Artificial boundary method for two-dimensional Burgers’ equation

The numerical solution of the two-dimensional Burgers equation in unbounded domains is considered. By introducing a circular artificial boundary, we consider the initial-boundary problem on the disc enclosed by the artificial boundary. Based on the Cole–Hopf transformation and Fourier series expansion, we obtain the exact boundary condition and a series of approximating boundary conditions on the artificial boundary. Then the original problem is reduced to an equivalent problem on the bounded domain. Furthermore, the stability of the reduced problem is obtained. Finally, the finite difference method is applied to the reduced problem, and some numerical examples are given to demonstrate the feasibility and effectiveness of the approach.     

Discrete artificial boundary conditions for transient scalar wave propagation in a 2D unbounded layered media

This paper presents an efficient numerical method for direct time-domain solution of the transient scalar wave propagation in a two-dimensional unbounded multi-layer soil. The unbounded domain is truncated by an artificial boundary which demands the corresponding boundary conditions. In the new approach, only the artificial boundary is discretized into one-dimensional finite elements, yielding a new time-dependent partial differential equation (PDE) for displacements with respect to only one spatial coordinate. Factorization of the PDE and introduction of the residual radiation functions, there results a linear first-order ordinary differential equation (ODE). Its stability is ensured. The time-dependent discrete artificial boundary conditions are determined by the solution of the ODE. In general, it is local in time, but it is non-local in space. Several numerical examples are given to verify the superiority of the proposed method.     

A finite element scheme with a high order absorbing boundary condition for elastodynamics

A high-order absorbing boundary condition (ABC) is devised on an artificial boundary for time-dependent elastic waves in unbounded domains. The configuration considered is that of a two-dimensional elastic waveguide. In the exterior domain, the unbounded elastic medium is assumed to be isotropic and homogeneous. The proposed ABC is an extension of the Hagstrom–Warburton ABC which was originally designed for acoustic waves, and is applied directly to the displacement field. The order of the ABC determines its accuracy and can be chosen to be arbitrarily high. The initial boundary value problem including this ABC is written in second-order form, which is convenient for geophysical finite element (FE) analysis. A special variational formulation is constructed which incorporates the ABC. A standard FE discretization is used in space, and a Newmark-type scheme is used for time-stepping. A long-time instability is observed, but simple means are shown to dramatically postpone its onset so as to make it harmless during the simulation time of interest. Numerical experiments demonstrate the performance of the scheme.     

Mixed finite element method and high-order local artificial boundary conditions for exterior problems of elliptic equation

The mixed finite element method is used to solve the exterior elliptic problem with high-order local artificial boundary conditions. New unknowns are introduced to reduce the order of the derivatives to two. This leads to an equivalent mixed variational problem such that the normal finite element can be used and special finite elements are no longer needed on the adjacent layer of the artificial boundary. Error estimates are obtained for some local artificial boundary conditions with prescribed order. Numerical examples are presented and the results demonstrate the effectiveness of this method.     

Exact Non-Reflecting Boundary Conditions on Perturbed Domains and <Emphasis Type="Italic">hp</Emphasis>-Finite Elements

For exterior scattering problems one of the chief difficulties arises from the unbounded nature of the problem domain. Inhomogeneous obstacles may require a volumetric discretization, such as the Finite Element Method (FEM), and for this approach to be feasible the exterior domain must be truncated and an appropriate condition enforced at the far, artificial, boundary. An exact, non-reflecting boundary condition can be stated using the classical DtN-FE method if the Artificial Boundary’s shape is quite specific: circular or elliptical. Recently, this approach has been generalized to permit quite general Artificial Boundaries which are shaped as perturbations of a circle resulting in the “Enhanced DtN-FE” method. In this paper we extend this method to a two-dimensional FEM featuring high-order polynomials in order to realize a high rate of convergence. This is more involved than simply specifying high-order test and trial functions as now the scatterer shape and Artificial Boundary must be faithfully represented. This entails boundary elements which conform (to high order) to the true boundary shapes. As we show, this can be accomplished and we realize an arbitrary order FEM without spurious reflections.     

