一种守恒型间断跟踪法用于处理一维Euler方程组中的若干问题

1引言间断跟踪法(front-tracking)是数值求解双曲型守恒型方程(组)的一种重要的数值方法,其主要特点是把间断作为移动的内边界来处理,光滑区域中的数值解用计算光滑解有效的数值方法来求解,而间断的移动和间断两侧的数值解的修正要满足Rankine-Hugoniot条件．我们称这样的跟踪法为传统的间断跟踪法(见[3],[14])．本文的第二作者多年来研究设计了一种基于解的守恒性质的间断跟踪法(见[11],[12]),其最主要的特点是以解的守恒性作为跟踪的机制,而不是象传统的间断跟踪法那样利用     

A singular-perturbed two-phase Stefan problem

The asymptotic behavior of a singular-perturbed two-phase Stefan problem due to slow diffusion in one of the two phases is investigated. In the limit, the model equations reduce to a one-phase Stefan problem. A boundary layer at the moving interface makes it necessary to use a corrected interface condition obtained from matched asymptotic expansions. The approach is validated by numerical experiments using a front-tracking method.     

一种守恒型间断跟踪法中对任意多个间断的移动和相互作用的处理

对一种守恒型间断跟踪法设计了一种技巧来处理任意多个间断的移动和相互作用．由此技巧我们就可以建立一个“一般的强健的”间断跟踪法．由于采用了此技巧就会使得算法在某时刻在某网格上会存在非相容性并且会产生O(1)-强度的误差．但不管怎样,这些误差在后续的计算中会被算法的守恒性所消除．还给出了几个数值例子来显示这一技巧的有效性．     

Dynamic adaptation method in gasdynamic simulations with nonlinear heat conduction

A dynamic adaptation method is applied to gas dynamics problems with nonlinear heat conduction. The adaptation function is determined by the condition that the energy equation is quasi-stationary and the grid point distribution is quasi-uniform. The dynamic adaptation method with the adaptation function thus determined and a front-tracking technique are used to solve the model problem of a piston moving in a heat-conducting gas. It is shown that the results significantly depend on the thermal conductivity chosen. The numerical results obtained on a 40-node grid are compared with self-similar solutions to this problem.     

Isentropic fluid dynamics in a curved pipe

In this paper we study isentropic flow in a curved pipe. We focus on the consequences of the geometry of the pipe on the dynamics of the flow. More precisely, we present the solution of the general Cauchy problem for isentropic fluid flow in an arbitrarily curved, piecewise smooth pipe. We consider initial data in the subsonic regime, with small total variation about a stationary solution. The proof relies on the front-tracking method and is based on [1].     

On the Problem of Instability in the Dimensions of Spline Spaces over T-Meshes with T-Cycles

The T-meshes are local modification of rectangular meshes which allow T-junctions. The splines over T-meshes are involved in many fields, such as finite element methods, CAGD etc. The dimension of a spline space is a basic problem for the theories and applications of splines. However, the problem of determining the dimension of a spline space is difficult since it heavily depends on the geometric properties of the partition. In many cases, the dimension is unstable. In this paper, we study the instability in the dimensions of spline spaces over T-meshes by using the smoothing cofactor-conformality method. The modified dimension formulas of spline spaces over T-meshes with T-cycles are also presented. Moreover, some examples are given to illustrate the instability in the dimensions of the spline spaces over some special meshes.     

A Finite-element Front-tracking Enthalpy Method for Stefan Problems

A finite-element front-tracking method, which avoids explicittreatment of the jump condition on the phase boundary, is proposed.This method is based on the discretization of a weak enthalpyformulation with isoparametric space—time finite elements.A one-dimensional two-phase problem and a two-dimensional one-phaseproblem are solved by this method. Then the method is appliedto a generalized Stefan problem—the spot-welding problemand extended to the alloy-solidification problem. Numerical experiments show that this method is convergent andunconditionally stable and that second-order convergence canbe expected if the biquadratic elements are used for one-dimensionalproblems. The effectiveness of this method is, in particular,shown in solving the last two problems.     

