La limite pour la viscosité zéro des équations de Stokes dans un demi-espace

The time-dependent Stokes equations are considered. It is shown that the solution is the sum of an inviscid solution, a boundary layer solution, and a small (in the zero viscosity limit) correction. Bounds on these solutions are given, in the appropriate Sobolev spaces, in terms of the norms of the initial and boundary data.     

人工压缩方程的OBLC区域分解法

本文根据物理尺度的区域分解法[1],[2],从奇异摄动的观点把解分为外部解和边界层校正,两者在固定的物面边界上藕合,这样对不同的尺度区域,可用不同的简化方程及计算方法.本文给出了人工压缩N-S方程的特征性质及其合适的边界提法,计算的例子表明,精度和效率是满意的.     

A Quasisteady Stefan Problem with Curvature Correction and Kinetic Undercooling

A quasisteady Stefan problem with curvature correction and kinetic undercooling is considered. It is a problem with phase transition, in which not only the Stefan condition, but also the curvature correction and kinetic undercooling effect hold on the free boundary, and in phase regions elliptic equations are satisfied by the unknown temperature at each time. The existence and uniqueness of a local classical solution of this problem are obtained.     

Mono-implicit Runge–Kutta formulae for the numerical solution of second order nonlinear two-point boundary value problems

Mono-implicit Runge–Kutta (MIRK) formulae are widely used for the numerical solution of first order systems of nonlinear two-point boundary value problems. In order to avoid costly matrix multiplications, MIRK formulae are usually implemented in a deferred correction framework and this is the basis of the well known boundary value code TWPBVP. However, many two-point boundary value problems occur naturally as second (or higher) order equations or systems and for such problems there are significant savings in computational effort to be made if the MIRK methods are tailored for these higher order forms. In this paper, we describe MIRK algorithms for second order equations and report numerical results that illustrate the substantial savings that are possible particularly for second order systems of equations where the first derivative is absent.     

Numerical Solution of the Incompressible Navier-Stokes Equations with Singular Perturbation Considerations

A numerical method for coarse grids is proposed for the numerical solution of the incompressible Navier-Stokes equations. From singular perturbation considerations, we obtain partial differential equations and boundary conditions for the outer solution and the boundary layer correction. The former problem is solved with the finite difference method and the latter with the approximate method. Numerical experiments show that accurate outer flow and boundary flux result with little computational effort.     

Application of Neumann-Ulam scheme to solving the first boundary value problem for a parabolic equation

Statistical estimates of the solutions of boundary value problems for parabolic equations with constant coefficients are constructed on paths of random walks. The phase space of these walks is a region in which the problem is solved or the boundary of the region. The simulation of the walks employs the explicit form of the fundamental solution; therefore, these algorithms cannot be directly applied to equations with variable coefficients. In the present work, unbiased and low-bias estimates of the solution of the boundary value problem for the heat equation with a variable coefficient multiplying the unknown function are constructed on the paths of a Markov chain of random walk on balloids. For studying the properties of the Markov chains and properties of the statistical estimates, the author extends von Neumann-Ulam scheme, known in the theory of Monte Carlo methods, to equations with a substochastic kernel. The algorithm is based on a new integral representation of the solution to the boundary value problem.     

The application of iterated defect correction to theLod method for parabolic equations

The paper is concerned with the numerical solution of the initial boundary value problem for a class of multi-dimensional parabolic partial differential equations. In particular the time-integration of semi-discrete equations is investigated. An attempt is made to develop integration formulas being computationally attractive and of high accuracy, while possessing unconditional stability properties. To that end iterated defect correction is applied to the LOD method. The convergence properties of this process are investigated. Numerical experiments are reported.     

Modelling two-dimensional flow past arbitrary cylindrical bodies using boundary element formulations

This paper presents an efficient method of solving Queen's linearized equations for steady plane flow of an incompressible, viscous Newtonian fluid past a cylindrical body of arbitrary cross-section. The numerical solution technique is the well known direct boundary element method. Use of a fundamental solution of Oseen's equations, the ‘Oseenlet’, allows the problem to be reduced to boundary integrals and numerical solution then only requires boundary discretization. The formulation and solution method are validated by computing the net forces acting on a single circular cylinder, two equal but separated circular cylinders and a single elliptic cylinder, and comparing these with other published results. A boundary element representation of the full Navier-Stokes equations is also used to evaluate the drag acting on a single circular cylinder by matching with the numerical Oseen solution in the far field.     

屈曲特征值问题的边界元方法及收敛性分析

讨论了屈曲特征值问题的定解条件,建立了相应的具约束的积分方程组及带Lagrange乘子的边界变分方程,给出了解的存在唯一性定理。建立了相应的边界元方法并讨论了近似解的误差估计。文末给出了数值算例。     

Efficient iterative solvers for time-harmonic Maxwell equations using domain decomposition and algebraic multigrid

Paper presents a set of parallel iterative solvers and preconditioners for the efficient solution of systems of linear equations arising in the high order finite-element approximations of boundary value problems for 3-D time-harmonic Maxwell equations on unstructured tetrahedral grids. Balancing geometric domain decomposition techniques combined with algebraic multigrid approach and coarse-grid correction using hierarchic basis functions are exploited to achieve high performance of the solvers and small memory load on the supercomputers with shared and distributed memory. Testing results for model and real-life problems show the efficiency and scalability of the presented algorithms.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.     

