Computer simulation

Abstract: An exact method is presented for numerically calculating, within the framework of the stochastic formulation of chemical kinetics, the time evolution of any spatially homogeneous mixture of molecular species which interreact through a specified set of coupled chemical reaction channels. The method is a compact, computer-oriented, Monte Carlo simulation procedure. It should be particularly useful for modeling the transient behavior of well-mixed gas-phase systems in which many molecular species participate in many highly coupled chemical reactions. For “ordinary” chemical systems in which fluctuations and correlations play no significant role, the method stands as an alternative to the traditional procedure of numerically solving the deterministic reaction rate equations. For nonlinear systems near chemical instabilities, where fluctuations and correlations may invalidate the deterministic equations, the method constitutes an efficient way of numerically examining the predictions of the stochastic master equation. Although fully equivalent to the spatially homogeneous master equation, the numerical simulation algorithm presented here is more directly based on a newly defined entity called “the reaction probability density function.” The purpose of this article is to describe the mechanics of the simulation algorithm, and to establish in a rigorous, a priori manner its physical and mathematical validity; numerical applications to specific chemical systems will be presented in subsequent publications.

Abstract: The description of coherent features of fluid flow is essential to our understanding of fluid-dynamical and transport processes. A method is introduced that is able to extract dynamic information from flow fields that are either generated by a (direct) numerical simulation or visualized/measured in a physical experiment. The extracted dynamic modes, which can be interpreted as a generalization of global stability modes, can be used to describe the underlying physical mechanisms captured in the data sequence or to project large-scale problems onto a dynamical system of significantly fewer degrees of freedom. The concentration on subdomains of the flow field where relevant dynamics is expected allows the dissection of a complex flow into regions of localized instability phenomena and further illustrates the flexibility of the method, as does the description of the dynamics within a spatial framework. Demonstrations of the method are presented consisting of a plane channel flow, flow over a two-dimensional cavity, wake flow behind a flexible membrane and a jet passing between two cylinders.

Abstract: A lattice Boltzmann model is developed which has the ability to simulate flows containing multiple phases and components. Each of the components can be immiscible with the others and can have different mass values. The equilibrium state of each component can have a nonideal gas equation of state at a prescribed temperature exhibiting thermodynamic phase transitions. The scheme incorporated in this model is the introduction of an interparticle potential. The dynamical rules in this model are local so it is highly efficient to compute on massively parallel computers. This model has many application in large-scale numerical simulations of various types of fluid flows

Abstract: There are two fundamental ways to view coupled systems of chemical equations: as continuous, represented by differential equations whose variables are concentrations, or as discrete, represented by stochastic processes whose variables are numbers of molecules. Although the former is by far more common, systems with very small numbers of molecules are important in some applications (e.g., in small biological cells or in surface processes). In both views, most complicated systems with multiple reaction channels and multiple chemical species cannot be solved analytically. There are exact numerical simulation methods to simulate trajectories of discrete, stochastic systems, (methods that are rigorously equivalent to the Master Equation approach) but these do not scale well to systems with many reaction pathways. This paper presents the Next Reaction Method, an exact algorithm to simulate coupled chemical reactions that is also efficient: it (a) uses only a single random number per simulation event, and (b) ...

Abstract: The evaluation of Coulombic or gravitational interactions in large-scale ensembles of particles is an integral part of the numerical simulation of a large number of physical processes. Examples include celestial mechanics, plasma physics, the vortex method in fluid dynamics, molecular dynamics, and the solution of the Laplace equation via potential theory. In a typical application, a numerical model follows the trajectories of a number of particles moving in accordance with Newton's second law of motion in a field generated by the whole ensemble. In many situations, in order to be of physical interest, the simulation has to involve thousands of particles (or more), and the fields have to be evaluated for a large number of configurations. Unfortunately, an amount of work of the order $O(N\sp 2)$ has traditionally been required to evaluate all pairwise interactions in a system of N particles, unless some approximation or truncation method is used. As a result, large-scale simulations have been extremely expensive in some cases, and prohibitive in others. We present an algorithm for the rapid evaluation of the potential and force fields in large-scale systems of particles. In order to evaluate all pairwise Coulombic interactions of N particles to within round-off error, the algorithm requires an amount of work proportional to N, and this estimate does not depend on the statistics of the distribution. Both two and three dimensional versions of the algorithm have been constructed, and we will discuss their applications to several problems in physics, chemistry, biology, and numerical complex analysis.