The coupling of boundary integral and finite element methods for the exterior steady Oseen problem

In this article, we present a new numerical method for solving the steady Oseen equations in an unbounded plane domain. The technique consists in coupling the boundary integral and the finite element methods. An artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior homogeneous one. The solution in exterior region is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocity-pressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. Finally, the optimal error estimates of the numerical solution are derived.Computer results will be discussed in a forthcoming paper.     

A direct boundary integral equation method for the numerical construction of harmonic functions in three-dimensional layered domains containing a cavity

We consider boundary value problems for the Laplace equation in three-dimensional multilayer domains composed of an infinite strip layer of finite height and a half-space containing a bounded cavity. The unknown (harmonic) function satisfies the Neumann boundary condition on the exterior boundary of the strip layer (i.e. at the bottom of the first layer), the Dirichlet, Neumann or Robin boundary condition on the boundary surface of the cavity and the corresponding transmission (matching) conditions on the interface layer boundary. We reduce this boundary value problem to a boundary integral equation over the boundary surface of the cavity by constructing Green's matrix for the corresponding transmission problem in the domain consisting of the infinite layer and the half-space (not with the cavity). This direct integral equation approach leads, for any of the above boundary conditions, to boundary integral equations with a weak singularity on the cavity. The numerical solution of this equation is realized by Wienert's [Die Numerische approximation von Randintegraloperatoren für die Helmholtzgleichung im R ³, Ph.D. thesis, University of Göttingen, Germany, 1990] method. The reduction of the problem, originally set in an unbounded three-dimensional region, to a boundary integral equation over the boundary of a bounded domain, is computationally advantageous. Numerical results are included for various boundary conditions on the boundary of the cavity, and compared against a recent indirect approach [R. Chapko, B.T. Johansson, and O. Protsyuk, On an indirect integral equation approach for stationary heat transfer in semi-infinite layered domains in R ³ with cavities, J. Numer. Appl. Math. (Kyiv) 105 (2011), pp. 4–18], and the results obtained show the efficiency and accuracy of the proposed method. In particular, exponential convergence is obtained for smooth cavities.     

The Local Artificial Boundary Conditions for Numerical Simulations of the Flow Around a Submerged Body

We consider in this work the numerical approximations of the two-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface. Several vertical segments are introduced as the upstream and the downstream artificial boundaries, where a sequence of high-order local 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 problem with the finite element method. The numerical examples show that the artificial boundary conditions given in this work are very effective.     

Nonreflecting boundary condition for the Helmholtz equation

To solve the Helmholtz equation in an infinite three-dimensional domain a spherical artificial boundary is introduced to restrict the computational domain Ω. To determine the nonreflecting boundary condition on ∂Ω, we start with a finite number of spherical harmonics for the Helmholtz equation. With a precise choice of (primary) nodes on the sphere, the theorem on Gauss-Jordan quadrature establishes the discrete orthogonality of the spherical harmonics when summed over these nodes. An approximate nonreflecting boundary condition for the Helmholtz equation follows readily upon solving the exterior Dirichlet problem. The accuracy of the boundary condition is determined using a point source, and the computational results are presented for the scattering of a wave from a sphere.     

Outflow Boundary Conditions for the Fourier Transformed One-Dimensional Vlasov–Poisson System

In order to facilitate numerical simulations of plasma phenomena where kinetic processes are important, we have studied the technique of Fourier transforming the Vlasov equation analytically in the velocity space, and solving the resulting equation numerically. Special attention has been paid to the boundary conditions of the Fourier transformed system. By using outgoing wave boundary conditions in the Fourier transformed space, small-scale information in velocity space is carried outside the computational domain and is lost. Thereby the so-called recurrence phenomenon is reduced. This method is an alternative to using numerical dissipation or smoothing operators in velocity space. Different high-order methods are used for computing derivatives as well as for the time-stepping, leading to an over-all fourth-order method.     

