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All sourcesNASA NTRSDOE OSTI
NASA NTRS2014

Time and Frequency-Domain Cross-Verification of SLS 6DOF Trajectory Simulations

The SLS GNC team and its partners have developed several time- and frequency-based simulations for development and analysis of the proposed SLS launch vehicle. The simulations differ in fidelity and some have unique functionality that allows them to perform specific analyses. Some examples of the purposes of the various models are: trajectory simulation, multi-body separation, Monte Carlo, hardware in the loop, loads, and frequency domain stability analyses. While no two simulations are identical, many of the models are essentially six degree-of-freedom (6DOF) representations of the SLS plant dynamics, hardware implementation, and flight software. Thus at a high level all of those models should be in agreement. Comparison of outputs from several SLS trajectory and stability analysis tools are ongoing as part of the program's current verification effort. The purpose of these comparisons is to highlight modeling and analysis differences, verify simulation data sources, identify inconsistencies and minor errors, and ultimately to verify output data as being a good representation of the vehicle and subsystem dynamics. This paper will show selected verification work in both the time and frequency domain from the current design analysis cycle of the SLS for several of the design and analysis simulations. In the time domain, the tools that will be compared are MAVERIC, CLVTOPS, SAVANT, STARS, ARTEMIS, and POST 2. For the frequency domain analysis, the tools to be compared are FRACTAL, SAVANT, and STARS. The paper will include discussion of these tools including their capabilities, configurations, and the uses to which they are put in the SLS program. Determination of the criteria by which the simulations are compared (matching criteria) requires thoughtful consideration, and there are several pitfalls that may occur that can severely punish a simulation if not considered carefully. The paper will discuss these considerations and will present a framework for responding to these issues when they arise. For example, small event timing differences can lead to large differences in mass properties if the criteria are to measure those properties at the same time, or large differences in altitude if the criteria are to measure those properties when the simulation experiences a staging event. Similarly, a tiny difference in phase can lead to large gain margin differences for frequency-domain comparisons of gain margins.

VanZwieten, Tannen↗
NASA NTRS2017

Time and Frequency-Domain Cross-Verification of SLS 6DOF Trajectory Simulations

The Space Launch System (SLS) Guidance, Navigation, and Control (GNC) team and its partners have developed several time- and frequency-based simulations for development and analysis of the proposed SLS launch vehicle. The simulations differ in fidelity and some have unique functionality that allows them to perform specific analyses. Some examples of the purposes of the various models are: trajectory simulation, multi-body separation, Monte Carlo, hardware in the loop, loads, and frequency domain stability analyses. While no two simulations are identical, many of the models are essentially six degree-of-freedom (6DOF) representations of the SLS plant dynamics, hardware implementation, and flight software. Thus at a high level all of those models should be in agreement. Comparison of outputs from several SLS trajectory and stability analysis tools are ongoing as part of the program's current verification effort. The purpose of these comparisons is to highlight modeling and analysis differences, verify simulation data sources, identify inconsistencies and minor errors, and ultimately to verify output data as being a good representation of the vehicle and subsystem dynamics. This paper will show selected verification work in both the time and frequency domain from the current design analysis cycle of the SLS for several of the design and analysis simulations. In the time domain, the tools that will be compared are MAVERIC, CLVTOPS, SAVANT, STARS, ARTEMIS, and POST 2. For the frequency domain analysis, the tools to be compared are FRACTAL, SAVANT, and STARS. The paper will include discussion of these tools including their capabilities, configurations, and the uses to which they are put in the SLS program. Determination of the criteria by which the simulations are compared (matching criteria) requires thoughtful consideration, and there are several pitfalls that may occur that can severely punish a simulation if not considered carefully. The paper will discuss these considerations and will present a framework for responding to these issues when they arise. For example, small event timing differences can lead to large differences in mass properties if the criteria are to measure those properties at the same time, or large differences in altitude if the criteria are to measure those properties when the simulation experiences a staging event. Similarly, a tiny difference in phase can lead to large gain margin differences for frequency-domain comparisons of gain margins.

