NUMERICAL SOLUTIONS OF PARABOLIC PROBLEMS ON UNBOUNDED 3-D SPATIAL DOMAIN

In this paper,the numerical solutions of heat equation on 3-D unbounded spatial do-main are considered. n artificial boundary Γ is introduced to finite the computationaldomain.On the artificial boundary Γ,the exact boundary condition and a series of approx-imating boundary conditions are derived,which are called artificial boundary conditions.By the exact or approximating boundary condition on the artificial boundary,the originalproblem is reduced to an initial-boundary value problem on the bounded computationaldomain,which is equivalent or approximating to the original problem.The finite differencemethod and finite element method are used to solve the reduced problems on the finitecomputational domain.The numerical results demonstrate that the method given in thispaper is effective and feasible.     

ARTIFICIAL BOUNDARY METHOD FOR THE THREE-DIMENSIONAL EXTERIOR PROBLEM OF ELASTICITY

The exact boundary condition on a spherical artificial boundary is derived for thethree-dimensional exterior problem of linear elasticity in this paper. After this bound-ary condition is imposed on the artificial boundary, a reduced problem only defined in abounded domain is obtained. A series of approximate problems with increasing accuracycan be derived if one truncates the series term in the variational formulation, which isequivalent to the reduced problem. An error estimate is presented to show how the errordepends on the finite element discretization and the accuracy of the approximate problem.In the end, a numerical example is given to demonstrate the performance of the proposedmethod.     

Convergence of a difference scheme for the heat equation in a long strip by artificial boundary conditions

The numerical solution of the heat equation on a strip in two dimensions is considered. An artificial boundary is introduced to make the computational domain finite. On the artificial boundary, an exact boundary condition is proposed to reduce the original problem to an initial‐boundary value problem in a finite computational domain. A difference scheme is constructed by the method of reduction of order to solve the problem in the finite computational domain. It is proved that the difference scheme is uniquely solvable, unconditionally stable and convergent with the convergence order 2 in space and order 3/2 in time in an energy norm. A numerical example demonstrates the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

凹角区域双曲外问题的精确人工边界条件

研究一类凹角区域双曲型外问题的数值方法.先用Newmark方法对时间进行离散化,在每个时间步求解一个椭圆外问题.然后引入人工边界,并获得精确的人工边界条件.给出半离散化问题的变分问题,证明了变分问题的适定性,并给出了误差估计.最后给出数值例子,以示该方法的可行性与有效性.     

An Artificial Boundary Condition for the Vortex Movements in Two Dimensions

1 Motion of vortices and cloud in cell method The motion of incompressible inviscid ?ow in two dimensions can be described by the equations ?u ?t (u ?) ρ1 ?P = f (1) ?u = 0 , (2) where u = (u, v), ρ, P , and f = (f1, f2) denote ?uid velocity, densit…     

THE GLOBAL ARTIFICIAL BOUNDARY CONDITIONS FOR NUMERICAL SIMULATIONS OF THE 3D FLOW AROUND A SUBMERGED BODY

We consider the numerical approximations of the three-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface in a channel. One vertical side is introduced as the up-stream artificial boundary and two vertical sides are introduced as the downstream arti-ficial boundaries. On the artificial boundaries, a sequence of high-order global artificial boundary conditions are given. Then the original problem is reduced to a problem defined on a finite computational domain, which is equivalent to a variational problem. After solving the variational problem by the finite element method, we obtain the numerical approximation of the original problem. The numerical examples show that the artificial boundary conditions given in this paper are very effective.     

THE ARTIFICIAL BOUNDARY CONDITIONS FOR NUMERICAL SIMULATIONS OF THE COMPLEX AMPLITUDE IN A COUPLED BAY-RIVER SYSTEM

We consider the numerical approximations of the complex amplitude in a coupled bayriver system in this work. One half-circumference is introduced as the artificial boundary in the open sea, and one segment is introduced as the artificial boundary in the river if the river is semi-infinite. On the artificial boundary a sequence of high-order artificial boundary conditions are proposed. Then the original problem is solved in a finite computational domain, which is equivalent to a variational problem. The numerical approximations for the original problem are obtained by solving the variational probiem with the finite element method. The numerical examples show that the artificial boundary conditions given in this work are very effective.     

