I Mathematical Theory- 1 Introduction- 11 Conservation laws- 12 Applications- 13 Mathematical difficulties- 14 Numerical difficulties- 15 Some references- 2 The Derivation of Conservation Laws- 21 Integral and differential forms- 22 Scalar equations- 23 Diffusion- 3 Scalar Conservation Laws- 31 The linear advection equation- 311 Domain of dependence- 312 Nonsmooth data- 32 Burgers' equation- 33 Shock formation- 34 Weak solutions- 35 The Riemann Problem- 36 Shock speed- 37 Manipulating conservation laws- 38 Entropy conditions- 381 Entropy functions- 4 Some Scalar Examples- 41 Traffic flow- 411 Characteristics and "sound speed"- 42 Two phase flow- 5 Some Nonlinear Systems- 51 The Euler equations- 511 Ideal gas- 512 Entropy- 52 Isentropic flow- 53 Isothermal flow- 54 The shallow water equations- 6 Linear Hyperbolic Systems 58- 61 Characteristic variables- 62 Simple waves- 63 The wave equation- 64 Linearization of nonlinear systems- 641 Sound waves- 65 The Riemann Problem- 651 The phase plane- 7 Shocks and the Hugoniot Locus- 71 The Hugoniot locus- 72 Solution of the Riemann problem- 721 Riemann problems with no solution- 73 Genuine nonlinearity- 74 The Lax entropy condition- 75 Linear degeneracy- 76 The Riemann problem- 8 Rarefaction Waves and Integral Curves- 81 Integral curves- 82 Rarefaction waves- 83 General solution of the Riemann problem- 84 Shock collisions- 9 The Riemann problem for the Euler equations- 91 Contact discontinuities- 92 Solution to the Riemann problem- II Numerical Methods- 10 Numerical Methods for Linear Equations- 101 The global error and convergence- 102 Norms- 103 Local truncation error- 104 Stability- 105 The Lax Equivalence Theorem- 106 The CFL condition- 107 Upwind methods- 11 Computing Discontinuous Solutions- 111 Modified equations- 1111 First order methods and diffusion- 1112 Second order methods and dispersion- 112 Accuracy- 12 Conservative Methods for Nonlinear Problems- 121 Conservative methods- 122 Consistency- 123 Discrete conservation- 124 The Lax-Wendroff Theorem- 125 The entropy condition- 13 Godunov's Method- 131 The Courant-Isaacson-Rees method- 132 Godunov's method- 133 Linear systems- 134 The entropy condition- 135 Scalar conservation laws- 14 Approximate Riemann Solvers- 141 General theory- 1411 The entropy condition- 1412 Modified conservation laws- 142 Roe's approximate Riemann solver- 1421 The numerical flux function for Roe's solver- 1422 A sonic entropy fix- 1423 The scalar case- 1424 A Roe matrix for isothermal flow- 15 Nonlinear Stability- 151 Convergence notions- 152 Compactness- 153 Total variation stability- 154 Total variation diminishing methods- 155 Monotonicity preserving methods- 156 l1-contracting numerical methods- 157 Monotone methods- 16 High Resolution Methods- 161 Artificial Viscosity- 162 Flux-limiter methods- 1621 Linear systems- 163 Slope-limiter methods- 1631 Linear Systems- 1632 Nonlinear scalar equations- 1633 Nonlinear Systems- 17 Semi-discrete Methods- 171 Evolution equations for the cell averages- 172 Spatial accuracy- 173 Reconstruction by primitive functions- 174 ENO schemes- 18 Multidimensional Problems- 181 Semi-discrete methods- 182 Splitting methods- 183 TVD Methods- 184 Multidimensional approaches

Numerical methods for conservation laws

The three‐dimensional (3-D) electromagnetic scattering problem is first formulated in the frequency domain in terms of an electric field volume integral equation. Three‐dimensional responses are then Fourier transformed with sine and cosine digital filters or with the decay spectrum. The digital filter technique is applied to a sparsely sampled frequency sounding, which is replaced by a cubic spline interpolating function prior to convolution with the digital filters. Typically, 20 to 40 frequencies at five to eight points per decade are required for an accurate solution. A calculated transient is usually in error after it has decayed more than six orders in magnitude from early to late time. The decay spectrum usually requires ten frequencies for a satisfactory solution. However, the solution using the decay spectrum appears to be less accurate than the solution using the digital filters, particularly after early times. Checks on the 3-D solution include reciprocity and convergence checks in the frequenc...

/pdf/transient-electromagnetic-response-of-a-three-dimensional-2b5uxzvjrw.pdf

Transient electromagnetic response of a three-dimensional body in a layered earth

http://www.math.pku.edu.cn/teachers/zhangpw/private/homepage/publication/2002_MMFEASPSD.pdf

A Moving Mesh Finite Element Algorithm for Singular Problems in Two and Three Space Dimensions

The effect of varying the Knudsen number Kn in microchannel e ows was simulated using the direct simulation Monte Carlo method (DSMC) combined with the monotonic Lagrangian grid (MLG). The DSMC‐ MLG, a method that provides automatic grid ree nement according to number density, has been optimized for massively parallel computation and provides a fast, highly resolved description of the e ow. New oute ow boundary conditions, consistent with the DSMC ‐ MLG algorithm, were developed to allow the user to specify the oute ow pressures. The effect of varying Kn was examined for three different values of Kn (0.07, 0.14, and 0.19 ) for a high-speed ine ow by varying the channel height. A Navier ‐ Stokes computation was also performed to show continuum regime e ow. The computations provide contours of pressure, temperature, and Mach number to show complex interactions among oblique shocks and boundary layers, and how these change with the Kn. Temperature jumps and slip velocities as functions of position along the wall are compared for all cases. The computations show that the velocity slip is approximately constant behind the shock, while the temperature jump is reduced.

