Finite Difference Theory Research Articles

A Novel Connection Element Method for Multiscale Numerical Simulation of Two-Phase Flow in Fractured Reservoirs

Summary This paper presents a novel approach to the numerical simulation of fractured reservoirs, called the connection element method (CEM), which differs from traditional grid-based methods. The reservoir computational domain is discretized into a series of nodes, and a system of connection elements is constructed based on the given connection lengths and angles. The pressure diffusion term is approximated using generalized finite difference theory. Meanwhile, the transmissibility and volume of the connection elements are determined, and pressure equations are solved discretely to obtain pressure at nodes to approximate the upstream flux along connection elements. Then, we solve the transport equation to obtain oil saturation profiles with low numerical diffusion, utilizing the discontinuous Galerkin (DG) method. Moreover, the flow path tracking algorithm is introduced to quantify the flow allocation factors between wells. In all, the pressure equation can be solved at a global coarse-scale point cloud and the saturation equation is calculated at a local fine-scale connection element. In other words, CEM is of multiscale characteristics relatively. Finally, several numerical examples are implemented to demonstrate that CEM can achieve a relatively better balance between computational accuracy and efficiency compared with embedded discrete fracture modeling (EDFM). Furthermore, CEM adopts flexible meshless nodes instead of grids with strong topology, making it more practical to handle complex reservoir geometry such as fractured reservoirs.

Heat Transfer-Inspired Network for Image Super-Resolution Reconstruction.

Image super-resolution (SR) is a critical image preprocessing task for many applications. How to recover features as accurately as possible is the focus of SR algorithms. Most existing SR methods tend to guide the image reconstruction process with gradient maps, frequency perception modules, etc. and improve the quality of recovered images from the perspective of enhancing edges, but rarely optimize the neural network structure from the system level. In this article, we conduct an in- depth exploration for the inner nature of the SR network structure. In light of the consistency between thermal particles in the thermal field and pixels in the image domain, we propose a novel heat-transfer-inspired network (HTI-Net) for image SR reconstruction based on the theoretical basis of heat transfer. With the finite difference theory, we use a second-order mixed-difference equation to redesign the residual network (ResNet), which can fully integrate multiple information to achieve better feature reuse. In addition, according to the thermal conduction differential equation (TCDE) in the thermal field, the pixel value flow equation (PVFE) in the image domain is derived to mine deep potential feature information. The experimental results on multiple standard databases demonstrate that the proposed HTI-Net has superior edge detail reconstruction effect and parameter performance compared with the existing SR methods. The experimental results on the microscope chip image (MCI) database consisting of realistic low-resolution (LR) and high-resolution (HR) images show that the proposed HTI-Net for image SR reconstruction can improve the effectiveness of the hardware Trojan detection system.

Implementation of a constitutive model for anisotropic rocks based on modified Lade failure criterion

The strength anisotropy for inherent anisotropic rocks presents a challenge to simulate considering the influence of intermediate principal stress. Based on the modified Lade failure criterion, a new anisotropic modified Lade criterion combining an empirical equation is established and verified using published experimental datasets of anisotropic rocks. The incremental constitutive model for anisotropic rocks is derived by using finite difference theory. The dynamic link library (DLL) module of the model is obtained by using the VC++ program, and then the model is validated against various laboratory test results. It is concluded that the proposed criterion and derived model can describe the strength anisotropy of inherent anisotropic rocks well.

Structure preserving algorithms for mathematical model of auto-catalytic glycolysis chemical reaction and numerical simulations

This paper aims to develop positivity preserving splitting techniques for glycolysis reaction–diffusion chemical model. The positivity of state variables in the glycolysis model is an essential property that must be preserved for all choices of parameters. We propose two splitting methods that remain dynamically consistent with the continuous glycolysis reaction–diffusion model. The proposed methods converge to a true steady-state or fixed point under the given condition. On contrary to the classical operator splitting finite difference methods, we use nonstandard finite difference theory to propose a new class of operator splitting techniques.

