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NASA NTRS2002

A Numerical Scheme for Ordinary Differential Equations Having Time Varying and Nonlinear Coefficients Based on the State Transition Matrix

A variable order method of integrating initial value ordinary differential equations that is based on the state transition matrix has been developed. The method has been evaluated for linear time variant and nonlinear systems of equations. While it is more complex than most other methods, it produces exact solutions at arbitrary time step size when the time variation of the system can be modeled exactly by a polynomial. Solutions to several nonlinear problems exhibiting chaotic behavior have been computed. Accuracy of the method has been demonstrated by comparison with an exact solution and with solutions obtained by established methods.

Bartels, Robert E.↗
NASA NTRS2004

Comparison of Numerical Schemes for a Realistic Computational Aeroacoustics Benchmark Problem

In this work, a nonlinear structured-multiblock CAA solver, the NASA GRC BASS code, will be tested on a realistic CAA benchmark problem. The purpose of this test is to ascertain what effect the high-accuracy solution methods used in CAA have on a realistic test problem, where both the mean flow and the unsteady waves are simultaneously computed on a fully curvilinear grid from a commercial grid generator. The proposed test will compare the solutions obtained using several finite-difference methods on identical grids to determine whether high-accuracy schemes have advantages for this benchmark problem.

Hixon, R.↗
NASA NTRS1993

Solution of the linear shallow water equations by the fourth-order leapfrog scheme

Numerical schemes of the first and second order of approximation introduce numerical distortion when the wave propagation over a long distance is investigated. To alleviate this problem, the fourth-order leapfrog scheme is constructed. The standard leapfrog method is based on the truncated Taylor series expansion which depicts an error proportional to the second-order terms. In the proposed method the numerical solution is corrected for these terms. The space and time corrections work well in diminishing numerical dispersion and dissipation.

Kowalik, Z.↗
NASA NTRS1999

Implicit Space-Time Conservation Element and Solution Element Schemes

Artificial numerical dissipation is in important issue in large Reynolds number computations. In such computations, the artificial dissipation inherent in traditional numerical schemes can overwhelm the physical dissipation and yield inaccurate results on meshes of practical size. In the present work, the space-time conservation element and solution element method is used to construct new and accurate implicit numerical schemes such that artificial numerical dissipation will not overwhelm physical dissipation. Specifically, these schemes have the property that numerical dissipation vanishes when the physical viscosity goes to zero. These new schemes therefore accurately model the physical dissipation even when it is extremely small. The new schemes presented are two highly accurate implicit solvers for a convection-diffusion equation. The two schemes become identical in the pure convection case, and in the pure diffusion case. The implicit schemes are applicable over the whole Reynolds number range, from purely diffusive equations to convection-dominated equations with very small viscosity. The stability and consistency of the schemes are analysed, and some numerical results are presented. It is shown that, in the inviscid case, the new schemes become explicit and their amplification factors are identical to those of the Leapfrog scheme. On the other hand, in the pure diffusion case, their principal amplification factor becomes the amplification factor of the Crank-Nicolson scheme.

Chang, Sin-Chung↗
NASA NTRS1997

The Implicit and Explicit alpha-mu Schemes

Artificial numerical dissipation is an important issue in large Reynolds number computations. In such computations, the artificial dissipation inherent in traditional numerical schemes can overwhelm the physical dissipation and yield inaccurate results on meshes of practical size. In the present work, the space-time conservation element and solution element method is used to construct new and accurate numerical schemes such that artificial numerical dissipation will not overwhelm physical dissipation. Specifically, these schemes have the property that numerical dissipation vanishes when the physical viscosity goes to zero. These new schemes therefore accurately model the physical dissipation even when it is extremely small. The method of space-time conservation element and solution element, currently under development, is a nontraditional numerical method for solving conservation laws. The method is developed on the basis of local and global flux conservation in a space-time domain, in which space and time are treated in a unified manner. Explicit solvers for model and fluid dynamic conservation laws have previously been investigated. In this paper, we introduce a new concept in the design of implicit schemes, and use it to construct two highly accurate solvers for a convection-diffusion equation. The two schemes become identical in the pure convection case, and in the pure diffusion case. The implicit schemes are applicable over the whole Reynolds number range, from purely diffusive equations to purely inviscid (convective) equations. The stability and consistency of the schemes are analyzed, and some numerical results are presented. It is shown that, in the inviscid case, the new schemes become explicit and their amplification factors are identical to those of the Leapfrog scheme. On the other hand, in the pure diffusion case, their principal amplification factor becomes the amplification factor of the Crank-Nicolson scheme. We also construct an explicit solver with the treatment of diffusion being based on that in the implicit solvers. The explicit solver has only a CFL stability limitation on the Courant number, yet it retains the second-order spatial accuracy of the implicit schemes.

