A research into the numerical method of Dirichlet's problem of complex Monge-Ampère equation on Cartan-Hartogs domain of the third type

Monge-Ampère equation is a nonlinear equation with high degree, therefore its numerical solution is very important and very difficult. In present paper the numerical method of Dirichlet's problem of Monge-Ampère equation on Cartan-Hartogs domain of the third type is discussed by using the analytic method. Firstly, the Monge-Ampère equation is reduced to the nonlinear ordinary differential equation, then the numerical method of the Dirichlet problem of Monge-Ampère equation becomes the numerical method of two point boundary value problem of the nonlinear ordinary differential equation. Secondly, the solution of the Dirichlet problem is given in explicit formula under the special case, which can be used to check the numerical solution of the Dirichlet problem.     

发展并行数值方法

1 引言 本世纪40年代中期至50年代初,第一台电子计算机和第一批存储程序计算机即vonNeumann计算机相继问世 。此后,计算机新陈代谢异常迅速,大约每隔5年运算速度增加10倍.50年代的计算机是串行结构,每一时刻只能按照一条指令对一个数据进行操作。由于电子信息传输速度以光速为极限,单靠改进线路已难于得到所期望的计算性能,串行计算机性能已接近了物理极限。为了克服传统计算机结构对提高运行速度的限制,从60年代起人们开始探索将并行性引入计算机结构设计,提出了研制并行计算机的设想。1972年单指令流多数据流并行计算机Illiac Ⅳ投入运行;1976年向量计算机Cray—1投入运行。在整个80年代,具有共享存储的并行向量计算机研制、生产和商售都获得了很大成功。当代高     

Eigenvalues of the discretized Navier-Stokes equation with application to the detection of Hopf bifurcations

This paper is concerned with a numerical approach to the problem of finding the leftmost eigenvalues of large sparse nonsymmetric generalised eigenvalue problems which arise in stability studies of incompressible fluid flow problems. The matrices have a special block structure that is typical of mixed finite element discretizations for such problems. The numerical approach is an extension of the hybrid technique introduced by Saad [22] and utilizes the idea of preconditioning the eigenvalue problem before applying Arnoldi's method. Two preconditioners, one a modified Cayley transform, the other a Chebyshev polynomial transform, are compared in numerical experiments on a double diffusive convection problem and the Cayley transform proves superior. The Cayley transform is then used to provide numerical results for the finite Taylor problem.     

Numerical solution of an equilibrium problem of based on the generalized level method

An equilibrium problem is studied whose special case is finding a Nash point in a noncooperative multiperson game. A numerical algorithm for solving this problem is described. Conditions on the problem are stated under which an estimate is obtained for the convergence rate of the algorithm to a unique solution of the problem. The results are used for a numerical analysis of noncooperative games.     

A finite volume element formulation and error analysis for the non-stationary conduction–convection problem

The non-stationary conduction–convection problem including the velocity vector field and the pressure field as well as the temperature field is studied with a finite volume element (FVE) method. A fully discrete FVE formulation and the error estimates between the fully discrete FVE solutions and the accuracy solution are provided. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary conduction–convection problem and is one of the most effective numerical methods by comparing the results of the numerical simulations of the FVE formulation with those of the numerical simulations of the finite element method and the finite difference scheme for the non-stationary conduction–convection problem.     

A computation of the dimensions of an underground cylindrical tank—an undergraduate modelling exercise

This paper develops the solution of a ‘real’ problem which involves simple numerical techniques. The numerical solution is interesting because of possible divergence and round‐off error growth. The problem makes an ideal case study for undergraduates following a course in numerical mathematics.     

The construction of numerical integration rules of degree three for product regions

Numerical integration formulas in n-dimensional Euclidean space of degree three are discussed. In this paper, for the product regions a method is presented to construct numerical integration formulas of degree three with 2n real points and positive weights. The presented problem is a little different from those dealt with by other authors. All the corresponding one-dimensional integrals can be different from each other and they are also nonsymmetrical. In this paper an n-dimensional numerical integration problem is turned into n one-dimensional moment problems, which simplifies the construction process. Some explicit numerical formulas are given. Furthermore, a more generalized numerical integration problem is considered, which will shed light on the final solution to the third degree numerical integration problem.     

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.     

求解非线性互补问题的微分方程方法

本文构造了一种求解非线性互补问题的微分方程方法.在一定条件下,证明了微分方程系统的平衡点是非线性互补问题的解并且基于一般微分方程系统的数值积分建立了一个数值算法.在适当的条件下,证明了此算法产生的序列解是收敛的.本文最后给出了数值结果,该结果表明了此微分方程方法的有效性.     