海水入侵危险程度的非线性映射分析

海水入侵是我国沿海地区出现的一典型资源与环境问题.作者选用地质、气候、地理环境和人为因素等四类指标来反映海水入侵存在的风险,采用非线性映射分析将高维空间问题转化为低维空间问题.根据离差极小、最大欧氏距离等原理提出了海水入侵风险阈值求解方法与海水入侵风险判定方法,使存在海水入侵风险的区域可被及时、准确地判定出来.实例分析表明,该方法使分析结果真实、准确,为沿海地区海水入侵风险管理开辟了一条新途径.     

An analytical solution of the diffusion- convection equation over a finite domain

Numerical solutions to the diffusion-convection equation are usually evaluated through comparison with analytical solutions in one dimension. Literature survey indicates the most frequently used analytical solution is one derived for asemi-infinite domain. This paper presents an analytical solution to this problem over a finite domain. Comparison is made with a solution to a mathematically similar problem in heat conduction with radiation.     

A front-tracking model to predict solidification macrostructures and columnar to equiaxed transitions in alloy castings

The solidified grain structure (macrostructure) of castings is predicted by process simulation using a newly extended front-tracking technique which models the growth of solid dendritic fronts through undercooled liquid during metallic alloy solidification. Such fronts are either constrained, as is the case with directed columnar growth from mould walls, or unconstrained, as is the case for multiple equiaxed growth from individual nucleating particles distributed throughout the liquid. Non-linear latent heat evolution is treated by incorporating the Scheil equation. Thermal conductivity changes with the solid fraction. A log-normal distribution of activation undercooling to initiate free growth from equiaxed nuclei is used, and the routines to deal with such growth followed by impingement of dendritic grains upon one another are verified by comparison with the results of analytical studies of simplified systems. The extensions to the model enable the predictions of equiaxed grain structure and, importantly, the columnar to equiaxed transition in inoculated alloy castings. The model is validated via comparison with experimental results. The front-tracking method is proposed as a suitable formulation for modelling alloy castings that solidify with a dendritic structure, either columnar, equiaxed, or both.     

Overlapping Schwarz waveform relaxation method for the solution of the reaction-diffusion equation

We are interested in solving time dependent problems using domain decomposition methods. In the classical approach, one discretizes first the time dimension and then one solves a sequence of steady problems by a domain decomposition method. In this article, we treat directly the time dependent problem and we study a Schwarz waveform relaxation algorithm for the convection diffusion equation. We study the convergence of the overlapping Schwarz waveform relaxation method for solving the reaction-diffusion equation over multi-overlapped subdomains. Also we will show that the method converges linearly and superlinearly over long and short time intervals, and the convergence depends on the size of overlap. Numerical results are presented from solutions of a specific model problems to demonstrate the convergence, linear and superlinear, and the role of the overlap size.     

Minimizing Fusion Frame Potential

Fusion frames are an emerging topic of frame theory, with applications to encoding and distributed sensing. However, little is known about the existence of tight fusion frames. In traditional frame theory, one method for showing that unit norm tight frames exist is to characterize them as the minimizers of an energy functional, known as the frame potential. We generalize the frame potential to the fusion frame setting. In particular, we introduce the fusion frame potential, and show how its minimization is equivalent to the minimization of the traditional frame potential over a particular domain. We then study this minimization problem in detail. Specifically, we show that if the desired number of fusion frame subspaces is large, and if the desired dimension of these subspaces is small compared to the dimension of the underlying space, then a tight fusion frame of those dimensions will necessarily exist, being a minimizer of the fusion frame potential.     

Stroboscopic averaging of highly oscillatory nonlinear wave equations

In this paper, we are concerned with stroboscopic averaging for highly oscillatory evolution equations posed in a Banach space. Using Taylor expansion, we construct a non‐oscillatory high‐order system whose solution remains exponentially close to the exact one over a long time. We then apply this result to the nonlinear wave equation in one dimension. We present the stroboscopic averaging method, which is a numerical method introduced by Chartier, Murua and Sanz‐Serna, and apply it to our problem. Finally, we conclude by presenting the qualitative and quantitative efficiency of this numerical method for some nonlinear wave problem. Copyright © 2014 John Wiley & Sons, Ltd.     

