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The one-way wave equations occurring in geophysics, underwater acoustics, and numerical studies involving absorbing boundary conditions are characterized analytically. The conditions under which such equations are well posed are obtained by examining the rational functions used to reduce them. Results are presented for Chebyshev, Pade, least-squares, and symmetric least-squares approximations.
The stability and accuracy of various boundary treatments are analyzed for the two-step Richtmyer and MacCormack methods. Special attention is paid to ways of imposing the extra boundary conditions after the first step of the two-step process. The theory of Kreiss is used to study stability properties for both scalar and vector equations. The theory of Skollermo is used to compare accuracies of the various methods. Computations were also performed on both wavelike equations and on systems that approach a steady state. Several suggestions are given for more reliable boundary treatments.
While numerically solving a problem initially formulated on an unbounded domain, one typically truncates this domain, which necessitates setting the artificial boundary conditions (ABC's) at the newly formed external boundary. The issue of setting the ABC's appears to be most significant in many areas of scientific computing, for example, in problems originating from acoustics, electrodynamics, solid mechanics, and fluid dynamics. In particular, in computational fluid dynamics (where external problems present a wide class of practically important formulations) the proper treatment of external boundaries may have a profound impact on the overall quality and performance of numerical algorithms. Most of the currently used techniques for setting the ABC's can basically be classified into two groups. The methods from the first group (global ABC's) usually provide high accuracy and robustness of the numerical procedure but often appear to be fairly cumbersome and (computationally) expensive. The methods from the second group (local ABC's) are, as a rule, algorithmically simple, numerically cheap, and geometrically universal; however, they usually lack accuracy of computations. In this paper we first present a survey and provide a comparative assessment of different existing methods for constructing the ABC's. Then, we describe a relatively new ABC's technique of ours and review the corresponding results. This new technique, in our opinion, is currently one of the most promising in the field. It enables one to construct such ABC's that combine the advantages relevant to the two aforementioned classes of existing methods. Our approach is based on application of the difference potentials method attributable to V. S. Ryaben'kii. This approach allows us to obtain highly accurate ABC's in the form of certain (nonlocal) boundary operator equations. The operators involved are analogous to the pseudodifferential boundary projections first introduced by A. P. Calderon and then also studied by R. T. Seeley. The apparatus of the boundary pseudodifferential equations, which has formerly been used mostly in the qualitative theory of integral equations and PDE'S, is now effectively employed for developing numerical methods in the different fields of scientific computing.
Attention is given to the application of the adjoint variable method under the special combination of displacement and stress boundary conditions which occur when static test conditions are simulated. In the two-step problem-solving method employed, calculation of internal stresses, displacements and total reaction force at the boundary is followed by the calculation of the applied load and the total reaction force due to unit displacements.
Radiative heat transfer in plate-type fins with unsymmetric thermal boundary conditions
Measured shapes of vibrational modes used to update mathematical models. Report extends discussion of multiple-boundary-condition vibrational testing techniques. Emphasis on further refinement of mathematical model by use of differences between measured eigenvectors and eigenvectors predicted by model to be refined.
Stress trajectories used to calculate stresses in irregular-shaped bodies - solutions of elasticity problems using boundary conditions
Spacecraft structures tested by sine-sweep vibration in boundary-condition simulation
Differential operators and adjoints defined on Hilbert space of n x 1 matrices under integral and multiple point boundary conditions
A fully implicit set of boundary conditions is developed for the solution of the parabolized Navier-Stokes equations for supersonic flow in two dimensions. Shock fitting is employed at the shock and the body has no-slip and specified temperature conditions. A specified heat transfer condition at the wall can be handled in a similar manner. In addition, the shock location is advanced in space in a fully implicit manner by utilizing the Rankine-Hugoniot conditions along with global conservation of mass.
The present examination of an extrapolation of velocity along an arbitrary direction in the case of inviscid solid boundary conditions demonstrates the error associated with the arbitrary direction and establishes that the most appropriate procedure lies in the extrapolation of the velocity components that are tangential to the body surface. It is noted that, in a typical calculation, mesh lines generally cluster together and are therefore parallel or nearly parallel to the body surface.
The extended boundary condition method is applied to Monte Carlo simulations of two-dimensional random rough surface scattering. The numerical results are compared with one-dimensional random rough surfaces obtained from the finite-element method. It is found that the mean scattered intensity from two-dimensional rough surfaces differs from that of one dimension for rough surfaces with large slopes.
The practical limit of airfoil thickness ratio for which acceptable engineering results are obtainable with the Dirichlet boundary-condition-based numerical methods is investigated. This is done by studying the effect of thickness on the calculated pressure distribution near the trailing edge and by comparing the aerodynamic coefficients with available exact solutions. The first objective of this study, owing to the wide use of such computational methods, is to demonstrate the numerical symptoms that occur when the body or wing thickness approaches zero and to increase the awareness of potential users of these methods. Additionally, an effort is made to obtain the practical limits of the trailing-edge thickness where such problems will appear in the flow solution, and to propose some possible cures for very thin airfoils or those with cusped trailing edges.
The problem of the specification of artificial outflow conditions in flow problems is studied. It is shown that for transport type equations incorrect outflow conditions will adversely affect the solution only in a small region near the outflow boundary, while for elliptic equations, e.g. those governing the streamfunction or pressure, a correct boundary specification is essential. In addition, integral outflow boundary conditions for fluid dynamical problems are considered. It is shown that such conditions are well posed, and their effect on the solutions of the Navier-Stokes equations is also considered.
The influence on a model-generated January climate of various surface boundary conditions, as well as initial conditions, was studied by using the GISS coarse-mesh climate model. Four experiments - two with water planets, one with flat continents, and one with mountains - were used to investigate the effects of initial conditions, and the thermal and dynamical effects of the surface on the model generated-climate. However, climatological mean zonal-symmetric sea surface temperature is used in all four runs over the model oceans. Moreover, zero ground wetness and uniform ground albedo except for snow are used in the last experiments.
From the meteoritic results presented in the preceding chapters, a set of boundary conditions is identified that can be applied, with variable degrees of confidence, as constraints to models that attempt to describe the origin and early evolution of the solar system. Those constraints bear upon the time scale, physical and chemical environment, and processes that characterized the early solar system.
Reflection laws of high order elliptic differential equations in two independent variables with constant coefficients and unequal characteristics across analytic boundary conditions
Rigorous uniform geometrical theory of diffraction (UGTD) diffraction coefficients are presented for a coated convex cylinder simulated with generalized impedance boundary conditions. In particular, ray solutions are obtained which remain valid in the transition region and reduce uniformly to those in the deep lit and shadow regions. These involve new transition functions in place of the usual Fock-type integrals, characteristic to the impedance cylinder. A uniform asymptotic solution is also presented for observations in the close vicinity of the cylinder. As usual, the diffraction coefficients for the convex cylinder are obtained via a generalization of the corresponding ones for the circular cylinder.