Computations of High-Speed, High Knudsen Number Microchannel Flows

Results are presented of direct numerical simulations (DNS) of a viscous, laminar ring. The effects of different generator configurations and velocity programs on the formation and post‐formation characteristics of the ring are studied. It is shown that during the formation phase of the ring, total circulation and impulse in the flowfield are approximately the same for the ‘‘nozzle’’ and ‘‘orifice’’ generators. It is also found that throughout this period the slug flow model under‐predicts the total circulation in the flow. During the formation phase, the simulation results for the time evolution of total circulation and location of the vortex spiral center are in agreement with the experimental findings of Didden [J. Appl. Mech. Phys. (ZAMP) 30, 101 (1979)]. The results of the flow visualization studies show that during the post‐formation phase a vortex bubble is formed. As the bubble propels itself forward a wake is formed in the rear of the bubble. The impulse and vorticity from the bubble are continuo...

Direct numerical simulation of a laminar vortex ring

The numerical simulation of compressible reactive flows

Computer implemented methods and systems using a multidimensional Monotonic Lagrangian Grid (MLG) to sort and order locations of aircraft, are used for simulating, controlling, and optimizing aircraft traffic systems. The MLG is combined with algorithms for collision detection and resolution, separation assurance, and updating aircraft trajectories, and applied to test cases of the air transportation system. Physical locations describing aircraft traffic moving in complex paths are stored into a data structure of the multidimensional MLG algorithm. The moving aircraft traffic platforms are sorted and ordered on a grid structure in real space and in an indexing space, causing a monotonic mapping between indices of the grid structure and locations describing the plurality of moving aircraft. Computer operations using data stored in the multidimensional MLG determine control strategies for aircraft separation assurance and optimum routes to circumvent blockages in transport paths.

Fast tracking methods and systems for air traffric modeling using a monotonic lagrangian grid

: The Monotonic Lagrangian Grid (MLG) is a data structure in which nodes are ordered in a monotonic way such that those nodes which are close in physical space also have nearby indices in the data structure arrays. An MLG ordering for a given system of nodes, as defined by the monotonicity constraints, is not unique. For all but the smallest systems, the number of allowed orderings is extremely large with many of the possible MLG's so badly structured that they lead to poor results when used in physical calculations. A well-structured MLG ordering is one that corresponds well to the physical ordering of the system. This paper shows that the majority of the MLG's for a given set of node locations are poorly structured, but that the small fraction which are well-structured tend to be extremely stable against perturbations of the node positions. It is this extreme stability of the well-structured MLG's that is responsible for both the utility of this approach in particle-based simulations and the success of stochastic grid regularization, a technique for restructuring from a poorly structured to a well-structured MLG. The high probability of encountering a well-structured MLG when the node dynamics is complex, even without stochastic grid regularization, is a result of this relative stability.

http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA325866&Location=U2&doc=GetTRDoc.pdf

The Stability and Multiplicity of the Monotonic Lagrangian Grid

: In this paper, time-dependent results obtained from a simplified but nonlinear analytic similarity solution and a detailed numerical simulation are used to study the relations between the fundamental processes occurring in the very early stages of flame ignition in homogeneous premixed gases. The parameters which may be varied are the composition of the mixture, the initial radius of energy deposition R sub 0, the duration of the heating TAN SUB 0, and the total energy deposited in the system E sub 0. The similarity solution plus the ignition delay times Tau sub c for the fuel-oxidizer mixture as a function of temperature can be used to calculate whether or not a given energy source is adequate to ignite the system. This simple procedure may then be calibrated using the time-dependent NRL detailed reactive flow models. The models include the thermophysical properties of the mixture, a full chemical kinetics scheme, the nonlinear convection of self-consistent fluid dynamics and the matrix molecular diffusion coefficients for the individual species. Results are presented for a selected mixture of H2-O2-N2 for two values of R sub 0 which show that a model must be constructed for a quench volume in order to complete the similarity solution calibration. (Author)

Theoretical and Computational Approach to Modeling Flame Ignition.

: We examine, using computational molecular dynamics, shocks launched in two-dimensional crystals by a flying plate. The interaction of the shock with various lattice defects is observed, and is seen to create sites of rapidly growing, thermalized, hot fluid-like phases included within the crystal lattice. We hypothesize that these included fluid-like regions are the sites of the initial chemical reactions leading to detonation in energetic materials, and that crystallographic defects therefore control the sensitivity of single crystal high explosives to shock-initiation. The computations are carried out on the massively parallel Connection Machine using a parallelized version of the MLG algorithm.

Elaine S. Oran

Papers

The numerical simulation of compressible reactive flows

Fast tracking methods and systems for air traffric modeling using a monotonic lagrangian grid

The Stability and Multiplicity of the Monotonic Lagrangian Grid

Theoretical and Computational Approach to Modeling Flame Ignition.

Molecular Dynamics of Shocks in Crystals with Defects