The artificial boundary conditions for incompressible materials on an unbounded domain

Summary. In this paper we consider the numerical simulations of the incompressible materials on an unbounded domain in . A series of artificial boundary conditions at a circular artificial boundary for solving incompressible materials on an unbounded domain is given. Then the original problem is reduced to a problem on a bounded domain, which be solved numerically by a mixed finite element method. The numerical example shows that our artificial boundary conditions are very effective. ReceivedJune 7, 1995 / Revised version received August 19, 1996     

一类非线性外问题的数值解法

本文利用FEM-BEM方法研究平面上一类非线性外问题数值方法, 给出了基于非线性人工边界条件的耦合问题收敛性结果和误差估计.数值算例验证了我们的理论分析结果. 最后, 我们提出求解其耦合问题的一种区域分解算法.     

Spectral Method for the Black-Scholes Model of American Options Valuation

In this paper, we devote ourselves to the research of numerical methods for American option pricing problems under the Black-Scholes model. The optimal exercise boundary which satisfies a nonlinear Volterra integral equation is resolved by a high-order collocation method based on graded meshes. For the other spatial domain boundary, an artificial boundary condition is applied to the pricing problem for the effective truncation of the semi-infinite domain. Then, the front-fixing and stretching transformations are employed to change the truncated problem in an irregular domain into a one-dimensional parabolic problem in [−1,1]. The Chebyshev spectral method coupled with fourth-order Runge-Kutta method is proposed for the resulting parabolic problem related to the options. The stability of the semi-discrete numerical method is established for the parabolic problem transformed from the original model. Numerical experiments are conducted to verify the performance of the proposed methods and compare them with some existing methods.     

Local Artificial Boundary Conditions for the Incompressible Viscous Flow in a Slip Channel

1.IntroductionManyboundaxyvaJueproblemsofpartialdiffereotialequationsinvo1vingunboundeddomainoccurinmanyareasofapplications,e-g.lfluidflowaroundobstacles,couplingofstructureswithfoundationandsoon.Forgettingthenumericalsolutionsoftheproblemsonunboundeddomian,anaturalapproachistocutoffanunboundedpartofthedomainbyintroducinganartificialboundaryandsetupanaPpropriatear-tificialboundaryconditiononthearti%ialboundaryThentheoriginalproblemisapproximatedbyaproblemonbou.d.dfdomain.Inthelastteny6aJrs,b…     

Solution of exterior problem using ellipsoidal artificial boundary

In this paper, we present a general ellipsoidal artificial boundary method for three-dimensional exterior problem. The exact artificial boundary condition, which is expressed explicitly by the series concerning the ellipsoidal harmonic functions, is derived and then an equivalent problem in a bounded domain is presented. The error estimates show that the convergence rate depends on the mesh parameter, the number of terms used in the exact artificial boundary condition, and the location of the artificial boundary.     

Analysis of artificial boundary conditions for exterior boundary value problems in three dimensions

Summary. Simple boundary conditions on an artificial boundary are discussed, then an exact boundary condition on the artificial boundary is obtained. Approximation to this boundary condition with high accuracy is given, and the error estimates are obtained. A numerical example is presented, and the numerical results are compared with the exact solution. Received January 27, 1997 / Revised version received May 14, 1999 / Published online February 17, 2000     

A System of Plane Elasticity Canonical Integral Equations and Its Application

In this paper, we obtain a new system of canonical integral equations for the plane elasticity problem over an exterior circular domain, and give its numerical solution. Coupling with the classical finite element method, it can be used for solving general plane elasticity exterior boundary value problems. This system of highly singular equations is also an exact boundary condition on the artificial boundary. It can be approximated by a series of nonsingular integral boundary conditions.     

