enabled by progress in Internet Protocols starting from the early 1970's (ARPAnet research). A key architectural thrust of this effort is to deploy persistent infrastructure incrementally, using a layered service model, where later higher-layer capabilities (such as adaptive science planning) are enabled by earlier lower-layer services (such as automated routing of object-based messages). In practice, there is also a mind shift needed from an engineering culture raised on point-to-point single-function communications (command uplink, telemetry downlink), to one in which assets are only indirectly accessed, via well-defined interfaces. We are aiming to foster a 'community of access' both among space assets and the humans who control them. This enables appropriate (perhaps eventually optimized) sharing of services and resources to the greater benefit of all participants. We envision such usage to be as automated in the future as using a cell phone is today - with all the steps in creating the real-time link being automated.

interplanetary network (IPN)↗
NASA NTRS1978

ATS-6 flight accelerometers

Five accelerometers mounted near the adapter base of the Titan 3-C launch vehicle and three on the hub of the ATS-F spacecraft provided (1) data for verifying basic spacecraft mode shapes and frequencies during powered flight while attached to the launch vehicle; (2) failure mode detection and diagnostic information on in-flight anomalies; and (3) data to be used in the design of future spacecraft to be flown on the Titan 3-C. Because data from the instruments mounted on the spacecraft hub passed through an in-flight disconnect at the separation plane between the transtage and ATS-F, the moment this connector was broken, the signal to the telemetry system showed a step function change. By monitoring these telemetry traces on the ground at appropriate times during flight sequences, a positive indication of spacecraft separation was obtained. Flight data showing dynamic response at spacecraft launch vehicle interface and at the top of ATS spacecraft during significant launch events are presented in tables.

Mattson, R.↗
NASA NTRSDate not supplied

Field Assessment of Sensorimotor Function Following Long-Duration Spaceflight

INTRODUCTION: Field assessments of functional task performance following long duration spaceflight are critical to characterize the risk associated with sensorimotor adaptation. A portable test battery involving sit-to-stand, prone-to-stand, walk and turn with obstacle, and tandem walking has been implemented during pre- and postflight testing to provide Sensorimotor Standard Measures that could be implemented in remote test locations. METHODS: To date, 19 astronauts (12 males, 7 females) participated in this study before and after 6–8-month expeditions to the International Space Station (198 ± 70 days, mean ±std). Ethics approvals were obtained, and all subjects provided informed consent. Tests were conducted preflight, within a few hours after landing, and then 1 day and 6–11 days later. Time to stability was the outcome measure for both standing tasks, time to completion and turn rate for the walk and turn task, and percent complete steps for the tandem walking (eyes open and closed). Statistical analyses included mixed effects (multi-level) generalized linear models. RESULTS: Consistent with previous Field Tests, significant effects of spaceflight were observed during the initial testing including longer times to stabilize posture when standing, longer times to complete the short obstacle walk, and fewer correct steps during tandem walking. The recovery timeline varied with task complexity, generally taking longer when either the basis of support was limited (e.g., tandem walk) and/or visual cues were deprived (eyes closed). DISCUSSION: These data suggest that additional sensorimotor-based countermeasures may be necessary to maintain functional performance during long-duration spaceflight. Maintaining core measures as new countermeasures are implemented during future missions will be instrumental in assessing their efficacy. This test battery will also serve as the basis for developing sensorimotor assessments during future space exploration.

Scott Jonathan Wood↗
NASA NTRS1996

Astrobiology Workshop: Leadership in Astrobiology

Astrobiology is defined in the 1996 NASA Strategic Plan as 'The study of the living universe.' At NASA's Ames Research Center, this endeavor encompasses the use of space to understand life's origin, evolution, and destiny in the universe. Life's origin refers to understanding the origin of life in the context of the origin and diversity of planetary systems. Life's evolution refers to understanding how living systems have adapted to Earth's changing environment, to the all-pervasive force of gravity, and how they may adapt to environments beyond Earth. Life's destiny refers to making long-term human presence in space a reality, and laying the foundation for understanding and managing changes in Earth's environment. The first Astrobiology Workshop brought together a diverse group of researchers to discuss the following general questions: Where and how are other habitable worlds formed? How does life originate? How have the Earth and its biosphere influenced each other over time? Can terrestrial life be sustained beyond our planet? How can we expand the human presence to Mars? The objectives of the Workshop included: discussing the scope of astrobiology, strengthening existing efforts for the study of life in the universe, identifying new cross-disciplinary programs with the greatest potential for scientific return, and suggesting steps needed to bring this program to reality. Ames has been assigned the lead role for astrobiology by NASA in recognition of its strong history of leadership in multidisciplinary research in the space, Earth, and life sciences and its pioneering work in studies of the living universe. This initial science workshop was established to lay the foundation for what is to become a national effort in astrobiology, with anticipated participation by the university community, other NASA centers, and other agencies. This workshop (the first meeting of its kind ever held) involved life, Earth, and space scientists in a truly interdisciplinary sharing of ideas related to life in the universe, and by all accounts was a resounding success.

DeVincenzi, D.↗
NASA NTRS2017

LPV Modeling of a Flexible Wing Aircraft Using Modal Alignment and Adaptive Gridding Methods

One of the earliest approaches in gain-scheduling control is the gridding based approach, in which a set of local linear time-invariant models are obtained at various gridded points corresponding to the varying parameters within the flight envelop. In order to ensure smooth and effective Linear Parameter-Varying control, aligning all the flexible modes within each local model and maintaining small number of representative local models over the gridded parameter space are crucial. In addition, since the flexible structural models tend to have large dimensions, a tractable model reduction process is necessary. In this paper, the notion of s-shifted H2- and H Infinity-norm are introduced and used as a metric to measure the model mismatch. A new modal alignment algorithm is developed which utilizes the defined metric for aligning all the local models over the entire gridded parameter space. Furthermore, an Adaptive Grid Step Size Determination algorithm is developed to minimize the number of local models required to represent the gridded parameter space. For model reduction, we propose to utilize the concept of Composite Modal Cost Analysis, through which the collective contribution of each flexible mode is computed and ranked. Therefore, a reduced-order model is constructed by retaining only those modes with significant contribution. The NASA Generic Transport Model operating at various flight speeds is studied for verification purpose, and the analysis and simulation results demonstrate the effectiveness of the proposed modeling approach.

LPV Modeling↗
DOE OSTI2024

Multilevel Parareal Algorithm with Averaging for Oscillatory Problems

The present study is an extension of the work done by Peddle, Haut, and Wingate and Haut and Wingate, where a two-level Parareal method with mapping and averaging is examined. The method proposed in this paper is a multilevel Parareal method with arbitrarily many levels, which is not restricted to the two-level case. We give an asymptotic error estimate which reduces to the two-level estimate for the case when only two levels are considered. Introducing more than two levels has important consequences for the averaging procedure, as we choose separate averaging windows for each of the different levels, which is an additional new feature of the present study. The different averaging windows make the proposed method especially appropriate for nonlinear multiscale problems, because we can introduce a level for each intrinsic scale of the problem and adapt the averaging procedure such that we reproduce the behavior of the model on the particular scale resolved by the level. The method is applied to nonlinear differential equations. The nonlinearities can generate a range of frequencies in the problem. The computational cost of the new method is investigated and studied on several examples.

97 MATHEMATICS AND COMPUTING↗