Stony Brook AMS - Downloadable Preprints, 1994

We examine the nonideal behavior of real vapor in the context of the theory of noninteracting molecular clusters. The vapor is treated as a perfect mixture of clusters, which in equilibrium attain a distribution in size determined by formation energies $\Delta G_{i}$, where $\Delta G_{i}$ is the energy required to form a cluster of $i$ molecules from $i$ molecules in bulk saturated liquid. A theory for the $\Delta G_{i}$ gives an equation of state that captures the nonideal behavior of the vapor; conversely, equation of state data provide a validation of the theory.

In this paper, we compare the predictions of this equation of state to experimental data. Utilizing the $\Delta G_{i}$ proposed by Dillmann and Meier and based on Fisher's droplet model, we compute the vapor compressibility along the saturation curve for several nonpolar substances and obtain excellent agreement with experiment. We also compute the third virial coefficient for these substances and observe correct qualitative behavior; in the case of benzene and {\em n}-octane, for which some data is available, we find rough agreement with experiment.

The conventional kinetic theory of homogeneous nucleation, which is based on the assumption of noninteracting clusters, demonstrates that the cluster series equation of state can be continued past the saturation point to describe metastable vapor, a claim that no non-virial equation of state can make {\em a priori}. Furthermore, the noninteracting cluster theory readily accommodates results of more detailed calculations of molecular clusters ({\em e.g.}, results of Monte Carlo or molecular dynamics studies). These considerations and the success of the simple Fisher-Dillmann-Meier model in predicting the behavior of nonideal vapor suggest possible avenues of investigation in equation of state research.

We consider a one-parameter family of nonstrictly hyperbolic systems of conservation laws modeling three-phase flow in a porous medium. For a particular value of the parameter, the model has a shock wave solution that undergoes several bifurcations upon perturbation of its left and right states and the parameter. In this paper we use singularity theory and bifurcation theory of dynamical systems, including Melnikov's method, to find all nearby shock waves that are admissible according to the viscous profile criterion. We use these results to construct a unique solution of the Riemann problem for each left and right state and parameter value in a neighborhood of the unperturbed shock wave solution; together with previous numerical work, this construction completes the solution of the three-phase flow model. In the bifurcation analysis, the unperturbed shock wave acts as an organizing center for the waves appearing in Riemann solutions.

This manuscript, Practical Parallel Processing, outgrows from the lectures I gave at Stony Brook in a graduate research course, ``large-scale scientific computing'', and at the Hong Kong University of Science and Technology in a summer course, ``applied parallel computing''. Majority of the attendees are advanced undergraduate and graduate students, and postdoctoral fellows, and some faculty members specializing in Applied Mathematics, Mathematics, Physics, Chemistry, Electrical Engineering, and Mechanical Engineering, from both institutions, in addition to HKU, CUHK, HK Polytechnic, and HKBC. Recently, this course was further refined and was taught again at Stony Brook as a workshop. People from Columbia University, Brookhaven National Laboratory, Northrop-Grumann Cooperation, and other Long Island engineering firms participated in the program.

This manuscript trys to address an audience with scattered interests and diverse background in scientific computing. The algorithmic topics included are calculus, linear algebra, PDEs, ODEs, FFT, and molecular dynamics. In each of these topics, the focus is placed at designing and analyzing parallel algorithms.

This book is still unfinished, but will be updated periodically. Please check with the author (deng@ams.sunysb.edu) for the latest version.

It is shown that the central core mathematical models of the mammalian kidney can be expressed as functions of core solute concentrations $z$ and tube water and solute permeabilities $h$. A set of nonlinear algebraic equations are derived. These correspond to the model differential equations that $h$ and $z$ satisfy at a properly chosen discrete set of points.

In mathematical models of the kidney concentrating mechanism, it is often necessary to simultaneously vary all the parameters to modify a given concentration profile to get a desired profile. We show how the changes can be made in the given profile to achieve realistic simultaneous variation of all the parameters. Results of some computational experiments are given.

