Showing papers on "Boundary value problem published in 1985"

Elliptic Problems in Nonsmooth Domains

Abstract: Foreword Preface 1. Sobolev spaces 2. Regular second-order elliptic boundary value problems 3. Second-order elliptic boundary value problems in convex domains 4. Second-order boundary value problems in polygons 5. More singular solutions 6. Results in spaces of Holder functions 7. A model fourth-order problem 8. Miscellaneous Bibliography Index.

The boundary value problems of mathematical physics

Abstract: I Preliminary Considerations.- II Equations of Elliptic Type.- III Equations of Parabolic Type.- IV Equations of Hyperbolic Type.- V Some Generalizations.- VI The Method of Finite Differences.

A nonreflecting boundary condition for discrete acoustic and elastic wave equations

Abstract: I J ~ One of the nagging problems which arises in application of discrete solution methods for wave-propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh. These undesired events eventually over­ ride the actual seismic signals which propagate in the modeled region. The solution to avoiding boundary effects used to be to enlarge the numerical mesh, thus delaying the side reflections and wraparound longer than the range of times involved in the modeling. Obviously this solution considerably increases the expense of computation. More recently, nonreflecting bound­ ary conditions were introduced for the finite-difference method (Clayton and Enquist, 1977: Reynolds, (978). These boundary conditions are based on replacing the wave equation in the boundary region by one-way wave equations which do not permit energy to propagate from the boundaries into the nu­ merical mesh. This approach has been relatively successful, except that its effectiveness degrades for events which impinge on the boundaries at shallow angles. It is also not clear how to apply this type of boundary condition to global discrete meth­ ods such as the Fourier method for which all grid points are coupled. In this note we describe an alternative scheme for construct­ ing a nonreflecting boundary condition. It is based on gradual reduction of the amplitudes in a strip of nodes along the bound­ aries of the mesh. The method appears extremely simple and robust, and it can be applied to a wide variety of time­ dependent problems. Unlike other methods, the effectiveness of this boundary condition does not decrease for shallow angles of incidence.

Numerical Treatment of Cross-Shelf Open Boundaries in a Barotropic Coastal Ocean Model

Abstract: Using a barotropic coastal ocean model with a straight coastline and uniform cross-shelf bottom slope, seven different cross-shelf open boundary conditions (four of which are applied in either implicit or explicit form) are compared in three numerical experiments 1) A mound of water is allowed to collapse and radiate waves toward the open boundaries 2) A uniform alongshelf wind stress is applied at zero time over the entire shelf and held constant for the duration of the experiment 3) A uniform cross-shelf wind stress is applied at zero time over the entire shelf and held constant for the duration of the experiment The boundary condition which is most transparent to waves consists of a sponge at the outer edge of the model domain with an Orlanski radiation condition at the outer edge of the sponge Several open boundary conditions perform adequately in the wind stress experiments, but the Orlanski radiation condition alone (without a sponge) appears to give the best total performance (of thes

Uniform decay estimates and the lorentz invariance of the classical wave equation

Abstract: On etudie le comportement asymptotique des solutions de □u=0 ou □=∂ t 2 −∂ 1 2 ...−∂ n 2 pour des conditions initiales u=0, u t =g(x) en t=0, avec g reguliere a support compact dans R n

The derivation and numerical solution of the equations for zero mach number combustion

Abstract: We present a limiting system of equations to describe combustion processes at low Mach number in either confined or unbounded regions and numerically solve these equations for the case of a flame propagating in a closed vessel. This system allows for large heat release, substantial temperature and density variations, and substantial interaction with the hydrodynamic flow field, including the effects of turbulence. This limiting system is much simpler than the complete system of equations of compressible reacting gas flow since the detailed effects of acoustic waves have been removed. Using a combination of random vortex techniques and flame propagation algorithms specially designed for turbulent combustion, we describe a numerical method to solve these zero Mach number equations. We use this method to analyze the competing effects of viscosity, exothermicity, boundary conditions and pressure on the rate of combustion for a flame propagating in a swirling flow inside a square.