Johnson, Matthew↗
NASA NTRS2018

Modernizing and Expanding the NASA Space Geodesy Network to Meet Future Geodetic Requirements

NASA maintains and operates a global network of Very Long Baseline Interferometry (VLBI), Satellite Laser Ranging (SLR), and Global Navigation Satellite System ground stations as part of the NASA Space Geodesy Program. The NASA Space Geodesy Network (NSGN) provides the geodetic products that support Earth observations and the related science requirements as outlined by the US National Research Council (NRC in Precise geodetic infrastructure: national requirements for a shared resource, National Academies Press, Washington, 2010. http://nap.edu/12954, Thriving on our changing planet: a decadal strategy for Earth observation from space, National Academies Press, Washington, 2018. http://nap.edu/24938). The Global Geodetic Observing System (GGOS) and the NRC have set an ambitious goal of improving the Terrestrial Reference Frame to have an accuracy of 1 mm and stability of 0.1 mm per year, an order of magnitude beyond current capabilities. NASA and its partners within GGOS are addressing this challenge by planning and implementing modern geodetic stations colocated at existing and new sites around the world. In 2013, NASA demonstrated the performance of its next-generation systems at the prototype next-generation core site at NASA’s Goddard Geophysical and Astronomical Observatory in Greenbelt, Maryland. Implementation of a new broadband VLBI station in Hawaii was completed in 2016. NASA is currently implementing new VLBI and SLR stations in Texas and is planning the replacement of its other aging domestic and international legacy stations. In this article, we describe critical gaps in the current global network and discuss how the new NSGN will expand the global geodetic coverage and ultimately improve the geodetic products. We also describe the characteristics of a modern NSGN site and the capabilities of the next-generation NASA SLR and VLBI systems. Finally, we outline the plans for efficiently operating the NSGN by centralizing and automating the operations of the new geodetic stations.

Merkowitz, S. M.↗
NASA NTRSDate not supplied

Neuro-Vestibular Examination During and Following Spaceflight (Vestibular Health)

BACKGROUND Adaptation to microgravity during spaceflight causes neurological disturbances that are either directly or indirectly mediated by the vestibular system. These disturbances could include space motion sickness, spatial disorientation, and cognitive impairment, as well as changes in head-eye coordination, vestibulo-ocular reflexes, and control of posture and locomotion. Otolith-mediated reflex gains appear to adapt rapidly during spaceflight and after landing. However, animal studies have shown that structural modifications of the vestibular sensory apparatus develop during long-duration spaceflight. To date, no studies have characterized the severity of vestibular syndromes experienced by astronauts as a function of the duration of spaceflight or whether the effects are caused by changes at the peripheral end organs, midbrain, cerebellum, or vestibular cortex. OBJECTIVES We will investigate temporal vestibular changes in crewmembers of short, 6-month, and one-year missions to identify trends in adaptation of vestibular health and performance in orbit and after landing. We will also determine whether the vestibular organs and/or the central vestibular system undergo structural changes during long-duration exposure to microgravity, which could cause vestibular disorders when transitioning to a different gravitational environment. METHODS Recordings of eye, head, and body movements, as well as subjective reports of perception of motion, will be used to determine the presence of abnormal eye movements, dysmetria, motion sickness symptoms, and illusions of motion during head or body movements. This includes characterization of temporal trends in central compensation for vestibular (otolith) asymmetry. Pre-flight data will be collected 90 days before the flight. In-flight tests will be performed early in the mission and once every one or two months thereafter. Post-flight examinations will be performed on the following days after return (R) from the mission: R+0, R+4, R+9, and R+30. Ground-based control tests will be performed on healthy volunteers in the laboratory to estimate mean normative responses. CONTROL RESULTS Thirty-two healthy (non-astronaut) control subjects performed the same ground test procedures as planned for crewmembers. In addition to establishing a normative database, these data were used to calculate vestibular asymmetries from perceptual reports during unilateral centrifugation, oculomotor responses during visual-alignment tasks, vestibulo-ocular reflex gain during head-impulse tests, and body rotation during stepping tests. A significant correlation was observed between asymmetries of subjective visual vertical and verbal report during unilateral centrifugation. Another significant correlation was observed between the asymmetries of ocular alignment, vestibulo-ocular reflex gain, and body rotation. These findings in a healthy cohort may help us better understand changes in vestibular asymmetries in crewmembers during and following spaceflight. RELEVANCE If the observed symptoms in crewmembers are more deleterious after the year-long missions than those documented after 6-month missions, then relevant countermeasures will be required to maintain the health and operational performance of astronauts during longer missions. Depending on the etiology of the vestibular syndrome revealed by these tests, countermeasures will be proposed based on vestibular rehabilitation therapies currently used in patients with vestibular disorders, such as habituation, gaze stabilization, and/or balance training exercises. ACKNOWLEDGEMENT This work is supported by the NASA’s Human Research Program Human Health Countermeasures Element.