Деякі застосування елементів теорії скінченних границь до розв'язування задач з математичного аналізу

У роботі проаналізовано доцільність використання апарату теорії скінченних різниць для обчислення сум. Наведено приклади знаходження сум, що ґрунтуються на застосуванні властивостей різницевого та антирізницевого оператора. Вказано відмінності та спільні риси між властивостями розв'язків найпростіших різницевих та диференціальних рівнянь. З’ясовано переваги та недоліки знаходження загального члена послідовності чисел Фібоначчі за допомогою рекурентного співвідношення та як розв'язку відповідного різницевого рівняння

Multi-hole seismic modeling in 3-D space and cross-hole seismic tomography analysis for boulder detection

A boulder stone, a common geological feature in south China, is referred to the remnant of a granite body which has been unevenly weathered. Undetected boulders could adversely impact the schedule and safety of subway construction when using tunnel boring machine (TBM) method. Therefore, boulder detection has always been a key issue demanded to be solved before the construction. Nowadays, cross-hole seismic tomography is a high resolution technique capable of boulder detection, however, the method can only solve for velocity in a 2-D slice between two wells, and the size and central position of the boulder are generally difficult to be accurately obtained. In this paper, the authors conduct a multi-hole wave field simulation and characteristic analysis of a boulder model based on the 3-D elastic wave staggered-grid finite difference theory, and also a 2-D imaging analysis based on first arrival travel time. The results indicate that (1) full wave field records could be obtained from multi-hole seismic wave simulations. Simulation results describe that the seismic wave propagation pattern in cross-hole high-velocity spherical geological bodies is more detailed and can serve as a basis for the wave field analysis. (2) When a cross-hole seismic section cuts through the boulder, the proposed method provides satisfactory cross-hole tomography results; however, when the section is closely positioned to the boulder, such high-velocity object in the 3-D space would impact on the surrounding wave field. The received diffracted wave interferes with the primary wave and in consequence the picked first arrival travel time is not derived from the profile, which results in a false appearance of high-velocity geology features. Finally, the results of 2-D analysis in 3-D modeling space are comparatively analyzed with the physical model test vis-a-vis the effect of high velocity body on the seismic tomographic measurements.

Design of a Simulator for Enhanced Oil Recovery Process Using a Nigerian Reservoir as a Case Study

This study involves the applications of different numerical techniques in a more general way to the design of a simulator for an enhanced oil recovery process with surfactant assisted water flooding. The data from a hypothetical oil well and a Nigerian oil well were used for the validation of the simulator. The process is represented by a system of nonlinear partial differential equations: the continuity equation for the transport of the components and Darcy’s equation for the phase flow. The orthogonal collocation, finite difference and coherence theory techniques were used in solving the equations that characterized the multidimensional, multiphase and multicomponent flow problem. Matlab computer programs were used for the numerical solution of the model equ- ations. The predicted simulator, obtained from the resulting numerical exercise confers uncondi- tional stability and more insight into the physical reservoir description. The results of the ortho- gonal collocation solution were compared with those of finite difference and coherence solutions. The results indicate that the concentration of surfactants for orthogonal collocation show more features than the predictions of the coherence solution and the finite difference, offering more opportunities for further understanding of the physical nature of the complex problem. We have found out that the partition of the three components between the two-phases determines other physical property data and hence the oil recovery. The oil recovery for the Nigerian oil reservoir is higher than the recovery predicted for the hypothetical crude. The displacement mechanism for the multicomponent and multiphase system is stable in the Nigerian oil reservoir due to the mod- erate value of the oil/water viscosity instead of the hypothetical reservoir with high value of oil/water ratio.

Study on the Rockmass Instability of Open-Pit Mine by Block Theory and Numerical Simulation Methods

On the background of rock masses and field engineering geology of a mine, the feature of rock mechanics and rock mass structure of surrounding rock at mining district are analyzed. Based on the finite difference theory and block theory, FLAC3D program (Fast Lagrangian Analysis for Continuum), rock mechanic and rock mass structure results are used to construct the finite difference mechanical model, which reflected the surrounding rock stability when mining. By the numerical simulation, the mechanical effect is studied by the process of mining and its results can be used to produce some theory and actual basis.

Non-probabilistic reliability sensitivity analysis of the model of structural systems with interval variables whose state of dependence is determined by constraints

Non-probabilistic reliability sensitivity analysis for structural systems plays an important role in determining key design variables that affect structural reliability strongly. Traditional non-probabilistic model assumes that all interval variables are mutually independent. However, this assumption may not be true in practical engineering. In this article, the dependency of interval variables is introduced into the non-probabilistic model by using both inequality and equality constraints. The non-probabilistic index model and optimization method for structural systems with interval variables, whose state of dependence is determined by constraints, are proposed on the basis of the existing non-probabilistic index theory. The linear optimization model is alternative when nonlinear optimization model cannot find any solution. Non-probabilistic reliability sensitivity analysis model and optimization method for structural systems, with the interval variables whose state of dependence is determined by constraints, are established based upon the finite difference theory. The proposed method is demonstrated via several examples.