Computing Three-Dimensional Incompressible Flows with Vortex Elements

Abstract: The techniques, capabilities and applicability of numerical models of three-dimensional, unsteady vortical flows with high Re are assessed. Vorticity is calculated only in appropriate regions and the velocity field is derived from the boundary conditions. Vorticity is assumed to take the shape of tubes with uniform core structures in the case of turbulence. The efforts being made to simplify equations for dense collections of vortex filaments in order to make them tractable to computer simulations are described. The effectiveness of vorticity arrow representations for accurately describing vorticity fields near surfaces is discussed, along with Lagrangian vortex elements, which may be of use in modelling the rotational part of flows around bluff bodies, nonuniform density flows and chemically reacting flows.

On the solution of integral equations with strong ly singular kernels

Abstract: In this paper some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m, m or = 1 Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations A mixed boundary value problem from the theory of elasticity is considered as an example Particularly for integral equations where the kernel contains, in addition to the dominant term (t,x) sup-m, terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results

The Global Cauchy Problem for the Non Linear Klein-Gordon Equation.

Abstract: On demontre l'unicite de solutions faibles et l'existence et l'unicite de solutions globales fortement continues dans l'espace energie

Fundamental Solutions Method for Elliptic Boundary Value Problems

Abstract: We consider a procedure for solving boundary value problems for elliptic homogeneous equations, known as the fundamental solutions method. We prove its applicability for some second order operators as well as for fourth order ones. The boundary conditions of an elliptic problem are approximated by using fundamental solutions of the corresponding operator with singularities located outside the domain of interest. In a specific case of the Laplacian an estimate shows that the method discussed possesses convergence properties as good as those of any method using harmonic polynomials as trial functions. Numerical examples are given, among which are simple Signorini’s problems for harmonic and Lame operators.

Spherical cap harmonic analysis

Abstract: The solution of Laplace's equation, in spherical coordinates, is developed for the boundary value problem appropriate to fitting the geomagnetic field over a spherical cap. The solution involves associated Legendre functions of integral order but nonintegral degree. The basis functions comprise two infinite sets, within each of which the functions are mutually orthogonal. The series for the expansion of the potential can by design be differentiated term by term to yield uniformly convergent series for the field components. The method is demonstrated by modeling the International Geomagnetic Reference Field 1980 at the earth's surface and upward continuing it to 300 and 600 km. The rate of convergence of the series is rapid, and standard errors of fit as low as the order of a nanotesla can be obtained with a reasonable number of coefficients. Upward continuation suffers from not considering data outside the cap, the deterioration being confined to the boundary at low continuation altitudes but spreading inward over the cap with increasing altitudes. At 600 km the standard error of upward continuation is about 5 times the standard error of fit. Both the fit and the upward continuation can be greatly improved at a given truncation level by subtraction of a known spherical harmonic potential determined from data from the whole earth.

Self-similar solutions for elastohydrodynamic cavity flow

Abstract: The enlargement of a lens-shaped cavity lying in a plane of cleavage between two elastic half spaces and filling with viscous fluid from a source on the axis of symmetry is considered. The internal flow is modelled by lubrication theory, which gives a nonlinear partial differential equation connecting the pressure to the cavity shape, and the same two quantities are also related by the singular integral equation of linear elasticity. If the total volume of fluid Q ( t ) in the cavity at time t is proportional either to t α or to exp (α t ) the resulting boundary value problem can be reduced to a self-similar form in which time does not appear explicitly. The solution in non-dimensional terms depends on a single parameter, which may be interpreted as the stress-intensity factor K at the tip. Calculations have been made for the two-dimensional version of the problem for a range of values of α and for a range of stress intensities. The numerical method is to expand the cavity height in a Chebyshev series, the coefficients being found by a nonlinear optimization technique to yield a least squares fit to the Reynolds equation. These lead to expressions for the rate of cavity growth and other quantities of physical interest.

On convergence of computation of chemically reacting flows

Abstract: The computational problems associated with high-temperature flows undergoing finite-rate ionization reactions is investigated. The conservation equations governing chemical species and vibrational and electron energies are solved simultaneously with those for overall mass, momentum, and energy for a one-dimensional subsonic flow, through a constant-area duct, originating behind a normal shock wave, using an implicit time-marching technique. Boundary conditions are imposed in the form of characteristic wave variables accounting for the effects of chemical reactions on the speed of sound. Converging solutions are obtained for cases in which chemical reactions are weak, but difficulty is encountered in other cases. The cause of the difficulty is investigated and shown to be the sharp pressure disturbances produced by such reactions.