Chang, Sin-Chung↗
NASA NTRS1992

Numerical experiments on a new class of nonoscillatory schemes

Numerical experiments for the SONIC schemes on 2D inviscid, compressible, steady, and unsteady problems are presented. These schemes belong to a new class of uniformly second-order accurate nonoscillatory schemes introduced by Huynh, with the well known UNO2 scheme of Harten and Osher being the most 'diffusive' in this class. The SONIC schemes can also be considered as uniformly second order accurate extensions of the popular TVD schemes. For simplicity, a MUSCL approach for spatial discretization and a Runge-Kutta method for time integration are used. Test problems include steady oblique shock reflection and the well known unsteady double Mach reflection problem. Results confirm that the SONIC schemes are more accurate than their TVD counterparts.

Suresh, Ambady↗
NASA NTRS1984

A numerical transport scheme which avoids negative mixing ratios

A new scheme for numerically integrating the transport equation has been developed. This square root method avoids the problem of negative mixing ratios by using the square root of the concentration instead of the concentration itself as an advective time step. Conservation of total mass at every time step is achieved by extending the concept of quadratic conservation to the numerical time integration method. A few time schemes that fulfill the requirement of step-by-step quadratic conservation are described, the simplest of which is the modified Lax-Wendroff method.

Schneider, H.-R.↗
NASA NTRS2015

Benefits of a 4th Ice Class in the Simulated Radar Reflectivities of Convective Systems Using a Bulk Microphysics Scheme

Numerous cloud microphysical schemes designed for cloud and mesoscale models are currently in use, ranging from simple bulk to multi-moment, multi-class to explicit bin schemes. This study details the benefits of adding a 4th ice class (hail) to an already improved 3-class ice bulk microphysics scheme developed for the Goddard Cumulus Ensemble model based on Rutledge and Hobbs (1983,1984). Besides the addition and modification of several hail processes from Lin et al. (1983), further modifications were made to the 3-ice processes, including allowing greater ice super saturation and mitigating spurious evaporationsublimation in the saturation adjustment scheme, allowing graupelhail to become snow via vapor growth and hail to become graupel via riming, and the inclusion of a rain evaporation correction and vapor diffusivity factor. The improved 3-ice snowgraupel size-mapping schemes were adjusted to be more stable at higher mixing rations and to increase the aggregation effect for snow. A snow density mapping was also added. The new scheme was applied to an intense continental squall line and a weaker, loosely-organized continental case using three different hail intercepts. Peak simulated reflectivities agree well with radar for both the intense and weaker case and were better than earlier 3-ice versions when using a moderate and large intercept for hail, respectively. Simulated reflectivity distributions versus height were also improved versus radar in both cases compared to earlier 3-ice versions. The bin-based rain evaporation correction affected the squall line case more but did not change the overall agreement in reflectivity distributions.

bulk microphysics schemes↗
NASA NTRS1990

Numerical experiments on the accuracy of ENO and modified ENO schemes

Numerical experiments have been performed using different ENO schemes and different time discretizations, in order to assess accuracy-degeneracy phenomena of the type described by Rogerson and Meiburg (1990). A modified ENO scheme recovers the correct order of accuracy for all the test problems with smooth initial conditions and gives results comparable to the original ENO schemes for discontinuous problems. It is concluded that ENO schemes may lose the full high-order accuracy predicted by local truncation-error analysis for scalar linear conservation laws with smooth initial conditions. A modified ENO scheme can overcome this accuracy-degeneracy problem for the test problems without increasing the computational cost.