We study the structure of solutions of Riemann problems for systems of two conservation laws. Such a solution comprises a sequence of elementary waves, viz., rarefaction and shock waves of various types; shock waves are required to have viscous profiles. We construct a Riemann solution by solving a system of equations characterizing its component waves. A Riemann solution is ``structurally stable'' if the number and types of its component waves are preserved when the initial data and the flux function are perturbed.

Under the assumption that rarefaction waves and shock states lie in the strictly hyperbolic region, we characterize Riemann solutions for which the defining equations have maximal rank and we prove that such solutions are structurally stable. Structurally stable Riemann solutions cannot contain overcompressive shock waves, but they can contain transitional shock waves, including doubly sonic transitional shock waves that have not been observed before.

A critical analysis of the mass conservation properties of the jump discontinuity propagating algorithms in the front-tracking method of Glimm et al. is performed in the context of miscible, two phase, incompressible flow in porous media. These algorithms do not enforce the conservation of mass properties of the hyperbolic system on any grid of finite discretization size. For the curve propagation algorithm, which is the core of the suite of discontinuity movement algorithms, we show that mass conservation errors vanish linearly with maximum mesh size of the moving grids. We present new curve propagation and redistribution algorithms which conserve mass for any grid of finite spacing. Analogously mass conserving untangling routines have also been developed. We investigate the performance of these new algorithms for diagonal five-spot computations.

Front tracking simulations of the Richtmyer-Meshkov instability produce significantly better agreement with experimentally measured growth rates than previously reported. Careful analysis of the early stages of the shock acceleration process show that nonlinearity and compressibility play a critical role in the behaviour of the shocked interface and are responsible for the deviations from Richtmyer's impulsive model. The late time behaviour of the interface growth rate is compared to with a nonlinear potential flow model of Hecht, et al.

We present a fully conservative, high resolution approach to front tracking for nonlinear systems of conservation laws in two space dimensions. An underlying uniform Cartesian grid is used, with some cells cut by the front into two subcells. The front is moved by solving a Riemann problem normal to each segment of the front and using the motion of the strongest wave to give an approximate location of the front at the end of the time step. A high resolution finite volume method is then applied on the resulting slightly-irregular grid to update all cell values. A "large time step" wave propagation algorithm is used that remains stable in the small cut cells with a time step that is chosen with respect to the uniform grid cells. Numerical results on a radially symmetric problem show that pointwise convergence with order between 1 and 2 is obtained in both the cell values and location of the front. Other computations are also presented.

We are concerned with global solutions and corresponding shock capturing approximations to the Euler equations of compressible gas dynamics with geometrical structure. Recent developments in this direction are reviewed and the role of the shock capturing approach in solving the Euler equations is discussed. Some efficient shock capturing methods as well as their convergence are analyzed to compute the corresponding compressible flows and to construct correct approximate solutions.

We prove the existence of global solutions to the Euler equations of compressible isentropic gas dynamics with geometrical structure, including transonic nozzle flow, spherically symmetric flow, and cylindrically symmetric rotating flow. Due to the presence of the geometrical source terms, the existence results themselves are new, especially as they pertain to radial flow in an unbounded region, $|\vec{x}|\ge 1$, and to transonic nozzle flow. Arbitrary data with $L^\infty$ bounds are allowed in these results. A shock capturing numerical scheme is introduced to compute such flows and to construct approximate solutions. The convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework.

The purpose of this note is to propose an improvement to the widely-used Steinberg-Lund model for rate-dependent plasticity. The improved model is rate-dependent above, as well as below, the Peierls stress. As a result of this change, one material constant in the Steinberg-Lund model acquires a new value, and we thereby obtain a revised estimate for the dislocation density in tantalum.

Computed microtomography is applied to a piece of Fontainebleau sandstone in order to determine the geometrical structure of the pores. The topology of the void space is then derived from the tomographic image of the volume. Permeability and conductibity are computed and found in good agreement with experimental data. Perspectives offered by this new nondestructive method with a potential resolution of the order of one micrometer or less are analyzed.