Showing papers on "Boundary value problem published in 1982"

An Interior Penalty Finite Element Method with Discontinuous Elements

Abstract: A new semidiscrete finite element method for the solution of second order nonlinear parabolic boundary value problems is formulated and analyzed. The test and trial spaces consist of discontinuous piecewise polynomial functions over quite general meshes with interelement continuity enforced approximately by means of penalties. Optimal order error estimates in energy and $L^2$-norms are stated in terms of locally expressed quantities. They are proved first for a model problem and then in general.

Stochastic theory of a data-handling system with multiple sources

Abstract: In this paper we consider a physical model in which a buffer receives messages from a finite number of statistically independent and identical information sources that asynchronously alternate between exponentially distributed periods in the ‘on’ and ‘off’ states. While on, a source transmits at a uniform rate. The buffer depletes through an output channel with a given maximum rate of transmission. This model is useful for a data-handling switch in a computer network. The equilibrium buffer distribution is described by a set of differential equations, which are analyzed herein. The mathematical results render trivial the computation of the distribution and its moments and thus also the waiting time moments. The main result explicitly gives all the system's eigenvalues. While the insertion of boundary conditions requires the solution of a matrix equation, even this step is eliminated since the matrix inverse is given in closed form. Finally, the simple expression given here for the asymptotic behavior of buffer content is insightful, for purposes of design, and numerically useful. Numerical results for a broad range of system parameters are presented graphically.

Boundary conditions for the numerical solution of elliptic equations in exterior regions

Abstract: Elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the well-posedness of the problem. Examples of practical applications include the Helmholtz equation and Laplace's equation. Computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition on a finite artificial surface. Direct imposition of the condition at infinity along the finite boundary results in large errors. A sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain. Estimates of the error due to the finite boundary are obtained for several cases. Computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions. The examples are based on a finite element formulation but finite difference methods can also be used.

The initial growth of disturbances in a baroclinic flow

Abstract: The growth of perturbations in a baroclinic flow is examined as an initial value problem. Although the long time asymptotic behavior is dominated by discrete exponentially growing normal modes when they exist, these do not form a complete set and initial intensification is shown to be dependent on the continuous spectrum. The vertical structure of perturbations emerges as an important influence on initial growth, and physically realistic disturbances are shown to grow to amplitudes where nonlinear effects are important before obtaining normal mode form. Connection is made with the work of Arnol'd (1965) and Blumen (1968) and the numerical experiments of Simmons and Hoskins (1979). Application of these results to cyclogenesis in geographically fixed areas is suggested and implied constraints on numerical models discussed.

A stochastic‐convective transport representation of dispersion in one‐dimensional porous media systems

Abstract: A stochastic-convective transport formulation based upon solute travel time probability is presented and is shown to include Fickian transport as a special subcase. It is further demonstrated that a travel time probability associated with a lognormal distribution of hydraulic conductivity will yield concentration breakthrough curves nearly equivalent to those of Fickian transport when the coefficient of variation in travel time is sufficiently small, i.e., less than unity. When applied to column tracer experiments, the results suggest that typical laboratory-measured hydrodynamic dispersion may be ascribed to local variations in hydraulic conductivity. Moreover, stochastic-convective transport is shown to conserve solute mass under a flux boundary condition but to fail to do so under a held concentration condition. This result indicates the importance of boundary conditions for properly formulated stochastic transport. The travel time formulation is shown to provide a direct link between measured dispersivity and the autocovariance of local flow velocity variations that are a consequence of media inhomogeneity and the system boundary conditions. Dispersivity is shown to manifest a scale effect by increasing in proportion to the length of a system when the velocity correlation range is greater than that length. An expression for the effective dispersivity is derived for the case of long-range velocity correlations and is shown to represent non-Fickian behavior.

Singular Perturbation Analysis of Boundary Value Problems for Differential-Difference Equations

Abstract: A study is made of a class of boundary-value problems for linear second-order differential-difference equations in which the highest-order derivative is multiplied by a small parameter. Consideration of the exact solution of simple model equations provides insight into the appropriate use of singular perturbation techniques for more general problems. The resulting analysis leads to several novel features which are not present in problems without differences.

Prediction of Subsonic Aerodynamic Characteristics: A Case for Low-Order Panel Methods

Abstract: A low-order panel method is presented for the calculation of subsonic aerodynamic characteristics of general configurations. The method is based on piecewise constant doublet and source singularities. Two forms of the internal Dirichlet boundary condition are discussed and the source distribution is determined by the external Neumann boundary condition. Calculations are compared with higher-order solutions for a number of cases. It is demonstrated that for comparable density of control points where the boundary conditions are satisfied, the low-order method gives comparable accuracy to the higher-order solutions. It is also shown that problems associated with some earlier low-order panel methods, e.g., leakage in internal flows and junctions and also poor trailing-edge solutions, do not appear for the present method. Further, the application of the Kutta condition is extremely simple; no extra equation or trailing-edge velocity point is required. The method has very low computing costs and this has made it practical for application to nonlinear problems requiring iterative solutions and to three-dimensional unsteady problems using a time-stepping approach. In addition, the method has been extended to model separated flows in three dimensions, using free vortex sheets to enclose the separated zone.

Stochastically gated diffusion‐influenced reactions

Abstract: The theory of diffusion‐influenced reactions is extended to cases where the reactivity of the species fluctuates in time (e.g., the accessibility of a binding site of a protein is modulated by a gate). The opening and closing of the gate is assumed to be a stationary Markov process [i.e., it is described by the kinetic scheme (open) a⇄b (closed)]. When the reaction is described by suitable boundary conditions, by solving the appropriate reaction‐diffusion equations, it is shown that the stochastically gated association rate constant (kSG) is given by k−1SG=k−1∞ + [a−1 b(a+b)κu(a+b)]−1, where κu(s) is the Laplace transform of the time‐dependent rate constant of the ungated problem and k∞ is the corresponding steady‐state rate constant. The limits when the relaxation time for gate fluctuations is larger or smaller than the characteristic time for diffusion are considered. The relation to previous work is discussed. The theory is applied to three models: (i) a gated sphere, (ii) a gated disk on an infinite plane (e.g., a channel in a membrane), and (iii) a gated localized axially symmetric reactive site on the surface of a spherical macromolecule.

Grain boundary diffusion mechanisms in metals

Abstract: In recent years it has become well established that fast diffusion along grain boundaries plays a key role in many important metallurgical processes including cases where net mass is transported along boundaries which act as sources and/or sinks for the fluxes of atoms. In addition, considerable advances have been made in understanding grain boundary structure, and new techniques have become available for studying kinetic phenomena in grain boundaries. This lecture will attempt to review our current knowledge of the atomistic mechanisms responsible for these grain boundary diffusion phenomena. Relevant aspects of the structure of grain boundaries and the point and line defects which may exist in grain boundaries are described first. The important experimental observations are then discussed. Diffusion models are then taken up, and it is concluded that the atomic migration occurs by a point defect exchange mechanism which, in at least the vast majority of boundaries in simple metals, most likely involves grain boundary vacancies. The grain boundary sources and/or sinks required to support divergences in the atomic (vacancy) fluxes are grain boundary dislocations. Phenomena therefore occur which resemblethe Kirkendall Effect in the bulk lattice in certain respects. Additional topics are discussed which include effects of boundary structure on boundary diffusion and the question of whether or not boundary diffusion is faster along migrating than stationary boundaries.