Showing papers on "Free boundary problem published in 1982"
706 citations
TL;DR: In this paper, a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain and estimates of the error due to the finite boundary are obtained for several cases.
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.
603 citations
TL;DR: In this article, the principle of global conservation of mass is used to define a unique direction for the outward pointing normal vector at any node on an irregular boundary of a domain containing an incompressible fluid.
Abstract: SUMMARY Various techniques for implementing normal and/or tangential boundary conditions in finite element codes are reviewed. The principle of global conservation of mass is used to define a unique direction for the outward pointing normal vector at any node on an irregular boundary of a domain containing an incompressible fluid. This information permits the consistent and unambiguous application of essential or natural boundary conditions (or any combination thereof) on the domain boundary regardless of boundary shape or orientation with respect to the co-ordinate directions in both two and three dimensions. Several numerical examples are presented which demonstrate the effectiveness of the recommended technique.
170 citations
144 citations
TL;DR: In this article, the authors derived a nonlinear free boundary problem, the boundary being between the saturated and the unsaturated regions in the medium, and proved the existence, uniqueness, a maximum principle and regularity properties for weak solutions of a Cauchy-Dirichlet problem in the cylinder {(x,t): 0≦x≦1, t≧0}.
Abstract: From the mathematical formulation of a one-dimensional flow through a partially saturated porous medium, we arrive at a nonlinear free boundary problem, the boundary being between the saturated and the unsaturated regions in the medium. In particular we obtain an equation which is parabolic in the unsaturated part of the domain and elliptic in the saturated part. Existence, uniqueness, a maximum principle and regularity properties are proved for weak solutions of a Cauchy-Dirichlet problem in the cylinder {(x,t): 0≦x≦1, t≧0} and the nature, in particular the regularity, of the free boundary is discussed. Finally, it is shown that solutions of a large class of Cauchy-Dirichlet problems converge towards a stationary solution as t → ∞ and estimates are given for the rate of convergence.
114 citations
TL;DR: In this article, the relation between the number of linearly independent solutions of the electro- and magnetostatic boundary value problems and topological characteristics of the underlying domain in 3-dimensional euclidean space is investigated in the framework of Hilbert space theory.
Abstract: The classically well-known relation between the number of linearly independent solutions of the electro- and magnetostatic boundary value problems (harmonic Dirichlet and Neumann vector fields) and topological characteristics (genus and number of boundaries) of the underlying domain in 3-dimensional euclidean space is investigated in the framework of Hilbert space theory. It can be shown that this connection is still valid for a large class of domains with not necessarily smooth boundaries (segment property). As an application the inhomogeneous boundary value problems of electro- and magnetostatics are discussed.
112 citations
TL;DR: In this paper, it was shown that the stability of the initial-boundary value problem for a scalar equation does not necessarily imply stability for a vector equation with a similar boundary treatment.
Abstract: It is well known that the stability of the initial-boundary value problem for a scalar equation does not necessarily imply stability for a vector equation with a similar boundary treatment. In fact, we show that for any boundary treatment one can construct systems for which the boundary treatment on the “natural” variables is not stable. Our main theorem is that one can introduce a treatment of the boundaries for which the stability of the system follows immediately from the stability for a scalar equation. This is accomplished by operating on the characteristic variables for those quantities that are not specified on the boundary. In a working code this can be accomplished by adding a correction term to the existing boundary algorithm. The analysis and computational results are presented for both finite differences and semi-discrete Galerkin methods.
99 citations
IBM1
TL;DR: In this paper, the authors proposed a method to solve singular perturbation problems by partitioning the original problem into inner and outer solution differential equation systems, where the inner solution problem is solved as a two-point boundary problem and the outer solution problems are solved as an initial value system.
TL;DR: In this article, a transonic flow over a supercritical swept wing is considered, and a wave-drag expression is derived for the conservation form of the momentum equation, expressed in terms of a small-disturbance equation.