A semidefinite programming approach for solving fractional optimal control problems

This paper presents a numerical scheme for solving fractional optimal control. The fractional derivative in this problem is in the Riemann–Liouville sense. The proposed method, based upon the method of moments, converts the fractional optimal control problem to a semidefinite optimization problem; namely, the nonlinear optimal control problem is converted to a convex optimization problem. The Grunwald–Letnikov formula is also used as an approximation for fractional derivative. The solution of fractional optimal control problem is found by solving the semidefinite optimization problem. Finally, numerical examples are presented to show the performance of the method.     

A numerical method for determining the localized initial condition in the FitzHugh-Nagumo and Aliev-Panfilov models

The inverse problem for the FitzHugh-Nagumo and Aliev-Panfilov models describing wave propagation in excitable media is considered. The problem lies in determining a localized initial condition from measurements on the external boundary of a plane region. A numerical method for solving the inverse problem is proposed, and the results from a numerical solution of the inverse problem for regions similar to different sections of a heart are presented.     

Parameter identification in multistage population dynamics model

We present a numerical analysis to solve a parameter identification problem. We identify the demographical parameters of a multistage population dynamics model (Ainseba et al., 2011 [12]). Our nonlinear optimization problem with constraints is solved by a Quasi-Newton method. The convergence proof of this numerical method is performed here. Some numerical applications of it are also given at the end of the paper.     

A NUMERICAL PERTURBATION EXPANSION METHOD FOR THE SOLUTION OF A LIENARD-TYPE INITIAL-VALUE PROBLEM WITH PERIODIC DECAYING FORCING TERM

A numerical perturbation expansion method is developed, analysed and implemented for the numerical solution of a second-order initial-value problem. The differential equation in this problem exhibits cubic damping, a cubic restoring force and a decaying forcing-term which is periodic with constant frequency. The method is compared with the numerical method by Twizell [1]. In fact, the later is first perturbation approximate solution in the present paper.     

Numerical solution to a stokes interface problem

A numerical algorithm is described for solving the generalized eigenvalue problem arising in the study of the spectrum of a preconditioned operator in the pressure equation derived from a Stokes interface problem. The algorithm is implemented for two finite element schemes. It is tested for a problem with an analytical solution and is applied to spectrum computations in the case of a piecewise constant viscosity. A large number of numerical experiments are analyzed, and recommendations are given for solving the Stokes interface problem in practice.     

NUMERICAL RECONSTRUCTION OF INTERNAL STATES BY BOUNDARY MEASUREMENTS FOR THE LAPLACE EQUATION

The purpose of the paper is to present a unified numerical method for problems consisting of the conventional boundary value problem. Cauchy problem, under-determined problem, and over-determined problem. The method is based on the direct variational approach, which paraphrases the problems into the primary and adjoint boundary value problems that can be tackled by commonly used computer programs for the numerical solution of the Laplace equation.     

Regularized state-constrained boundary optimal control of the Navier-Stokes equations

The numerical solution of the Dirichlet boundary optimal control problem of the Navier-Stokes equations in presence of pointwise state constraints is investigated. Two different regularization techniques are considered. First, a Moreau-Yosida regularization of the problem is studied. Optimality conditions are derived and the convergence of the regularized solutions towards the original one is proved. A source representation of the control combined with a Lavrentiev type regularization strategy is also presented. The analysis concerning optimality conditions and convergence of the regularized solutions is carried out. In the last part of the paper numerical experiments are presented. For the numerical solution of each regularized problem a semi-smooth Newton method is applied.     

Numerical solutions of the m-membranes problem

In this paper numerical approximation for the m-membrane problem is considered. We make a change of variables that leads to a different expression of the quadratic functional that allows after discretizing the problem to reformulate it as finite dimensional bound constrained quadratic problem. To our knowledge this is the first paper on numerical approximation of the m-membrane problem. We reformulate the m-membrane problem as a bound constraint quadratic minimization problem. The bound constraint quadratic form is solved with the gradient projection method.     

实对称五对角矩阵逆特征值问题

1 引 言 对于n阶实对称矩阵A=(aij),r是一个正整数,且1≤r≤n-1,当|i-j|>r时,aij=0(i,j=1,2,…,n),至少有一个i使得ai,i+r≠0,则称矩阵A是带宽为2r+1的实对称带状矩阵.特别地,当r=1时,称A为实对称三对角矩阵;当r=2时,称A为实对称五对角矩阵. 实对称带状矩阵逆特征值问题应用十分广泛,这类问题不仅来自微分方程逆特征值问     

基于高斯伪谱的最优控制求解及其应用

研究一种基于高斯伪谱法的具有约束受限的最优控制数值计算问题.方法将状态演化和控制规律用多项式参数化近似,微分方程用正交多项式近似.将最优控制问题求解问题转化为一组有约束的非线性规划求解.详细论述了该种近似方法的有效性.作为该种方法的应用,讨论了一个障碍物环境下的机器人最优路径生成问题.将机器人路径规划问题转化为具有约束条件最优控制问题,然后用基于高斯伪谱的方法求解,并给出了仿真结果.