Consistent Initial Conditions Research Articles

Positively invariant sets for singular discrete-time systems

Abstract The aim of this paper is to present some results allowing positively invariant and asymptotically stable polyhedral sets to be determined for the linear singular discrete-time system ExK+1 = AxK; the regular and singular cases of pencil (E, A) are considered. Necessary and sufficient algebraic conditions guaranteeing that some sets are positively invariant and asymptotically stable are given. Also discussed are some issues relating to the positive invariance of some domains and consistent initial conditions. Finally, as an application, the proposed results are used to determine a stability domain for a state feedback regulator with constraints on controls

Diffusive Limits for Linear Transport Equations

The authors show that the Hibert and Chapman-Enskog asymptotic treatments that reduce the nonlinear Boltzmann equation to the Euler and Navier-Stokes fluid equations have analogs in linear transport theory. In this linear setting, these fluid limits are described by diffusion equations, involving familiar and less familiar diffusion coefficients. Because of the linearity extant, one can carry out explicitly the initial and boundary layer analyses required to obtain asymptotically consistent initial and boundary conditions for the diffusion equations. In particular, the effects of boundary curvature and boundary condition variation along the surface can be included in the boundary layer analysis. A brief review of heuristic (nonasymptotic) diffusion description derivations is also included in our discussion.

Diffusive transport in binary anisotropic stochastic mixtures

It has recently been shown that a linear kinetic theory model for particle transport in a binary Markovian stochastic mixture possesses several different diffusive limits. This work was restricted to isotropic mixing statistics and a common scale length for the chord lengths of the two immiscible components of the mixture. Here we allow anisotropic statistics as well as independent length scales for the two components. The formalism used is that of asymptotic expansions, and we find an additional richness of asymptotic diffusive limits by relaxing the prior limitations on the mixing. A short discussion of the derivation of asymptotically consistent initial and boundary conditions for the diffusion equations is also given.

Decomposition of Hessenberg DAE systems to state space form

An algorithm is given for symbolically decoupling the solutions to a linear, time dependent differential-algebraic equation Ez′ = A(t)z + ⨍(t), z(t)ϵR s , in Hessenberg form into state and algebraic components. The state variables are the solutions to an ordinary differential equation with initial conditions restricted to a subspace of R s , while the algebraic components are linear functions of the state variables involving derivatives of the coefficients and input functions up to order r − 1, where r is the index of the system. This decomposition provides closed form solutions to linear Hessenberg DAEs in terms of the fundamental solutions of the state variable system. The implications of the algorithm for computing consistent initial conditions, for certain singular optimal control problems, and for numerical solutions are briefly discussed.

Computing consistent initial conditions for differential-algebraic equations

To solve differential-algebraic equation systems (DAEs) successfully initial conditions must be consistent, i.e. they must satisfy the corresponding extended system of the DAEs. For index one DAEs they can be obtained by either taking an implicit Euler step or solving the extended system together with a feasible set of specifications. We present computational algorithms for determining consistent initial conditions based on either approach. In addition, a purely numerical algorithm reported recently for computing initial values for higher index DAEs is modified here to reduce the computational complexity by using structural information on the DAEs.

Diffusive limits for particle transport in stochastic mixtures

Under the assumption of weak spatial and temporal gradients as well as weak absorption and sources, it is well known that the classic kinetic equation describing linear particle transport has a diffusion equation as an asymptotic limit. Considered here is such particle transport in a stochastic Markovian mixture of two immiscible fluids. It is shown that a variety of asymptotic diffusive limits exist under the scaling indicated above, depending upon the additional scaling introduced for the Markov transition lengths. These lengths are indicative of the mean chord lengths of the fluid packets in the stochastic mixture, and it is thus found that the details of the diffusive limits are strongly dependent upon the presumed sizes of the fluid packets. Asymptotically consistent initial and boundary conditions are also derived for each diffusion description obtained.