A Petrov-Galerkin method and boundary approximations for hyperbolic initial boundary-value problems

The Petrov-Galerkin method is used to construct a semidiscrete approximation for hyperbolic systems of equations in one space dimension. Stability is analyzed for the Cauchy problem and for an initial and boundary-value problem with positive and negative characeristic speeds of unequal magnitude. Numerical experiments are conducted on a linear problem to support the stability analysis and to compare the accuracies of various boundary approximations. Experiments on a nonlinear problem check the viability of the best boundary method.     

Study of weak solutions for a fourth-order parabolic equation with variable exponent of nonlinearity

The aim of this paper is to study the existence and uniqueness of weak solutions for an initial boundary problem of a fourth-order parabolic equation with variable exponent of nonlinearity. First, the authors of this paper apply Leray-Schauder’s fixed point theorem to prove the existence of solutions of the corresponding nonlinear elliptic problem and then obtain the existence of weak solutions of nonlinear parabolic problem by combining the results of the elliptic problem with Rothe’s method. In addition, the authors also discuss the regularity of weak solutions in the case of space dimension one.     

Additive splitting methods for elliptic-parabolic problems

Three finite-difference splitting schemes are proposed for numerical solution of the nonlinear 3D parabolic-elliptic problem. Adaptive front-tracking and time-stepping strategies are included into the algorithms. Parallelization of the algorithms is done using the domain decomposition method. The 1D decomposition of the computational domain is used in order to obtain the optimal computational load balancing among processors and to minimize the frequency of data communications. A redistribution of the computational domain among processors is done dynamically during computations.

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Sunto In questo lavoro vengono proposti tre diversi schemi di decomposizione alle differenze finite per la risoluzione numerica di problemi ellittico-parabolici non lineari in 3D. Negli algoritmi sono incluse strategie adattative front-tracking e time-stepping. La parallelizzazione degli algoritmi è realizzata usando il metodo della decomposizione di domini. Viene impiegata la decomposizione 1D del dominio computazionale per ottenere il bilanciameto ottimale del carico computazionale tra i processori e per minizzare la frequenza della comunicazione dei dati. Durante le computazioni, infine, viene realizzata dinamicamente la ridistribuzione dei domini computazionali.

     

The numerical solution of Stefan problems with front-tracking and smoothing methods

The numerical performance of some computer methods for heat transfer with change of phase is discussed. For one-dimensional problems the application of invariant imbedding to time-discretized problems is suggested. For some multidimensional problems an absorption of the phase transition process into the diffusion equation through the so-called enthalpy transformation is advocated. If this transformation is not applicable, a locally one-dimensional Gauss-Seidel-type front-tracking method coupled with invariant imbedding is effective.     

多介质流体界面的守恒型跟踪法

本文中,我们对茅德康在【D.K.Mao,Towards front-tracking based on conservation in two space dimensionsⅡ,tracking discontinuities in capturing fashion,J.Comput.Phys.,226,(2007),pp 1550-1588】一文中所建立的守恒型跟踪法进行了改进,成功将算法运用到多介质流体界面的计算中.最后,我们使用改进后的算法计算了两个数值算例,验证了算法的有效性.     

On some boundary element methods for the heat equation

Summary The object of this paper is to study some boundary element methods for the heat equation. Two approaches are considered. The first, based on the heat potential, has been studied numerically by previous authors. Here the convergence analysis in one space dimension is presented. In the second approach, the heat equation is first descretized in time and the resulting elliptic problem is put in the boundary formulation. A straight forward implicit method and Crank-Nicolson's method are thus studied. Again convergence in one space dimension is proved.     

Components identification based method for box constrained variational inequality problems with almost linear functions

We present a method for solving a class of box constrained variational inequality problems. The method makes use of a procedure for identifying some components of the solution by bounding it with an interval vector. It is shown that the method computes an approximate solution of the variational inequality problem by solving at most n reduced systems of equations, where n is the dimension of the problem. Among those systems, only the one of the smallest dimension has to be solved with high accuracy. The others are solved merely to identify some components of the solution, and so the computation can be done under a very mild requirement of accuracy. Numerical results are presented for the obstacle problem, to illustrate the efficiency of the method. AMS subject classification (2000)  90C33, 65G30, 65K10