A COLLOCATION METHOD FOR THE CONDUCTIVITY PROBLEM WITH DISCONTINUOUS COEFFICIENT

In this paper, a new collocation BEM for the Robin boundary value problem of the conductivity equation ▽(γ▽u) = 0 is discussed, where the 7 is a piecewise constant function. By the integral representation formula of the solution of the conductivity equation on the boundary and interface, the boundary integral equations are obtained. We discuss the properties of these integral equations and propose a collocation method for solving these boundary integral equations. Both the theoretical analysis and the error analysis are presented and a numerical example is given.     

A Stefan Problem with Curvature Correction in a Concentrated Capacity

We consider a Stefan problem with the curvature correction in a concentrated capacity. It is a kind of phase transition problems, in which the unknown temperature satisfies both the Stefan condition and the curvature correction on the free boundary, and also fulfills a kind of heat equations with special inner heat source. The inner beat source is related to the derivative of the temperature with respect to the direction vertical to the phase regions. The result established in this paper is the existence of a global weak solution of this problem.     

Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1], [2].The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary.On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points.The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.     

Local Well-posedness for Linearized Degenerate MHD Boundary Layer Equations in Analytic Setting

In this note we derive MHD boundary layer equations according to viscosity and resistivity coefficients. Especially, when these viscosity and resistivity coefficients are of different orders, it leads to degenerate MHD boundary layer equations. We prove these degenerate boundary layers are stable around a steady solution.     

Convergence of the alternating direction method for the numerical solution of a heat conduction equation with delay

Two-dimensional parabolic equations with delay effects in the time component are considered. An alternating direction scheme is constructed for the numerical solution of these equations. The question on the reduction of a problem with inhomogeneous boundary conditions to a problem with homogeneous boundary conditions is considered. The order of approximation error for the alternating direction scheme, stability, and convergence order are investigated.     

NUMERICAL SOLUTION OF THE UNSTEADY INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON THE CURVILINEAR HALF-STAGGERED MESH NUMERICAL SOLUTION OF THE UNSTEADY INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON THE CURVILINEAR HALF-STAGGERED MESH

1. IntroductionLet us consider the unsteady incompressible Navier--Stokes equatinns (qSE)on a twedimensional reginn n with boundary On. Here w is the velocity vector; interms of its Caxtesian,components w = (u, v)"; p is the Pressure. The initial conditionis given assatisheng (1.2). We are concerned mainly with the solid wall boundary conditiollbut we will also briefly discuss the outflow boundary conditinn. Note the convectionterm can ajBo be written in conservative form (w' glad)w = div(…     

弹性地基上四边自由矩形薄板的解析解

将弹性地基用Winkler模型来代替,并首先把弹性地基上薄板弯曲问题的控制方程表示成为Hamilton正则方程,然后利用辛几何方法对全状态相变量进行分离变量,求出其本征值后,再按本征函数展开的方法求出弹性地基上四边自由矩形薄板的解析解．由于在求解过程中不需要事先人为的选取挠度函数,而是从弹性地基上薄板弯曲的基本方程出发,直接利用数学的方法求出可以满足四边自由边界条件的解析解,使得问题的求解更加理论化．还给出了计算实例来验证所采用的方法以及所推导出的公式的正确性．     

A parametrization method for solving nonlinear two-point boundary value problems

A sharper version of the local Hadamard theorem on the solvability of nonlinear equations is proved. Additional parameters are introduced, and a two-parameter family of algorithms for solving nonlinear two-point boundary value problems is proposed. Conditions for the convergence of these algorithms are given in terms of the initial data. Using the right-hand side of the system of differential equations and the boundary conditions, equations are constructed from which initial approximations to the unknown parameters can be found. A criterion is established for the existence of an isolated solution to a nonlinear two-point boundary value problem. This solution is shown to be a continuous function of the data specifying the problem.     

On the general linear coupled system for diffusion in media with two diffusivities

For the general linear coupled system of partial differential equations arising in the theory of diffusion in media with double diffusivity, simple uniqueness criteria, and a method of solution of boundary value problems are established. The equations studied retain the so-called cross terms which have been neglected in all previous investigations. Moreover, these equations arise as generalizations of a number of existing theories; for example, heat flow in heterogeneous multicomponent systems, flow of water in fissured rocks and a model of an arms race. The simple inequalities obtained on the various constants of the theory which guarantee uniqueness of solutions and existence of source solutions might serve as guidelines in an experimental determination of these constants. The solution procedure involves solving two boundary value problems for the classical diffusion equation and the formulae given mean that closed form expressions can be deduced for a number of commonly occurring boundary value problems. The paper emphasizes the general equations without special reference to particular physical applications or boundary value problems.