Consistent initial conditions of nonlinear networks with switches

In the analysis of switched networks discontinuity of network variables and the presence of impulsive voltages and currents at the switching instants are often encountered. A theory for obtaining consistent initial conditions of electrical networks containing ideal switches and impulsive sources is proposed. The method is based on charge and flux conservation, and requires the solution of three one-element-kind networks. It is applicable immediately to networks with linear resistors, linear controlled sources, and linear or nonlinear capacitors and inductors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Consistent initial conditions of linear switched networks

The analysis of switched networks encounters a major difficulty due to the discontinuity of network variables and the presence of impulsive voltages and currents at the switching instants. The problem of obtaining consistent initial conditions for linear networks with ideal switched is addressed. The method used is based on numerical Laplace transform theory and is applicable to any circuit configuration. Capacitor loops, inductor cutsets, ideal switches with zero impedance or admittance, and impulsive sources do not present problems. Although initial conditions are normally required for time-domain analysis, an even more important application is in the frequency-domain analysis of general periodically switched linear networks. The algorithms are easily implemented in a computer program. >

Recent developments in the numerical solution of differential/algebraic systems

In this paper we survey some recent developments in the numerical solution of nonlinear differential/algebraic equation (DAE) systems of the form 0 = F( t, y, y'), where ∂F ∂y' may be singular. Initial value problems in DAEs arise in a wide variety of applications, including circuit and control theory, chemical kinetics, modeling of constrained mechanical systems, fluid dynamics and robotics. DAE systems include standard form ODEs as a special case, but they also include problems which are in many ways quite different from ODEs. We explore some classes of initial value problems which can be solved by backward differentiation formulas, and discuss some results on the order of convergence of implicit Runge-Kutta methods applied to DAE systems. Finding a consistent set of initial conditions is often a problem for DAE systems arising in applications. We outline some recent work on a general algorithm for finding consistent initial conditions. Finally, we discuss some new developments in the numerical solution of DAE boundary value problems.

Adaptation of initial conditions for internal waves in a slightly compressible fluid

Consequently, the Cauchy problem for (1.2) requires "consistent" initial data that satisfy the conditions (1.3). At the same time the initial data for the original system (i.i) can be given arbitrarily. Then for c >> i, when the solution of the system (i.i) is close to the solution of the limit system (1.2) while the initial data u c, v c, w c, pC, pc are arbitrary and do not satisfy conditions (1.3), a process of "adaptation" of the initial data to these conditions should occur. It is natural ~o propose that this transition process include sound wave radiation, occur at the times x = L/c (L is the characteristic dimension of the problem), and result in consistent initial conditions u, v, w, P, P for which the velocity field is made vortical while the pressure and density turn out to be related by relationship (l. Bb).

A General Form for Solvable Linear Time Varying Singular Systems of Differential Equations

A canonical form is derived for all linear solvable systems $E(t)x'(t) + F(t)x(t) = f(t)$ with sufficiently smooth coefficients E, F Using this form it is shown that for all smooth enough solvable systems a class of recently defined numerical imbedding methods and an algorithm to compute the manifold of consistent initial conditions always work. In addition, necessary and sufficient conditions are given on $E(t)$, $F(t)$ to insure solvability in the case when $E(t)$, $F(t)$ are infinitely differentiable.