Abstract: Computational fluid dynamics are used to discuss problems inherent to transonic three-dimensional flow past supercritical swept wings. The formulation for a boundary value problem for the flow past the wing is provided, including consideration of weak shock waves and the use of parabolic coordinates. A swept wing code is developed which requires a mesh of 152 x 10 x 12 points and 200 time cycles. A formula for wave drag is calculated, based on the idea that the conservation form of the momentum equation becomes an entropy inequality measuring the drag, expressible in terms of a small-disturbance equation for a potential function in two dimensions. The entropy inequality has been incorporated in a two-dimensional code for the analysis of transonic flow over airfoils. A method of artificial viscosity is explored for optimum pressure distributions with design, and involves a free boundary problem considering speed over only a portion of the wing.
TL;DR: In this paper, a three-dimensional eddy current problem is formulated as a boundary value problem for electric and magnetic strengths and the boundary integral equations for the densities of these surface sources are derived, these integral equations are of minimum order.
Abstract: A three-dimensional eddy current problem is formulated as a boundary value problem for electric and magnetic strengths. Some peculiarities of this boundary value problem are emphasized. The most important of them is the possibility to split this boundary value problem into two boundary value problems which can be solved in succession one after another: 1) the boundary value problem for the magnetic strength in the whole space and 2) the boundary value problem for the electric strength in outer air space. From the practical point of view it is enough to find the solution only of the first boundary value problem. The principle of separate surface imaginary sources is proposed for the solution of this boundary value problem and the boundary integral equations for the densities of these surface sources are derived, These integral equations are of minimum order.
TL;DR: In this article, the authors studied boundary value problems in a domain with periodically varying cross-section, and proved the unique solvability of these problems in function spaces with weighted norms, and theorems on the Noether property and on the asymptotics of the solutions of the problems with exponentially small perturbations of the coefficients.
Abstract: In a domain with periodically varying cross-section, this paper studies boundary value problems, elliptic in the Douglis-Nirenberg sense, in which the coefficients are periodic functions with the same period. Necessary and sufficient conditions for the unique solvability of these problems in function spaces with weighted norms are proved, and theorems on the Noether property and on the asymptotics of the solutions of boundary value problems with exponentially small perturbations of the coefficients are adduced. Bibliography: 15 titles.
01 Jan 1982
TL;DR: In this paper, a phase field model is derived for free boundary problems where the effects of supercooling and surface tension are present and a scheme for obtaining numerical approximations is derived, and sample numerical results are presented.
Abstract: A phase field model is derived for free boundary problems where the effects of supercooling and surface tension are present. A scheme for obtaining numerical approximations is derived, and sample numerical results are presented. This research was supported in part by the ARO Contract No. DAAG 29-80-C-0081. Partial support was also provided under NASA Contract No. NAS1-15810 while the author was in residence at the Institute for Computer Applications in Science and Engineering, Hampton, VA 23665. UNIVERSITY LIBRARIES CARNEGIE-MELLON UNIVERSITY PITTSBURGH, PENNSYLVANIA 15213 1. The H-method for Stefan Problems The standard description of a solidification process is captured in the classical Stefan problem [1]. In this context T « T(x,t) denotes a temperature field with T* denoting the phase transition temperature. In particular, points x in the material ft are in the liquid phase when T > TA, and conversely, they are in the solid phase when T < T^. At points ££ft where there is no phase transition, i.e., TOc,t) + T^, the following diffusion equation is valid: (1.1) || div(D grad T) x £ ft, t > 0, T(x,t) + T*. The transition region is defined by (1.2) f(t) {x €R : T(x,t) T*}, and for points in this region (1.3) Xv + [D*grad T*vJ* 0, x £ T(t). Here X is the latent heat, v the normal velocity of F(t), \^ the normal to F(t), and [•]_ denotes the jump across F(t). To complete the specification of the problem v» specify initial conditions, e.g., (1.4) T(x,0) » TQ(x) x €ft, for a given initial temperature field TQ, and boundary conditions. For simplicity we use Dirichlet type conditions, namely (1-5) T(x,t) Tx(x), x £ 3ft, t > 0, where T^ is a given temperature field defined on the boundary 3ft of ft.
TL;DR: In this paper, the authors consider the solvability of nonlinear boundary value problems for differential equations where the nonlinearity is bounded and study the asymptotic behavior of certain multivalued functionals.
Abstract: We consider the solvability of some nonlinear boundary value problems for differential equations where the nonlinearity is bounded. This involves the study of the asymptotic behaviour of certain multivalued functionals.