T R Macaulay↗
NASA NTRSDate not supplied

Neuro-Vestibular Examination During and Following Spaceflight (Vestibular Health)

BACKGROUND Adaptation to microgravity during spaceflight causes neurological disturbances that are either directly or indirectly mediated by the vestibular system. These disturbances can include space motion sickness, spatial disorientation, and cognitive impairment, as well as changes in head-eye coordination, vestibulo-ocular reflexes, and control of posture and locomotion. Otolith-mediated reflex gains appear to adapt rapidly during spaceflight and after landing. However, animal studies have shown that structural modifications of the vestibular sensory apparatus develop during long-duration spaceflight. To date, no studies have characterized the severity of vestibular syndromes experienced by astronauts as a function of the duration of spaceflight or whether the effects are caused by changes at the peripheral end organs, midbrain, cerebellum, or vestibular cortex. OBJECTIVES We will investigate temporal vestibular changes in crewmembers of short, 6-month, and one-year missions to identify trends in adaptation of vestibular health and performance in orbit and after landing. We will also determine whether the vestibular organs and/or the central vestibular system undergo structural changes during long-duration exposure to microgravity, which could cause vestibular disorders when transitioning to a different gravitational environment. METHODS Recordings of eye, head, and body movements, as well as subjective reports of perception of motion, will be used to determine the presence of abnormal eye movements, dysmetria, motion sickness symptoms, and illusions of motion during head or body movements. This includes characterization of temporal trends in central compensation for vestibular (otolith) asymmetry. Pre-flight data will be collected 90 days before launch. In-flight examinations will be performed early in the mission (Flight Days 1 and 30) and once every two or three months thereafter. Post-flight examinations will be performed on the following days after return (R) from the mission: R+0, R+4, R+9, and R+30. Ground-based control tests have been performed on healthy volunteers in the laboratory to estimate mean normative responses. RESULTS Data collection for this study is ongoing. Data processing techniques are being refined. Our eye tracking method for measuring three-dimensional eye movement takes advantage of modern computer vision software (OpenCV) and advances in the field of iris recognition to improve measurement of ocular-counterolling. RELEVANCE If the observed symptoms in crewmembers are more deleterious after the year-long missions than those documented after 6-month missions, then relevant countermeasures will be required to maintain the health and operational performance of astronauts during longer missions. Depending on the etiology of the vestibular syndrome revealed by these tests, countermeasures will be proposed based on vestibular rehabilitation therapies currently used in patients with vestibular disorders, such as habituation, gaze stabilization, and/or balance training exercises.

T R Macaulay↗
NASA NTRSDate not supplied

Neuro-Vestibular Examination During and Following Spaceflight (Vestibular Health)

