Three Lectures: Nemd, Spam, and Shockwaves

We discuss three related subjects well suited to graduate research. The first, Nonequilibrium molecular dynamics or "NEMD", makes possible the simulation of atomistic systems driven by external fields, subject to dynamic constraints, and thermostated so as to yield stationary nonequilibrium states. The second subject, Smooth Particle Applied Mechanics or "SPAM", provides a particle method, resembling molecular dynamics, but designed to solve continuum problems. The numerical work is simplified because the SPAM particles obey ordinary, rather than partial, differential equations. The interpolation method used with SPAM is a powerful interpretive tool converting point particle variables to twice-differentiable field variables. This interpolation method is vital to the study and understanding of the third research topic we discuss, strong shockwaves in dense fluids. Such shockwaves exhibit stationary far-from-equilibrium states obtained with purely reversible Hamiltonian mechanics. The SPAM interpolation method, applied to this molecular dynamics problem, clearly demonstrates both the tensor character of kinetic temperature and the time-delayed response of stress and heat flux to the strain rate and temperature gradients. The dynamic Lyapunov instability of the shockwave problem can be analyzed in a variety of ways, both with and without symmetry in time. These three subjects suggest many topics suitable for graduate research in nonlinear nonequilibrium problems.Comment: 40 pages, with 21 figures, as presented at the Granada Seminar on the Foundations of Nonequilibrium Statistical Physics, 13-17 September, as three lecture

A delay differential model of ENSO variability: Parametric instability and the distribution of extremes

We consider a delay differential equation (DDE) model for El-Nino Southern Oscillation (ENSO) variability. The model combines two key mechanisms that participate in ENSO dynamics: delayed negative feedback and seasonal forcing. We perform stability analyses of the model in the three-dimensional space of its physically relevant parameters. Our results illustrate the role of these three parameters: strength of seasonal forcing $b$, atmosphere-ocean coupling $\kappa$, and propagation period $\tau$ of oceanic waves across the Tropical Pacific. Two regimes of variability, stable and unstable, are separated by a sharp neutral curve in the $(b,\tau)$ plane at constant $\kappa$. The detailed structure of the neutral curve becomes very irregular and possibly fractal, while individual trajectories within the unstable region become highly complex and possibly chaotic, as the atmosphere-ocean coupling $\kappa$ increases. In the unstable regime, spontaneous transitions occur in the mean ``temperature'' ({\it i.e.}, thermocline depth), period, and extreme annual values, for purely periodic, seasonal forcing. The model reproduces the Devil's bleachers characterizing other ENSO models, such as nonlinear, coupled systems of partial differential equations; some of the features of this behavior have been documented in general circulation models, as well as in observations. We expect, therefore, similar behavior in much more detailed and realistic models, where it is harder to describe its causes as completely.Comment: 22 pages, 9 figure

Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics

Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit

Spatially partitioned embedded Runge-Kutta Methods

We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory

The role of the central chemoreceptor in causing periodic breathing.

In a previous publication (Fowler et aL, 1993), we reduced the classical cardiorespiratory control model of (Grodins et aL, 1967) to a much simpler form, which we then used to study the phenomenon of periodic breathing. In particular, cardiac output was assumed constant, and a single (constant) delay representing arterial blood transport time between lung and brain was included in the model. In this paper we extend this earlier work, both by allowing for the variability in transport delays, due to the dependence of cardiac output on blood gas concentrations, and also by including further delays in the system. In addition, we extensively discuss the physiological implications of parameter variations in the model; several novel mechanisms for periodic breathing in clinical situations are proposed. The results are discussed in the light of recent observational studies. Keywords: Periodic breathing; Cheyne-Stokes respiration; heart-rate variability*, differential-delay equations. 1

Investigation and modeling of space shuttle main engine shutdown transient chugging

The space shuttle main engines experience a low frequency pressure pulsation in both the fuel and oxidizer preburners during the shutdown transient. This pressure pulsation, called chugging, has been linked to undesirable bearing loads and possible damage to the spark ignitor supply piping for the fuel preburner. The problem is briefly described and a model is proposed that includes: (1) a transient stirred tank reactor model for the combustion chamber, (2) a resistance capacitance model for the supply piping and (3) purge gas/liquid oxygen interface tracking

Implicit second and third orders runge-kutta for handling discontinuities in delay differential equations

Implicit Runge-Kutta (RK) methods have been developed and implemented in solving Delay Differential Equations (DDEs) systems which often encounter discontinuities. These discontinuities might occur after and even before the initial solution. The methods are chosen because they can be modified to handle discontinuities by means of mapping of past values and they are in fact the most well-organized way to handle the so-called stiff differential equations, which are differential equations usually characterized by a rapidly decaying solution. The advantage of implicit Runge-Kutta methods is in their superior stability compared to the explicit methods, more so when applied to stiff equations. Our objective is to develop a scheme for solving DDEs using implicit RK2 and RK3. Our numerical scheme is able to successfully handle discontinuities in the system and produces results with acceptable error. We compare the result from [1] which used explicit RK2 and RK4 with our findings. Our result is markedly better than [1] even in the lower order RK

Numerical Solution of ODEs and the Columbus' Egg: Three Simple Ideas for Three Difficult Problems

On computers, discrete problems are solved instead of continuous ones. One must be sure that the solutions of the former problems, obtained in real time (i.e., when the stepsize h is not infinitesimal) are good approximations of the solutions of the latter ones. However, since the discrete world is much richer than the continuous one (the latter being a limit case of the former), the classical definitions and techniques, devised to analyze the behaviors of continuous problems, are often insufficient to handle the discrete case, and new specific tools are needed. Often, the insistence in following a path already traced in the continuous setting, has caused waste of time and efforts, whereas new specific tools have solved the problems both more easily and elegantly. In this paper we survey three of the main difficulties encountered in the numerical solutions of ODEs, along with the novel solutions proposed.Comment: 25 pages, 4 figures (typos fixed