Modeling of Geothermal Systems

Summary During the last decade, the use of numerical modeling for geothermal resource evaluation has grown significantly, and new modeling approaches have been developed. In this paper, we present a summary of the current status in numerical modeling of geothermal systems, emphasizing recent developments. Different modeling approaches are described and their applicability discussed. The various modeling tasks-including natural-state, exploitation, injection, multicomponent, and subsidence modeling-are illustrated with geothermal field examples. Introduction A number of different methods for modeling the behavior of geothermal reservoirs are currently available to reservoir engineers. These methods vary widely in complexity and cost of application. In the selection of the proper method for a particular study, one must consider the amount and quality of field data available and the objectives of the study. Geothermal systems are generally very complex, exhibiting such features as fracture-dominated flow, phase change, chemical reactions, and thermal effects. Modeling studies must be carried out to analyze data from geothermal wells accurately and to estimate the generating potential of a system. When a model of a geothermal potential of a system. When a model of a geothermal system is developed, the existing field data must be carefully evaluated, and the important physical processes that occur in the system identified. After a plausible conceptual model of the field is developed, one must choose a mathematical (numerical) model that can realistically evaluate the performance of the geothermal reservoir and reliably predict its future behavior. We have found that modeling the natural state of a field before modeling the field under exploitation can give very valuable reservoir information. It not only tests the conceptual model qualitatively, but also gives estimates of mass and heat flow in the system. Furthermore, it provides consistent initial conditions for the exploitation provides consistent initial conditions for the exploitation models. The primary objective for geothermal reservoir modelinc, is to provide answers to important reservoir management questions relating to well decline, well spacing, generating capacity (power potential) of the reservoir, injection effects, and potential subsidence and scaling problems. These questions must be addressed by use of a problems. These questions must be addressed by use of a proper exploitation model that has evolved from the proper exploitation model that has evolved from the conceptual model and the natural-state modeling studies. In this paper, we present a brief review of geothermal reservoir modeling, emphasizing recent developments. The different modeling, approaches are described, and their advantages and limitations are discussed. We briefly describe the governing equations for mass and heat flow and discuss phase transitions and solution techniques. Examples illustrate the different methodologies for modeling of natural state, exploitation, injection, multicomponent flow, and subsidence. Finally, we identify problems of current interest in geothermal reservoir modeling. Earlier summaries of geothermal reservoir modeling are given by Witherspoon et al. and O'Sullivan. Physical Processes and Physical Processes and Conceptual Models In contrast to oil and gas reservoirs, geothermal systems are very dynamic in their natural state. There is continuous transport of fluid, heat, and chemical species. Important physical processes in geothermal systems include mass transport, convective and conductive heat transfer, phase change (boiling and condensation), dissolution and phase change (boiling and condensation), dissolution and precipitation of minerals, and stress change caused by precipitation of minerals, and stress change caused by pore-pressure changes. Most of these processes are pore-pressure changes. Most of these processes are strongly coupled; for example, phase change disturbs chemical equilibria, often resulting in precipitation/ dissolution of minerals that in time can alter porosities and permeabilities of the subsurface rocks. This in turn can permeabilities of the subsurface rocks. This in turn can affect the mass transport in the system. In modeling geothermal reservoirs, one must carefully evaluate which physical processes need to be considered in a specific modeling study. This will depend on the objectives of the study and the complexity of the geothermal system. Most currently available geothermal simulators consider only single-component mass and heat transport. In recent years, several simulators capable of modeling the transport of a second component, either a noncondensible gas or a dissolved solid, have been developed. Conceptual models of geothermal systems vary greatly in complexity. Perhaps the "simplest" geothermal systems are those created by hot water upflow through a single fault or at the intersection of two or more faults (e.g., Susanville and East Mesa, CA. ) A rather complex porous-medium-type geothermal reservoir is the Cerro Prieto field, Mexico (Fig. 1). The lithology consists of interlayered shale and sandstone beds. The detailed lithology shown in Fig. 1 has been determined mainly on the basis of wireline well logs. JPT P. 1007

Consistent initial conditions for singular nonlinear systems

A necessary condition for consistent initial conditions for singular nonlinear systems has been discussed. It is shown that for linear systems or systems of index less than three these conditions are equivlaent to previously reported results. However, for nonlinear systems of index greater than two these new conditions correct those previously reported. One consequence is that Euler's method may fail to estimate solutions for some semi-state equations.

A self-consistent model for high repetition rate copper vapor lasers

A computer model for a high repetition rate copper vapor laser is described. Equations for the discharge pulse, laser pulse, and interpulse afterglow period are simultaneously integrated over many discharge cycles until a consistent solution is obtained. In this way, consistent initial conditions are obtained and many scaling characteristics of the copper laser are successfully reproduced. This is a major improvement over models which consider only a single-laser pulse. By modeling both an ideal and real laser, and comparing the results to experimental data, we have determined that the peak-elctron temperature and the initial density of metastable copper are the factors which dominate the performance of the laser. Processes which effect these two quantities are therefore the most important in describing the laser.

Singular linear systems of differential equations with delays

The differential equation Ax + Bx = Cx(t-l) + f is studied where A, B, C are square matrices. All matrices are allowed to be singular. Examples are given to show that not all initial functions are consistent and that for consistent initial conditions, solutions may be continuous for only a finite time period. Solutions are given explicitly by a recursion formula and consistent initial conditions are explicitly characterized

Applications of the Drazin Inverse to Linear Systems of Differential Equations with Singular Constant Coefficients

Let A, B be $n \times n$ matrices, f a vector-valued function. A and B may both be singular. The differential equation $Ax' + Bx = f$ is studied utilizing the theory of the Drazin inverse. A closed form for all solutions of the differential equation is given when the equation has unique solutions for consistent initial conditions.