BACKGROUND Adaptation to microgravity during spaceflight causes neurological disturbances that are either directly or indirectly mediated by the vestibular system. These disturbances can include space motion sickness, spatial disorientation, and cognitive impairment, as well as changes in head-eye coordination, vestibulo-ocular reflexes, and control of posture and locomotion. Otolith-mediated reflex gains appear to adapt rapidly during spaceflight and after landing. However, animal studies have shown that structural modifications of the vestibular sensory apparatus develop during long-duration spaceflight. To date, no studies have characterized the severity of vestibular syndromes experienced by astronauts as a function of the duration of spaceflight or whether the effects are caused by changes at the peripheral end organs, midbrain, cerebellum, or vestibular cortex. OBJECTIVES We are investigating temporal vestibular changes in crewmembers of short, 6-month, and one-year missions to identify trends in adaptation of vestibular health and performance in orbit and after landing. We are also differentiating between peripheral and central vestibular forms of vertigo and oculomotor disorders. METHODS Recordings of eye, head, and body movements, as well as subjective reports of perception of motion, are being used to determine the presence of abnormal eye movements, dysmetria, motion sickness symptoms, and illusions of motion during head or body movements. This includes characterization of temporal trends in central compensation for vestibular (otolith) asymmetry. In-flight examinations are being performed early in the mission (Flight Days 1 and 30) and once every 2-3 months thereafter. Postflight examinations are performed after return (R) from the mission on R+0, R+4, R+9, and R+30. The inflight and postflight motion sickness questionnaires are customized to support data sharing across related studies. Ground-based control testing has been performed on healthy volunteers(18 females, 14 males;38.6 ± 9.2 years) in the laboratory to estimate mean normative responses, and on patients with bilateral vestibulopathy (BVP) (17 females, 13 males; 60.6 ± 13.0 years) at the University of Caen. RESULTS As of September 2024, two crewmembers have completed all preflight, inflight, and postflight testing. Additional crewmembers are currently enrolled and data collection is currently ongoing. For ground testing, BVP patients performed similarly to previous postflight astronauts on R+0 in various walking performance tasks. Three additional body movement perception tasks have been tested. For the Triangle Completion Task, BVP patients had a larger mean angle of deviation and longer mean distance than healthy controls. For the Self-Rotation Task and Distance Perception Task, BVP patients had larger errors than healthy controls. These data suggest that vestibular deficiencies impact all aspects of body movement perception tested; whereas previous studies suggest that vestibular deficiencies are only associated with directional errors, not with overall trajectories/path lengths. These data will be compared to those of crewmembers during early postflight readaptation. RELEVANCE If the observed symptoms in crewmembers are more deleterious after the year-long missions than those documented after 6-month missions, then relevant countermeasures will be required to maintain the health and operational performance of astronauts during longer missions. Depending on the etiology of the vestibular syndrome revealed by these tests, countermeasures will be proposed based on vestibular rehabilitation therapies currently used in patients with vestibular disorders, such as habituation, gaze stabilization, and/or balance training exercises. ACKNOWLEDGEMENT This work is supported by NASA’s Human Research Program Human Health Countermeasures Element.

T R Macaulay↗
NASA NTRS1992

Numerical Modeling of a Vortex Stabilized Arcjet

Arcjet thrusters are being actively considered for use in Earth orbit maneuvering applications. Experimental studies are currently the chief means of determining an optimal thruster configuration. Earlier numerical studies have failed to include all of the effects found in typical arcjets including complex geometries, viscosity, and swirling flow. Arcjet geometries are large area ratio converging nozzles with centerbodies in the subsonic portion of the nozzle. The nozzle walls serve as the anode while the centerbody functions as the cathode. Viscous effects are important because the Reynolds number, based on the throat radius, is typically less than 1,000. Experimental studies have shown that a swirl or circumferential velocity component stabilizes a constricted arc. This dissertation describes the equations governing flow through a constricted arcjet thruster. An assumption that the flowfield is in local thermodynamic equilibrium leads to a single fluid plasma temperature model. An order of magnitude analysis reveals the governing fluid mechanics equations are uncoupled from the electromagnetic field equations. A numerical method is developed to solve the governing fluid mechanics equations, the Thin Layer Navier-Stokes equations. A coordinate transformation is employed in deriving the governing equations to simplify the application of boundary conditions in complex geometries. An axisymmetric formulation is employed to include the swirl velocity component as well as the axial and radial velocity components. The numerical method is an implicit finite-volume technique and allows for large time steps to reach a converged steady-state solution. The inviscid fluxes are flux-split, and Gauss-Seidel line relaxation is used to accelerate convergence. Converging-diverging nozzles with exit-to-throat area ratios up to 100:1 and annular nozzles were examined. Quantities examined included Mach number and static wall pressure distributions, and oblique shock structures. As the level of swirl and viscosity in the flowfield increased the mass flow rate and thrust decreased. The technique was used to predict the flow through a typical arcjet thruster geometry. Results indicate swirl and viscosity play an important role in the complex geometry of an arcjet.

Pawlas, Gary E.↗
NASA NTRS1991

Modeling the growth phase of a substorm using the Tsyganenko model and multi-spacecraft observations - CDAW-9

The CDAW-9 Event C focused upon the early part of 3 May 1986 when a large substorm onset occurred at 0111 UT. By modifying the Tsyganenko 1989 magnetic field model, a model is constructed in which the near-earth current systems are enhanced with time to describe the observed development of the tail magnetic field during the growth phase. The cross-tail current intensity and the thickness of the current sheet are determined by comparison with three spacecraft in the near-earth tail. The location of the auroral bulge as recorded by the Viking imager is mapped to the equatorial current sheet. The degree of chaotization of the thermal electrons is estimated, and the consequences to the tail stability towards on tearing are discussed. It is concluded that the mapping of the brightening region in the auroral oval corresponds to the regions in the tail where the current sheet may be unstable towards ion tearing.