A combined finite and infinite element approach for modeling spherically symmetric transient subsurface flow

This paper presents a finite element-infinite element coupling approach for modeling a spherically symmetric transient flow problem in a porous medium of infinite extent. A finite element model is used to examine the flow potential distribution in a truncated bounded region close to the spherical cavity. In order to give an appropriate artificial boundary condition at the truncated boundary, a transient infinite element, that is developed to describe transient flow in the exterior unbounded domain, is coupled with the finite element model. The coupling procedure of the finite and infinite elements at their interface is described by means of the boundary integro-differential equation rather than through a matrix approach. Consequently, a Neumann boundary condition can be applied at the truncated boundary to ensure the C¹-continuity of the solution at the truncated boundary. Numerical analyses indicate that the proposed finite element-infinite element coupling approach can generate a correct artificial truncated boundary condition to the finite element model for the unbounded flow transport problem.     

Absorbing boundary conditions for time harmonic wave propagation in discretized domains

While many successful absorbing boundary conditions (ABCs) are developed to simulate wave propagation into unbounded domains, most of them ignore the effect of interior discretization and result in spurious reflections at the artificial boundary. We tackle this problem by developing ABCs directly for the discretized wave equation. Specifically, we show that the discrete system (mesh) can be stretched in a non-trivial way to preserve the discrete impedance at the interface. Similar to the perfectly matched layers (PML) for continuous wave equation, the stretch is designed to introduce dissipation in the exterior, resulting in a PML-type ABC for discrete media. The paper includes detailed formulation of the new discrete ABC, along with the illustration of its effectiveness over continuous ABCs with the help of error analysis and numerical experiments. For time-harmonic problems, the improvement over continuous ABCs is achieved without any computational overhead, leading to the conclusion that the discrete ABCs should be used in lieu of continuous ABCs.     

A compression technique for the boundary integral equation reduced from Helmholtz equation

Applying the trigonometric wavelets and the multiscale Galerkin method, we investigate the numerical solution of the boundary integral equation reduced from the exterior Dirichlet problem of Helmholtz equation by the potential theory. Consequently, we obtain a matrix compression strategy, which leads us to a fast algorithm. Our truncated treatment is simple, the computational complexity and the condition number of the truncated coefficient matrix are bounded by a constant. Furthermore, the entries of the stiffness matrix can be evaluated from the Fourier coefficients of the kernel of the boundary integral equation. Examples given for demonstrating our numerical method shorten the runtime obviously.     

A Schwarz alternating algorithm for a three-dimensional exterior harmonic problem with prolate spheroid boundary

In this paper, we investigate a Schwarz alternating algorithm for a three-dimensional exterior harmonic problem with prolate spheroid boundary. Based on natural boundary reduction, the algorithm is constructed and its convergence is discussed. The finite element method and the natural boundary element method are alternatively applied to solve the problem in a bounded subdomain and a typical unbounded subdomain. The convergence rate is analyzed in detail for a typical domain. Two numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method.     

On the viscous–viscous and the viscous–inviscid interactions in Computational Fluid Dynamics

We consider here the exterior boundary value problem for compressible viscous flow around airfoils. In a first approximation, the viscosity effects are neglected at some distance to the airfoil. The unbounded domain is decomposed by an artificial boundary into a bounded computational domain (near field) and an associated far field. The complete system of conservation laws, modelling viscous flow in the near field is coupled with simplified models for inviscid flow in the far field. The use of the heterogeneous domain decomposition method including physically and mathematically justified transmission conditions at the artificial interface provides one with a quite accurate approximate solution, modelling the viscous–inviscid interaction between the two model zones. However, such a solution does not take into account the viscosity in the far field and does not satisfy the natural transmission conditions at the artificial interface (i.e. continuity of the solution and of the normal flux). In order to get some information for the a-posteriori improvement of this solution, we introduce one-dimensional transmission-boundary value problems, obtained by an appropriate dimensional reduction of the coupled problems from CFD. The one-dimensional problems are analyzed in the framework of singular perturbation theory. We consider formal asymptotic expansions to construct appropriate boundary layer corrections of the coupled problem modelling the viscous–inviscid interaction. Our one-dimensional analysis seems to allow an extension to higher dimensions and therefore could be used in the computation of the solution to the compressible Navier–Stokes problem by updating the solution of the approximation by a (degenerate) Navier–Stokes/Euler problem with boundary layer viscosity correction terms. Received: 22 February 1999 / Accepted: 17 June 1999     