Shu, Chi-Wang↗
NASA NTRS1985

A factored implicit scheme for numerical weather prediction

An implicit method is proposed to factor the nonlinear partial differential equations governing fast and slow modes of dynamic motion in numerical weather prediction schemes. The method permits separate factorization of the slow and fast modes of the implicit operator. A simple two-dimensional version of the system of three-dimensional equations governing atmospheric dynamics over shallow water was analyzed to assess the accuracy of the proposed method. It is shown that the method has a small error which is comparable to other discretization errors in the overall scheme.

Augenbaum, J. M.↗
NASA NTRS2005

3 Lectures: "Lagrangian Models", "Numerical Transport Schemes", and "Chemical and Transport Models"

The topics for the three lectures for the Canadian Summer School are Lagrangian Models, numerical transport schemes, and chemical and transport models. In the first lecture I will explain the basic components of the Lagrangian model (a trajectory code and a photochemical code), the difficulties in using such a model (initialization) and show some applications in interpretation of aircraft and satellite data. If time permits I will show some results concerning inverse modeling which is being used to evaluate sources of tropospheric pollutants. In the second lecture I will discuss one of the core components of any grid point model, the numerical transport scheme. I will explain the basics of shock capturing schemes, and performance criteria. I will include an example of the importance of horizontal resolution to polar processes. We have learned from NASA's global modeling initiative that horizontal resolution matters for predictions of the future evolution of the ozone hole. The numerical scheme will be evaluated using performance metrics based on satellite observations of long-lived tracers. The final lecture will discuss the evolution of chemical transport models over the last decade. Some of the problems with assimilated winds will be demonstrated, using satellite data to evaluate the simulations.

Anne Douglass↗
NASA NTRS1995

Some Numerical Experiments on Detonation Wave Propagation

In this paper we present the results of a series of numerical experiments done on the propagation and initiation of a detonation wave. The calculations are performed in one-dimension, with considerable grid resolution. Of particular interest are the following questions: (1) the nature of periodic and chaotic instabilities generated by the wave; (2) the influence of the grid resolution on these instabilities; (3) the influence of the 'quality' of the numerical scheme; and (4) the influence of 'noise'. In the calculations, we use a second-order Total Variation Diminishing (TVD) scheme as the basic numerical method, with grid spacings as low as a fraction of a micron. Detonations waves are generated at the closed end of a tube, and allowed to propagate for approximately 20 cm. The required energy for successful initiation of the detonation will be measured for different cases of grid resolution and numerical schemes. A modified version of the TVD scheme has also been devised, which allows for much lower numerical diffusion of the radical species in the exponentially growing region behind the shock. The effect of this modification will be demonstrated. Oscillations in peak pressure and induction length are seen to develop in some cases: the oscillations can go through a sequence of modes, from a regular, high frequency mode to a low frequency mode with period doubling. A chaotic regime can also be obtained. General conclusions on the quality of algorithms will be presented. We will also discuss the performance of a version of the code developed on the IBM SP2 parallel computer.

Cambier, Jean-Luc↗
NASA NTRS1993

Flow interaction experiment. Volume 1: Aerothermal modeling, phase 2

An experimental and computational study is reported for the flow of a turbulent jet discharging into a rectangular enclosure. The experimental configurations consisting of primary jets only, annular jets only, and a combination of annular and primary jets are investigated to provide a better understanding of the flow field in an annular combustor. A laser Doppler velocimeter is used to measure mean velocity and Reynolds stress components. Major features of the flow field include recirculation, primary and annular jet interaction, and high turbulence. A significant result from this study is the effect the primary jets have on the flow field. The primary jets are seen to create statistically larger recirculation zones and higher turbulence levels. In addition, a technique called marker nephelometry is used to provide mean concentration values in the model combustor. Computations are performed using three levels of turbulence closures, namely k-epsilon model, algebraic second moment (ASM), and differential second moment (DSM) closure. Two different numerical schemes are applied. One is the lower-order power-law differencing scheme (PLDS) and the other is the higher-order flux-spline differencing scheme (FSDS). A comparison is made of the performance of these schemes. The numerical results are compared with experimental data. For the cases considered in this study, the FSDS is more accurate than the PLDS. For a prescribed accuracy, the flux-spline scheme requires a far fewer number of grid points. Thus, it has the potential for providing a numerical error-free solution, especially for three-dimensional flows, without requiring an excessively fine grid. Although qualitatively good comparison with data was obtained, the deficiencies regarding the modeled dissipation rate (epsilon) equation, pressure-strain correlation model, and the inlet epsilon profile and other critical closure issues need to be resolved before one can achieve the degree of accuracy required to analytically design combustion systems.