Research on Very Fast Transient over-Voltages Distribution in Large Power Transformer Windings

To analyze the transient response of transformer windings under very fast transient over-voltage (VFTO), multi-conductor transmission line (MTL) model based on the representation of transformer windings by its individual turns are established. Space discretization is needed for solving the time-domain telegraph equations of MTL. To calculate the voltage distributions along transformer windings, through combining the compact finite difference (CFD) theory and the backward differentiation formulas (BDF). Simulation software ATP is introduced, and the simulation results demonstrate that the proposed approach is feasible.

Analysis of temperature field of milling insert with 3D complex groove using Cellular Automata

A numerical method for modelling the temperature field of milling insert with 3D complex groove is proposed based on the self-organising Cellular Automata. The self-organising local rule of Cellular Automata is derived to describe the heat transfer that affects temperature of milling insert using the finite difference theory. Consequently, a Cellular Automata model is built to predict the dynamic temperature field of the milling insert with 3D complex groove, which is also called wave-edge insert. Case studies are performed using sets of selected cutting conditions. Corresponding milling experiments and finite element analysis are also conducted to verify the simulation results of the Cellular Automata method. The comparison shows the validity and advantages of the proposed method.

A time integral formulation and algorithm for structural dynamics with nonlinear stiffness

A newly-developed numerical algorithm, which is called the new Generalized-α (G-α) method, is presented for solving structural dynamics problems with nonlinear stiffness. The traditional G-α method has undesired overshoot properties as for a class of α-method. In the present work, seven independent parameters are introduced into the single-step three-stage algorithmic formulations and the nonlinear internal force at every time interval is approximated by means of the generalized trapezoidal rule, and then the algorithm is implemented based on the finite difference theory. An analysis on the stability, accuracy, energy and overshoot properties of the proposed scheme is performed in the nonlinear regime. The values or the ranges of values of the seven independent parameters are determined in the analysis process. The computational results obtained by the new algorithm show that the displacement accuracy is of order two, and the acceleration can also be improved to a second order accuracy by a suitable choice of parameters. Obviously, the present algorithm is zero-stable, and the energy conservation or energy decay can be realized in the high-frequency range, which can be regarded as stable in an energy sense. The algorithmic overshoot can be completely avoided by using the new algorithm without any constraints with respect to the damping force and initial conditions.

The effect of very large perturbation flows on a perturbation viscometer

A perturbation viscometer is a flux response technology device that measures gradients on the viscosity–composition curve of a binary mixture or more generally on the viscosity–composition surface. It provides a method of calculating relative gas viscosities very precisely. In conventional perturbation viscometry a small perturbation flow of one component of the gas mixture is added to a main flow of known composition flowing through a capillary. The pressure upstream of the capillary changes sequentially due to the change in flowrate and viscosity of the mixture. The ratio of these pressure changes gives the gradient of the viscosity–composition curve. Integration of the viscosity gradients across the composition range yields the relative viscosities of the mixtures. This present paper gives new theoretical treatments for large perturbation flows up to 150% of the main flowrate. Corrections to the theory for small finite perturbations are proposed that account for the differences observed when larger perturbations are used. These correction terms are used to demonstrate how the logarithmic viscosity gradient–composition curve can be found by adding progressively bigger perturbations to each of the two pure components. A second method is proposed which demonstrates how the viscosity ratios can be found directly from the large perturbation measurements. Experimental studies of the mixture argon–nitrogen at 25 ∘ C and 1.1 bar are used to validate the new theoretical treatments. Results using both large and small perturbation measurements permit direct comparison between the different methods. In addition, the large perturbation data are used to define the optimum perturbation size for use with conventional finite difference theory.