TL;DR: Several different problems of determining unknown parameters or functions in parabolic partial differential equations from overspecified boundary data are treated in this paper, where explicit solutions are presented for various problems and the existence of a solution of various other problems is presented.
TL;DR: In this article, a method to obtain boundary conditions for the wave equation in one dimension, fitting to the discretization scheme and stable, was presented, and error estimates on the reflected part were given.
Abstract: When computing a partial differential equation, it is often necessary to introduce artificial boundaries. Here we explain a systematic method to obtain boundary conditions for the wave equation in one dimension, fitting to the discretization scheme and stable. Moreover, we give error estimates on the reflected part.
TL;DR: In this paper, a multidimensional, multiphase problem of Stefan type, involving quasilinear parabolic equations and nonlinear boundary conditions is considered, and the existence and uniqueness of a weak solution to the problem, as well as continuous and monotone dependence of the solution upon data are shown.
Abstract: A multidimensional, multiphase problem of Stefan type, involving quasilinear parabolic equations and nonlinear boundary conditions is considered. Regularization techniques and monotonicity methods are exploited. Existence and uniqueness of a weak solution to the problem, as well as continuous and monotone dependence of the solution upon data are shown.
TL;DR: In this article, an integral equation formulation for three dimensional anisotropic elastostatic boundary value problems is presented, where both the finite and infinite body with a crack are considered and the boundary integral equation can be solved numerically for the unknown surface tractions and displacements.
TL;DR: In this article, the authors prove the existence of smooth solutions of the open boundary problem for shallow water equations by the bounded derivative method, which requires that a number of initial time derivatives be of the order of the slow time scale and that the boundary data be smooth.
Abstract: The shallow water equations are a symmetric hyperbolic system with two time scales. In meteorological terms, slow and fast scale motions are referred to as Rossby and inertial/gravity waves, respectively. We prove the existence of smooth solutions (solutions with a number of space and time derivatives on the order of the slow time scale) of the open boundary problem for the shallow water equations by the bounded derivative method. The proof requires that a number of initial time derivatives be of the order of the slow time scale and that the boundary data be smooth. If the boundary data are smooth and only have small errors, then we show that the solution of the open boundary problem is smooth and that only small errors are produced in the interior. If the boundary data are smooth but have large errors, then we show that the solution of the open boundary problem is still smooth. Unfortunately the boundary error propagates into the interior at the speed associated with the fast time scale and destroys the solution in a short time. Thus it is necessary to keep the boundary error small if the solution is to be computed correctly. We show that this restriction can be relaxed so that only the large-scale boundary data need be correct. We demonstrate the importance of these conclusions in several numerical experiments.
TL;DR: In this article, the authors studied boundary value problems for singularly perturbed systems of partial differential equations and showed that in the limit of zero viscosity or infinitely large difference the behavior is described by a maximal positive boundary value problem in £2.
Abstract: We study three types of singular perturbations of a symmetric positive system of partial differential equations on a domain £2 C R". In all cases the limiting behavior is given by the solution of a maximal positive boundary value problem in the sense of Friedrichs. The perturbation is either a second order elliptic term or a term large on the complement of Q. The first corresponds to a sort of viscosity and the second to physical systems with vastly different properties in I2 and outside U. The results show that in the limit of zero viscosity or infinitely large difference the behavior is described by a maximal positive boundary value problem in £2. The boundary condition is determined in a simple way from the system and the singular terms. 1. Introduction. In this paper we study boundary value roblems for singularly perturbed systems of partial differential equations. The unperturbed system is of first order and positive symmetric in the sense of Friedrichs. We study three sorts of singular perturbations: (1) the addition of e times a positive second order elliptic system in a domain fi, (2,3) the addition of XP or \P(d/dt) with A » 1 and P strictly positive in the exterior of n and zero in S2. In the first case we study the solutions He to the Dirichlet problem in S2. Bardos, Brezis and Brezis (1) have shown that in this case ue^u weakly in L2(fi) where u is uniquely determined as the solution of a maximal positive boundary value problem for the unperturbed opera- tor. Here one has the phenomenon of loss of boundary conditions in the limit and the existence of boundary layers for uc near 3I2. We prove several more refined estimates on ue, in particular a uniform bound in //l/2_,,(fi) for any rj > 0. These estimates yield a proof of the strong convergence of uc to u in all Hs(u) for s < {-. This singular perturbation problem is well known and much studied, in contrast to the problems of the second sort mentioned above. In the latter problems we study the Cauchy problems
TL;DR: In this article, the authors examined the applicability of recent theoretical developments in the stability analysis of difference approximations for initial boundary-value problems of the hyperbolic type.