Analysis of a FEM/BEM coupling method for transonic flow computations

A sensitive issue in numerical calculations for exterior flow problems, e.g.around airfoils, is the treatment of the far field boundary conditions on a computational domain which is bounded. In this paper we investigate this problem for two-dimensional transonic potential flows with subsonic far field flow around airfoil profiles. We take the artificial far field boundary in the subsonic flow region. In the far field we approximate the subsonic potential flow by the Prandtl-Glauert linearization. The latter leads via the Green representation theorem to a boundary integral equation on the far field boundary. This defines a nonlocal boundary condition for the interior ring domain. Our approach leads naturally to a coupled finite element/boundary element method for numerical calculations. It is compared with local boundary conditions. The error analysis for the method is given and we prove convergence provided the solution to the analytic transonic flow problem around the profile exists.

     

A finite-element capacitance matrix method for exterior Helmholtz problems

Summary. We introduce an algorithm for the efficient numerical solution of exterior boundary value problems for the Helmholtz equation. The problem is reformulated as an equivalent one on a bounded domain using an exact non-local boundary condition on a circular artificial boundary. An FFT-based fast Helmholtz solver is then derived for a finite-element discretization on an annular domain. The exterior problem for domains of general shape are treated using an imbedding or capacitance matrix method. The imbedding is achieved in such a way that the resulting capacitance matrix has a favorable spectral distribution leading to mesh independent convergence rates when Krylov subspace methods are used to solve the capacitance matrix equation. Received May 2, 1995     

Polynomial and nonpolynomial spline methods for fractional sub-diffusion equations

We consider one-dimensional fractional sub-diffusion equations on an unbounded domain. For a problem of this type for which an exact or approximate artificial boundary condition is available we reduce it to an initial-boundary value problem on a bounded domain. We then analyze the numerical solution of the problem by polynomial and nonpolynomial spline methods. The consistency and the Von Neumann stability analysis of these methods are also discussed. Numerical experiments clarify the effectiveness and order of accuracy of the proposed methods.     

Asymptotics of solutions for a basic case of fluid–structure interaction

We consider the Navier–Stokes equations in a half-plane with a drift term parallel to the boundary and a small source term of compact support. We provide detailed information on the behavior of the velocity and the vorticity at infinity in terms of an asymptotic expansion at large distances from the boundary. The expansion is universal in the sense that it only depends on the source term through some constants. The expansion also applies to the problem of an exterior flow past a small body moving at constant velocity parallel to the boundary, and can be used as an artificial boundary condition on the edges of truncated domains for numerical simulations.     

三维Poisson方程外问题的高阶局部人工边界条件

１引言假设Ｒ３是一分片光滑的闭曲面．是以为边界的无界区域,=Ｒ3是以为边界的有界区域,并且存在球Ｂ0＝ｘｘＲ0我们考虑下面Ｐｏｉｓｓｏｎ方程的外问题：这里f(x）,ｇ（x）是,上的已知函数,f(x)的支集是紧的,即存在一个球面=x·x=R1,使得x＝xｘＲ１,有ｆx＝０．令＝,则ｆ（ｘ）的支集包含在中,令＝xx＝,表示u在上的外法向微商．用流量为零的条件代替无限远处条件（３）,则我们得到一个新的外问题：我们将分别讨论问题（１）－（３）和（４）－（７）的数值解．由于求解区域的无界性,给数值计算带来了本质性的困难．克服此…     

APPROXIMATION, STABILITY AND FAST EVALUATION OF EXACT ARTIFICIAL BOUNDARY CONDITION FOR THE ONE-DIMENSIONAL HEAT EQUATION

In this paper we consider the numerical solution of the one-dimensional heat equation on unbounded domains. First an exact semi-discrete artificial boundary condition is derived by discretizing the time variable with the Crank-Nicolson method. The semi-discretized heat equation equipped with this boundary condition is then proved to be unconditionally stable, and its solution is shown to have second-order accuracy. In order to reduce the computational cost, we develop a new fast evaluation method for the convolution operation involved in the exact semi-discrete artificial boundary condition. A great advantage of this method is that the unconditional stability held by the semi-discretized heat equation is preserved. An error estimate is also given to show the dependence of numerical errors on the time step and the approximation accuracy of the convolution kernel. Finally, a simple numerical example is presented to validate the theoretical results.