Scattering attenuation: 2-D and 3-D finite difference simulations vs. theory

Scattering of seismic waves is studied by producing synthetic vertical seismic profiling (VSP) seismograms with 2-D and 3-D finite difference modelling in random media. The random models used are Gaussian and band-limited self-similar, or fractal random media. The modelling is performed acoustically, but we believe that, considering the geometry of this study, the results obtained will hold for the elastic case as well. Properties of the random media are discussed, in particular the difference between discrete and continuous media, and the importance of this difference. We show that when using the band-limited Von Karman correlation function when generating the random medium, the size of the model should be greater than 2 πa, where a is the correlation distance, and the grid spacing should be less then a. If not, the medium will not have the proper characteristics. Analytical expressions for scattering attenuation, derived from single scattering theory, can be used to estimate scattering Q from borehole velocity logs, if it is known what minimum scattering angle, θ min, to use. θ min, the minimum angle energy, must be scattered to be regarded as not contributing to the propagating wave. We estimate θ min by comparing Q values estimated from our synthetic VSP seismograms with the analytical expressions. The comparison also shows when the assumption of single scattering is valid. Previous studies in 2-D give a θ min of ∼30°. In this paper, we make a comparison for both the 2-D and 3-D cases, and show that the Q estimate is highly sensitive to how the analysis is done. We show that single scattering theory agrees well with finite difference simulations for self-similar media with low Hurst numbers, but with a somewhat lower θ min of 10–20°. This holds for a range of correlation lengths, a, including the case of infinite, or absence of, a. For Gaussian and exponential media, simulations and theory agree as well with θ min of 10–20°, but only for ka<5, where k is the wave number of the source. For ka>5, simulations and theory diverge, and single scattering theory cannot explain the amplitude attenuation observed in the scattering simulations for these types of media, indicating that it may be difficult to estimate the fractal properties of a medium from seismic data alone. With the difficulties of characterizing the scattering medium, and to estimate the scattering attenuation in the simple case of synthetic data with pure scattering, we conclude that it may be difficult to separate scattering and intrinsic attenuation from real data.

On-line Cartesian trajectory control of mechanisms along complex curves

New methods have been developed to control a mechanism's realtime Cartesian motion along spatially complex curves such as Non-Uniform Rational B-splines (NURBS). The methods dynamically map the critical trajectory parameters between parameter space, Cartesian space, and joint space. Trajectory models that relate Cartesian tool speeds and accelerations to joint speeds and accelerations have been generalized so that they can be applied to most classes of robots and CNC mechanisms. A simple and efficient predictor-corrector method uses finite difference theory to predict the parametric changes required to generate the desired curvilinear distances along the trajectory, and then correct the erorrs arising from this prediction. Polynomial approximation methods successfully approximate joint speeds and accelerations rather than require a closed-form inverse Jacobian solution. The numerical algorithms prove to be time bounded (fixed number of computational steps), and the generated trajectories are smooth and continuous. Both simulation and physical experiments using an Open-Architecture Controller demonstrate the feasibility and usefulness of the developed trajectory generation algorithms and methods. The methods can be conducted at trajectory rates greater than 100 Hz, depending on mechanism complexity.

Numerical magetohydrodynamics in astrophysics: Algorithm and tests for one-dimensional flow`

view Abstract Citations (223) References (15) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Numerical Magnetohydrodynamics in Astrophysics: Algorithm and Tests for One-dimensional Flow Ryu, Dongsu ; Jones, T. W. Abstract We describe a numerical code to solve the equations for ideal magnetohydrodynamics (MHD). It is based on an explicit finite difference scheme on an Eulerian grid, called the Total Variation Diminishing (TVD) scheme, which is a second-order-accurate extension of the Roe-type upwind scheme. We also describe a nonlinear Riemann solver for ideal MHD, which includes rarefactions as well as shocks and produces exact solutions for two-dimensional magnetic field structures as well as for the three-dimensional ones. The numerical code and the Riemann solver have been used to test each other. Extensive tests encompassing all the possible ideal MHD structures with planar symmetries (\ie ~one-dimensional flows) are presented. These include those for which the field structure is two-dimensional ({\it i.e.}, those flows often called ``$1 + 1/2$ dimensional'') as well as those for which the magnetic field plane rotates ({\it i.e.,}, those flows often called ``$1 + 1/2 + 1/2$ dimensional''). Results indicate that the code can resolve strong fast, slow, and magnetosonic shocks within 2-4 cells while more cells are required if shocks become weak. With proper stiffening, rotational discontinuities are resolved within 3-5 cells. Contact discontinuities are also resolved within 3-5 cells with stiffening and 6-8 cells without stiffening, while the stiffening on contact discontinuities in some cases generates numerical oscillations. Tangential discontinuities spread over more than 10 cells. Our tests confirm that slow compound structures with two-dimensional magnetic field are composed of intermediate shocks (so called ``2-4'' Publication: The Astrophysical Journal Pub Date: March 1995 DOI: 10.1086/175437 arXiv: arXiv:astro-ph/9404074 Bibcode: 1995ApJ...442..228R Keywords: Algorithms; Computational Astrophysics; Magnetic Fields; Magnetohydrodynamics; One Dimensional Flow; Shock Waves; Computational Grids; Finite Difference Theory; Shock Tubes; Tvd Schemes; Upwind Schemes (Mathematics); Astrophysics; HYDROMAGNETICS; MAGNETOHYDRODYNAMICS: MHD; METHODS: NUMERICAL; SHOCK WAVES; Astrophysics E-Print: 32 pages with 14 figures (uuencoded compressed tar file of text and figures available via anonymous ftp from directory /pub at sirius.chungnam.ac.kr), submitted to The Astrophysical Journal, CNU-AST-94-1 full text sources arXiv | ADS |