Abstract: The applicability to practical calculations of recent theoretical developments in the stability analysis of difference approximations is examined for initial boundary-value problems of the hyperbolic type. For the numerical experiments the one-dimensional inviscid gasdynamic equations in conservation law form are selected. A class of implicit schemes based on linear multistep methods for ordinary differential equations is chosen and the use of space or space-time extrapolations as implicit or explicit boundary schemes is emphasized. Some numerical examples with various inflow-outflow conditions highlight the commonly discussed issues: explicit vs implicit boundary schemes, and unconditionally stable schemes. HEN finite-difference schemes are used to solve initial boundary-value problems for the equations of fluid dynamics, it is well known that most methods require more boundary conditions than those required by the governing partial differential equations. These additional boundary conditions for the finite-difference equations are often called "numerical boundary conditions." The numerical boundary conditions cannot be imposed arbitrarily but are determined, in general, using interior information, for example, by extrapolation or uncentered approximations. In this paper, any numerical procedure used to provide a numerical boundary condition will be called a "boundary scheme/' Whatever schemes are used for the numerical boundary conditions, it is a common practice to assume that the boundary scheme has a local effect and will not affect the solution globally. During the early 1970s, Kreiss,1'2 Osher,3 Gustafsson et al.,4 Varah,5 and Gustafsson6 published a series of papers establishing methods for checking the stability and accuracy of difference approximations with boundary schemes included. Since then, further progress has been made in the theory of linear difference approximations for initial boundary-value problems of the hyperbolic and parabolic type.7'12 Because improper treatment of the boundary conditions can lead to instability and inaccuracy, even though we start with a stable interior scheme (i.e., scheme for the interior points), it is appropriate to adopt an approach that includes the stability and accuracy of the combined interior and boundary schemes. Surveys of recent developments and extensive bibliographies are included in papers by Coughran13 and Yee.14 The purpose of this paper is to examine the applicability to practical calculations (for nonlinear gasdynamic problems) of recent theoretical stability analyses of implicit difference approximations for initial boundary-value problems of the hyperbolic type. As numerical computations have progressed, the use of the conservative form of the gasdynamic equations has gained popularity. For physical reasons it is sometimes desirable to specify boundary conditions in the nonconservative variables and to compute with conservative variables in the interior. We will consider the additional complications introduced by this procedure.
TL;DR: In this paper, Nagumo's boundary value theory is applied to several classes of singularly perturbed boundary value problems of higher order, including scalar nonlinear differential equations and their system analogues.
Abstract: Two differential inequality results of Nagumo on initial and boundary value problems for systems [Proc. Phys. Math. Soc. Japan, 19 (1937), pp. 861–866; 21 (1939), pp. 529–534] are combined to yield existence and comparison results on certain boundary value problems for nth order scalar nonlinear differential equations and their system analogues. This theory is then applied to several classes of singularly perturbed boundary value problems of higher order. Many examples are discussed in order to motivate the theory and indicate avenues of further study.
TL;DR: In this paper, the sensitivity of linear boundary value problems to perturbations of the boundary condition was investigated and a useful quantity to decide for well or ill conditioning was derived, which can be used to explain why the multiple shooting technique is stable for certain well posed problems having solutions with different growth behavior.
Abstract: Investigation is made into the sensitivity of solutions of linear boundary value problems to perturbations of the boundary condition We derive a useful quantity to decide for well or ill conditioning From this it is deduced which kind of requirements the boundary conditions should meet in order to have a well conditioned problem It is shown that this quantity can fruitfully be used to explain why, eg, the multiple shooting technique is stable (in contrast to the single shooting one) for certain well posed problems having solutions with different growth behavior The results are sustained by a number of examples