Pulkkinen, T. I.↗
NASA NTRS2020

Enhanced Eddy Activity in the Beaufort Gyre in Response to Sea Ice Loss

The Beaufort Gyre freshwater content has increased since the 1990s, potentially stabilizing in recent years. The mechanisms proposed to explain the stabilization involve either mesoscale eddy activity that opposes Ekman pumping or the reduction of Ekman pumping due to reduced sea ice?ocean surface stress. However, the relative importance of these mechanisms is unclear. Here, we present observational estimates of the Beaufort Gyre mechanical energy budget and show that energy dissipation and freshwater content stabilization by eddies increased in the late-2000s. The loss of sea ice and acceleration of ocean currents after 2007 resulted in enhanced mechanical energy input but without corresponding increases in potential energy storage. To balance the energy surplus, eddy dissipation and its role in gyre stabilization must have increased after 2007. Our results imply that declining Arctic sea ice will lead to an increasingly energetic Beaufort Gyre with eddies playing a greater role in its stabilization.

Armitage, Thomas W.K.↗
NASA NTRS1998

High Stability Engine Control (HISTEC): Flight Demonstration Results

Future aircraft turbine engines, both commercial and military, must be able to accommodate expected increased levels of steady-state and dynamic engine-face distortion. The current approach of incorporating sufficient design stall margin to tolerate these increased levels of distortion would significantly reduce performance. The High Stability Engine Control (HISTEC) program has developed technologies for an advanced, integrated engine control system that uses measurement- based estimates of distortion to enhance engine stability. The resulting distortion tolerant control reduces the required design stall margin, with a corresponding increase in performance and/or decrease in fuel burn. The HISTEC concept was successfully flight demonstrated on the F-15 ACTIVE aircraft during the summer of 1997. The flight demonstration was planned and carried out in two parts, the first to show distortion estimation, and the second to show distortion accommodation. Post-flight analysis shows that the HISTEC technologies are able to successfully estimate and accommodate distortion, transiently setting the stall margin requirement on-line and in real-time. Flight demonstration of the HISTEC technologies has significantly reduced the risk of transitioning the technology to tactical and commercial engines.

Delaat, John C.↗
NASA NTRS1980

Engine component improvement: Performance improvement, JT9D-7 3.8 AR fan

A redesigned, fuel efficient fan for the JT9D-7 engine was tested. Tests were conducted to determine the effect of the 3.8 AR fan on performance, stability, operational characteristics, and noise of the JT9D-7 engine relative to the current 4.6 AR Bill-of-Material fan. The 3.8 AR fan provides increased fan efficiency due to a more advanced blade airfoil with increased chord, eliminating one part span shroud and reducing the number of fan blades and fan exit guide vanes. Engine testing at simulated cruise conditions demonstrated the predicted 1.3 percent improvement in specific fuel consumption with the redesigned 3.8 AR fan. Flight testing and sea level stand engine testing demonstrated exhaust gas temperature margins, fan and low pressure compressor stability, operational suitability, and noise levels comparable to the Bill-of-Material fan.

Gaffin, W. O.↗
NASA NTRS1990

Science observations with the IUE using the one-gyro mode

The International Ultraviolet Explorer (IUE) attitude control system originally included an inertial reference package containing six gyroscopes for three axis stabilization. The science instrument includes a prime and redundant Field Error Sensor (FES) camera for target acquisition and offset guiding. Since launch, four of the six gyroscopes have failed. The current attitude control system utilizes the remaining two gyros and a Fine Sun Sensor (FSS) for three axis stabilization. When the next gyro fails, a new attitude control system will be uplinked which will rely on the remaining gyro and the FSS for general three axis stabilization. In addition to the FSS, the FES cameras will be required to assist in maintaining fine attitude control during target acquisition. This has required thoroughly determining the characteristics of the FES cameras and the spectrograph aperture plate as well as devising new target acquisition procedures. The results of this work are presented.

Imhoff, C.↗
NASA NTRS1990

Science observations with the IUE using the one-gyro mode

The International Ultraviolet Explorer (IUE) attitude control system originally included an inertial reference package containing six gyroscopes for three axis stabilization. The science instrument includes a prime and redundant Field Error Sensor (FES) camera for target acquisition and offset guiding. Since launch, four of the six gyroscopes have failed. The current attitude control system utilizes the remaining two gyros and a Fine Sun Sensor (FSS) for three axis stabilization. When the next gyro fails, a new attitude control system will be uplinked, which will relay on the remaining gyro and the FSS for general three axis stabilization. In addition to the FSS, the FES cameras will be required to assist in maintaining fine attitude control during target acquisition. This has required thoroughly determining the characteristics of the FES cameras and the spectrograph aperture plate as well as devising new target acquisition procedures. The results of this work are presented.