Well-posedness in the generalized sense for boundary layer suppressing boundary conditions

The time-dependent Navier-Stokes equation for incompressible fluid flow together with new boundary layer suppressing boundary conditions for open boundaries is investigated. In these new boundary conditions one typically prescribes a high-order derivative of some of the dependent variables. We prove that these boundary conditions give rise to a problem that is well posed in the generalized sense. This means that there exists a unique smooth solution of the linearized problem and that this solution can be estimated by data.     

The Dirichlet problem for the Laplacian with discontinuous boundary data in a 2D multiply connected exterior domain

The Dirichlet problem for Laplacian in a planar multiply connected exterior domain bounded by smooth closed curves is considered in case, when the boundary data is piecewise continuous, i.e. it may have jumps in certain points of the boundary. It is assumed that the solution to the problem may be not continuous at the same points. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, the integral representation for a classical solution is obtained. The problem is reduced to a uniquely solvable Fredholm integral equation of the second kind and of index zero. It is shown that a weak solution to the problem does not exist typically, though the classical solution exists.     

The boundary element-free method for 2D interior and exterior Helmholtz problems

The boundary element-free method (BEFM) is developed in this paper for numerical solutions of 2D interior and exterior Helmholtz problems with mixed boundary conditions of Dirichlet and Neumann types. A unified boundary integral equation is established for both interior and exterior problems. By using the improved interpolating moving least squares method to form meshless shape functions, mixed boundary conditions in the BEFM can be satisfied directly and easily. Detailed computational formulas are derived to compute weakly and strongly singular integrals over linear and higher order integration cells. Three numerical integration procedures are developed for the computation of strongly singular integrals. Numerical examples involving acoustic scattering and radiation problems are presented to show the accuracy and efficiency of the meshless method.     

High-order compact scheme for boundary points

In this paper, we introduce a new high-order scheme for boundary points when calculating the derivative of smooth functions by compact scheme. The primitive function reconstruction method of ENO schemes is applied to obtain the conservative form of the compact scheme. Equations for approximating the derivatives around the boundary points 1 and N are determined. For the Neumann (and mixed) boundary conditions, high-order equations are derived to determine the values of the function at the boundary points, 1 and N, before the primitive function reconstruction method is applied. We construct a subroutine that can be used with Dirichlet, Neumann, or mixed boundary conditions. Numerical tests are presented to demonstrate the capabilities of this new scheme, and a comparison to the lower-order boundary scheme shows its advantages.     

平面弹性方程外问题的非重叠型区域分解算法

１．引言 区域分解算法是八十年代兴起的偏微分方程求解新技术．基于有限元法的区域分解算法对求解有界区域问题行之有效［２,４,９］．边界元方法则是处理无界区域问题的强有力的工具［１,１０,１７］,有限元与边界元耦合法得到广泛应用 ［３,５,７］．近年又发展了基于自然边界归化的区域分解算法,特别适用于无界区域问题［８,１１,１２］．迄今这方面的文章主要是针对二维Ｐｏｉｓｓｏｎ方程及双调和方程的［１３－１６］． 本文讨论平面弹性方程的Ｄｉｒｉｃｈｌｅｔ外边值问题其中Ω是充分光滑闭曲线Г０之外的无界区域,ｕ…