Magnetoconvection dynamics in a stratified layer. 1: Two-dimensional simulations and visualization

view Abstract Citations (15) References (50) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Magnetoconvection Dynamics in a Stratified Layer. I. Two-dimensional Simulations and Visualization Lantz, Steven R. ; Sudan, R. N. Abstract To gain insight in the problem of fluid convection below solar photosphere, time-dependent magnetohydrodynamic convection is studied by numerical simulation to the magneto-anelastic equations, a model appropiate for low Mach numbers. Numerical solutions to the equations are generated on a two-dimensional Cartesian mesh by a finite-difference, predictor-corrector algorithm. The thermodynamic properties of the fluid are held constant at the rigid, stress-free top and bottom boundaries of the computational box, while lateral boundaries are treated as periodic. In most runs the background polytropic fluid configuration is held fixed at Rayleigh number R = 5.44 times the critical value, Prandtl number P = 1.8, and aspect ratio a = 1, while the magnetic parameters are allowed to vary. The resulting dynamical behavior is shown to be strongly influenced by a horizontal magnetic field which is imposed at the bottom boundary. As the field strength increases from zero, an initially unsteady 'single-roll' state, featuring complex time dependence is replaced by a steady 'traveling-wave tilted state; then, an oscillatory or 'sloshing' state; then, a steady two-poll state with no tilting; and finally, a stationary state. Because the magnetic field is matched onto a potential field at the top boundary, it can penetrate into the nonconducting region above. By varying a magnetic diffusivity, the concentrations of weak magnetic fields at the top of these flows can be shown to be explainable in terms of an advection-diffusion balance. Publication: The Astrophysical Journal Pub Date: March 1995 DOI: 10.1086/175411 Bibcode: 1995ApJ...441..903L Keywords: Magnetohydrodynamics; Solar Convection (Astronomy); Solar Interior; Solar Magnetic Field; Two Dimensional Models; Advection; Computerized Simulation; Diffusion; Finite Difference Theory; Magnetostriction; Predictor-Corrector Methods; Solar Physics; CONVECTION; MAGNETOHYDRODYNAMICS: MHD; SUN: INTERIOR; SUN: MAGNETIC FIELDS; VIDEOTAPES full text sources ADS | Related Materials (1) Part 2: 1995ApJ...441..925L