Nikjooy, M.↗
NASA NTRS1993

Flow interaction experiment. Volume 2: Aerothermal modeling, phase 2

An experimental and computational study is reported for the flow of a turbulent jet discharging into a rectangular enclosure. The experimental configurations consisting of primary jets only, annular jets only, and a combination of annular and primary jets are investigated to provide a better understanding of the flow field in an annular combustor. A laser Doppler velocimeter is used to measure mean velocity and Reynolds stress components. Major features of the flow field include recirculation, primary and annular jet interaction, and high turbulence. A significant result from this study is the effect the primary jets have on the flow field. The primary jets are seen to create statistically larger recirculation zones and higher turbulence levels. In addition, a technique called marker nephelometry is used to provide mean concentration values in the model combustor. Computations are performed using three levels of turbulence closures, namely k-epsilon model, algebraic second moment (ASM), and differential second moment (DSM) closure. Two different numerical schemes are applied. One is the lower-order power-law differencing scheme (PLDS) and the other is the higher-order flux-spline differencing scheme (FSDS). A comparison is made of the performance of these schemes. The numerical results are compared with experimental data. For the cases considered in this study, the FSDS is more accurate than the PLDS. For a prescribed accuracy, the flux-spline scheme requires a far fewer number of grid points. Thus, it has the potential for providing a numerical error-free solution, especially for three-dimensional flows, without requiring an excessively fine grid. Although qualitatively good comparison with data was obtained, the deficiencies regarding the modeled dissipation rate (epsilon) equation, pressure-strain correlation model, and the inlet epsilon profile and other critical closure issues need to be resolved before one can achieve the degree of accuracy required to analytically design combustion systems.

Nikjooy, M.↗
NASA NTRS1995

Modeling of nonequilibrium space plasma flows

Godunov-type numerical solution of the 20 moment plasma transport equations. One of the centerpieces of our proposal was the development of a higher order Godunov-type numerical scheme to solve the gyration dominated 20 moment transport equations. In the first step we explored some fundamental analytic properties of the 20 moment transport equations for a low b plasma, including the eigenvectors and eigenvalues of propagating disturbances. The eigenvalues correspond to wave speeds, while the eigenvectors characterize the transported physical quantities. In this paper we also explored the physically meaningful parameter range of the normalized heat flow components. In the second step a new Godunov scheme type numerical method was developed to solve the coupled set of 20 moment transport equations for a quasineutral single-ion plasma. The numerical method and the first results were presented at several national and international meetings and a paper describing the method has been published in the Journal of Computational Physics. To our knowledge this is the first numerical method which is capable of producing stable time-dependent solutions to the full 20 (or 16) moment set of transport equations, including the full heat flow equation. Previous attempts resulted in unstable (oscillating) solutions of the heat flow equations. Our group invested over two man-years into the development and implementation of the new method. The present model solves the 20 moment transport equations for an ion species and thermal electrons in 8 domain extending from a collision dominated to a collisionless region (200 km to 12,000 km). This model has been applied to study O+ acceleration due to Joule heating in the lower ionosphere.

Gombosi, Tamas↗
NASA NTRS1998

Effect of Swirl on Turbulent Structures in Supersonic Jets

Direct Numerical Simulation (DNS) is used to study the mechanism of generation and evolution of turbulence structures in a temporally evolving supersonic swirling round jet and also to examine the resulting acoustic radiations. Fourier spectral expansions are used in the streamwise and azimuthal directions and a 1-D b-spline Galerkin representation is used in the radial direction. Spectral-like accuracy is achieved using this numerical scheme. Direct numerical simulations, using the b-spline spectral method, are carried out starting from mean flow initial conditions which are perturbed by the most unstable linear stability eigenfunctions. It is observed that the initial helical instability waves evolve into helical vortices which eventually breakdown into smaller scales of turbulence. 'Rib' structures similar to those seen in incompressible mixing layer flow of Rogers and Moserl are observed. The jet core breakdown stage exhibits increased acoustic radiations.

Rao, Ram Mohan↗