Symmetric positive systems with boundary characteristic of constant multiplicity

Abstract: The theory of maximal positive boundary value problems for symmetric positive systems is developed assuming that the boundary is characteristic of constant multiplicity. No such hypothesis is needed on a neighborhood of the boundary. Both regularity theorems and mixed initial boundary value problems are discussed. Many classical ideas are sharpened in the process.

The penetration of a finger into a viscous fluid in a channel and tube

Abstract: The steady-state shape of a finger penetrating into a region filled with a viscous fluid is examined. The two-dimensional and axisymmetric problems are solved using Stokes equations for low Reynolds number flow. To solve the equations, an assumption for the shape of the finger is made and the normal-stress boundary condition is dropped. The remaining equations are solved numerically by covering the domain with a composite mesh composed of a curvilinear grid which follows the curved interface, and a rectilinear grid parallel to the straight boundaries. The shape of the finger is then altered to satisfy the normal-stress boundary condition by using a nonlinear least squares iteration method. The results are compared with the singular perturbation solution of Bretherton (J. Fluid Mech., 10 (1961), pp. 166–188). When the axisymmetric finger moves through a tube, a fraction $m$ of the viscous fluid is left behind on the walls of the tube. The fraction $m$ was measured experimentally by Taylor (J. Fluid Mech., 10 (1961), pp. 161–165) as a function of the dimensionless parameter µU/T. The numerical results are compared with the experimental results of Taylor.

Sensitivity Analysis for Steady State Groundwater Flow Using Adjoint Operators

Abstract: Adjoint sensitivity theory is currently being considered as a potential method for calculating the sensitivity of nuclear waste repository performance measures to the parameters of the system. For groundwater flow systems, performance measures of interest include piezometric heads in the vicinity of a waste site, velocities or travel time in aquifers, and mass discharge to biosphere points. The parameters include recharge-discharge rates, prescribed boundary heads or fluxes, formation thicknesses, and hydraulic conductivities. The derivative of a performance measure with respect to the system parameters is usually taken as a measure of sensitivity. To calculate sensitivities, adjoint sensitivity equations are formulated from the equations describing the primary problem. The solution of the primary problem and the adjoint sensitivity problem enables the determination of all of the required derivatives and hence related sensitivity coefficients. In this study, adjoint sensitivity theory is developed for equations of two-dimensional steady state flow in a confined aquifer. Both the primary flow equation and the adjoint sensitivity equation are solved using the Galerkin finite element method. The developed computer code is used to investigate the regional flow parameters of the Leadville Formation of the Paradox Basin in Utah. The results illustrate the sensitivity of calculated local heads to the boundary conditions. Alternatively, local velocity related performance measures are more sensitive to hydraulic conductivities.

Electric currents in the solar corona and the existence of magnetostatic equilibrium

Abstract: The random motions of magnetic field lines induced by convective flows below the solar surface cause braiding and twisting of the coronal magnetic field, and may be responsible for heating the solar corona The suggestion by Parker (1972) that the field does in general not attain equilibrium, and must develop current sheets in which the braiding patterns are dissipated (topological dissipation), is considered Using an analogy with two-dimensional flows, it is shown that invariance of the winding pattern in the general direction of the field is not a necessary requirement for equilibrium, as Parker suggested Discontinuities in the magnetic field (current sheets) arise only if the velocity field at the photospheric boundary is itself a discontinuous function of position This suggests that the corona field can simply adjust to the slowly changing boundary conditions in the photosphere, and that topological dissipation of the winding patterns does not take place Some implications for coronal heating are discussed 35 references

A method of calculating radiative heat diffusion in particle simulations

Abstract: A new method for solving heat diffusion in three dimensional particle simulations is described. The difficulties encounted by other authors is discussed, in particular the difficulty of including boundary conditions in particle simulations. One and three dimensional tests of the method are described.

Boundary conditions for lightness computation in Mondrian World

Abstract: Land's retinex theory of lightness computation explains how for a “Mondrian World” image, consisting of a number of patches each of uniform reflectance, the reflectances can be computed from an image of that object. Horn has shown that the computation can be realised as a parallel process performed by successive layers of cooperating computational cells, arranged on hexagonal grids. However, the layers will, in practice, be arrays of finite extent and it is shown to be critical that cells on array boundaries behave correctly. The computation is first analysed in continuous terms, expressed as the solution of a differential equation with certain boundary conditions, and proved to be optimal in a certain sense. The finite element method is used to derive a discrete algorithm.