Evolution of magnetized protoplanetary disks

view Abstract Citations (25) References (27) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Evolution of Magnetized Protoplanetary Disks Reyes-Ruiz, Mauricio ; Stepinski, Tomasz F. Abstract We investigate the global evolution of a turbulent protoplanetary disk in its viscous stage, incorporating the effects of Maxwell stress due to a large-scale magnetic field permeating disk. We assume that the viscous stress is given by an alpha model. A magnetic field is produced contemporaneously by an alpha omega dynamo mechanism and the resultant Maxwell stress assists the viscous stress in providing the means for disk evolution. The aim of this work is to compare the evolution of magnetized and nonmagnetized disks driven by turbulent viscosity of the same magnitude and thus assess the effects of a self-generated magnetic field on the structure and dynamical evolution of protoplanetary disks. Two illustrative examples corresponding to two different initial conditions are considered: a high-mass case that starts with a disk of 0.245 solar mass and angular momentum of 5.6 x 1052g sq cm/s, and a low-mass that case starts with a disk of 0.11 solar mass and angular momentum of 1.8 x 1052g sq cm/s. For each of these two cases the radial development of a disk is calculated numerically assuming a fiducial value of the dimensionless viscosity parameter alphass = 0.01, as well as alphass = 2 x 10-3. In all cases the central star has a mass equal to 1 solar mass. The most striking feature of magnetized disk evolution is the presence of the surface density bulge located in the region of the disk where the dynamo mechanism cannot support a magnetic field. The bulge persists for a time of the order of 105-106 yr. The presence and persistence of the surface density bulge may have important implications for the process of planet formation and the overall characteristics of resultant planetary systems. Publication: The Astrophysical Journal Pub Date: January 1995 DOI: 10.1086/175120 Bibcode: 1995ApJ...438..750R Keywords: Accretion Disks; Magnetic Fields; Planetary Evolution; Protoplanets; Viscosity; Finite Difference Theory; Ionization; Planetary Systems; Turbulence; Astrophysics; ACCRETION; ACCRETION DISKS; MAGNETOHYDRODYNAMICS: MHD; STARS: PLANETARY SYSTEMS full text sources ADS |

The piecewise-linear predictor-corrector code - A Lagrangian-remap method for astrophysical flows

view Abstract Citations (12) References (25) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Piecewise-Linear Predictor-Corrector Code: A Lagrangian-Remap Method for Astrophysical Flows Lufkin, Eric A. ; Hawley, John F. Abstract We describe a time-explicit finite-difference algorithm for solving the nonlinear fluid equations. The method is similar to existing Eulerian schemes in its use of operator-splitting and artificial viscosity, except that we solve the Lagrangian equations of motion with a predictor-corrector and then remap onto a fixed Eulerian grid. The remap is formulated to eliminate errors associated with coordinate singularities, with a general prescription for remaps of arbitrary order. We perform a comprehensive series of tests on standard problems. Self-convergence tests show that the code has a second-order rate of convergence in smooth, two-dimensional flow, with pressure forces, gravity, and curvilinear geometry included. While not as accurate on idealized problems as high-order Riemann-solving schemes, the predictor-corrector Lagrangian-remap code has great flexibility for application to a variety of astrophysical problems. Publication: The Astrophysical Journal Supplement Series Pub Date: October 1993 DOI: 10.1086/191833 Bibcode: 1993ApJS...88..569L Keywords: Gas Dynamics; H Ii Regions; Molecular Clouds; Predictor-Corrector Methods; Stellar Winds; Two Dimensional Flow; Euler-Lagrange Equation; Finite Difference Theory; Galactic Evolution; Hydrodynamics; Astrophysics; HYDRODYNAMICS; METHODS: NUMERICAL full text sources ADS |

A cosmological hydrodynamic code based on the total variation diminishing scheme

view Abstract Citations (229) References (25) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS A Cosmological Hydrodynamic Code Based on the Total Variation Diminishing Scheme Ryu, Dongsu ; Ostriker, Jeremiah P. ; Kang, Hyesung ; Cen, Renyue Abstract An explicitly second-order finite difference code based on a total variation diminishing scheme for self-gravitating cosmological hydrodynamic systems is described. The code was developed to follow correctly the adiabatic changes of extremely supersonic preshock flows with a Mach number larger than 100 as well as very strong shocks. In calculations of the formation of a 1D Zel'dovich pancake, an energy accuracy of 1 percent is obtained for 32 cells per unit wavelength, and the accuracy reaches 0.01 percent as the number of cells approach 1024. It is shown that shocks are well resolved and separate cleanly the hot, dense, collapsed peaks from the cold, low-density, expanding voids. Publication: The Astrophysical Journal Pub Date: September 1993 DOI: 10.1086/173051 Bibcode: 1993ApJ...414....1R Keywords: Cosmology; Dark Matter; Hydrodynamic Equations; Mach Number; Shock Waves; Tvd Schemes; Entropy; Finite Difference Theory; Supersonic Flow; Thermal Energy; Astrophysics; COSMOLOGY: THEORY; HYDRODYNAMICS; METHODS: NUMERICAL; SHOCK WAVES full text sources ADS |