Imhoff, C.↗
NASA NTRS1985

Effects of bearing deadbands on bearing loads and rotor stability

The Jeffcott model of a turbopump which was modified by adding deadband (clearance) effects along with fluid seal forces are currently understood. Equations of motion for the model were written in polar coordinates, and a constant side force was added to the model to account for the likely misalignment between bearings and seals. The force models and system equations of motion are described as well as limit cycle, and stability analyses. Studies of systems with rotor imbalance and with or without side effects were studied for three types of motion. Results show that: (1) deadband does not affect stability-in-the-large; (2) stability-in-the small is enhanced by deadband and side force; (3) bearings loads are highest for motion with synchronous or nonsynchronous periodic enclosing origin; (4) side force acting in concert with deadband effects may either increase or decrease bearing loads; and (5) bearing loads in a stable pump are determined primarily by rotor imbalance and side forces.

Glaese, J. R.↗
NASA NTRS2015

Functional Foods Baseline and Requirements Analysis

Current spaceflight foods were evaluated to determine if their nutrient profile supports positioning as a functional food and if the stability of the bioactive compound within the food matrix over an extended shelf-life correlated with the expected storage duration during the mission. Specifically, the research aims were: Aim A. To determine the amount of each nutrient in representative spaceflight foods immediately after processing and at predetermined storage time to establish the current nutritional state. Aim B. To identify the requirements to develop foods that stabilize these nutrients such that required concentrations are maintained in the space food system throughout long duration missions (up to five years). Aim C. To coordinate collaborations with health and performance groups that may require functional foods as a countermeasure.

Cooper, M. R.↗
NASA NTRS1996

High Stability Engine Control (HISTEC)

Future aircraft turbine engines, both commercial and military, must be able to successfully accommodate expected increased levels of steady-state and dynamic engine-face distortion. The current approach of incorporating a sufficient component design stall margin to tolerate these increased levels of distortion would significantly reduce performance. The objective of the High Stability Engine Control (HISTEC) program is to design, develop, and flight demonstrate an advanced, high-stability, integrated engine control system that uses measurement-based, real-time estimates of distortion to enhance engine stability. The resulting distortion tolerant control reduces the required design stall margin, with a corresponding increase in performance and decrease in fuel burn. The HISTEC concept, consisting of a Distortion Estimation System and a Stability Management Control, has been designed and developed. The Distortion Estimation System uses a small number of high-response pressure sensors at the engine face to calculate indicators of the type and extent of distortion in real time. The Stability Management Control, through direct control of the fan and compressor pressure ratio, accommodates the distortion by transiently increasing the amount of stall margin available based on information from the Distortion Estimation System. Simulation studies have shown the HISTEC distortion tolerant control is able to successfully estimate and accommodate time-varying distortion. Currently, hardware and software systems necessary for flight demonstration of the HISTEC concept are being designed and developed. The HISTEC concept will be flight tested in early 1997.

DeLaat, John C.↗
NASA NTRS1992

Numerical simulation of conservation laws

A new numerical framework for solving conservation laws is being developed. This new approach differs substantially from the well established methods, i.e., finite difference, finite volume, finite element and spectral methods, in both concept and methodology. The key features of the current scheme include: (1) direct discretization of the integral forms of conservation laws, (2) treating space and time on the same footing, (3) flux conservation in space and time, and (4) unified treatment of the convection and diffusion fluxes. The model equation considered in the initial study is the standard one dimensional unsteady constant-coefficient convection-diffusion equation. In a stability study, it is shown that the principal and spurious amplification factors of the current scheme, respectively, are structurally similar to those of the leapfrog/DuFort-Frankel scheme. As a result, the current scheme has no numerical diffusion in the special case of pure convection and is unconditionally stable in the special case of pure diffusion. Assuming smooth initial data, it will be shown theoretically and numerically that, by using an easily determined optimal time step, the accuracy of the current scheme may reach a level which is several orders of magnitude higher than that of the MacCormack scheme, with virtually identical operation count.

Chang, Sin-Chung↗