Numerical methods for coupled systems of quasilinear elliptic equations with nonlinear boundary conditions

This paper is concerned with numerical solutions of a coupled system of arbitrary number of quasilinear elliptic equations under combined Dirichlet and nonlinear boundary conditions. A finite difference system for a transformed system of the quasilinear equations is formulated, and three monotone iterative schemes for the computation of numerical solutions are given using the method of upper and lower solutions. It is shown that each of the three monotone iterations converges to a minimal solution or a maximal solution depending on whether the initial iteration is a lower solution or an upper solution. A comparison result among the three iterative schemes is given. Also shown is the convergence of the minimal and maximal discrete solutions to the corresponding minimal and maximal solutions of the continuous system as the mesh size tends to zero. These results are applied to a heat transfer problem with temperature dependent thermal conductivity and a Lotka-Volterra cooperation system with degenerate diffusion. This degenerate property leads to some interesting distinct property of the system when compared with the non-degenerate semilinear systems. Numerical results are given to the above problems, and in each problem an explicit continuous solution is constructed and is used to compare with the computed solution     

A finite element convergence analysis for 3D Stokes equations in case of variational crimes

We investigate a finite element discretization of the Stokes equations with nonstandard boundary conditions, defined in a bounded three-dimensional domain with a curved, piecewise smooth boundary. For tetrahedral triangulations of this domain we prove, under general assumptions on the discrete problem and without any additional regularity assumptions on the weak solution, that the discrete solutions converge to the weak solution. Examples of appropriate finite element spaces are given.     

Positive solutions and dynamics of a finite difference reaction–diffusion system

We investigate the dynamics and methods of computation for some nonlinear finite difference systems that are the discretized equations of a time-dependent and a steady-state reaction–diffusion problem. The formulation of the discrete equations for the time-dependent problem is based on the implicit method for parabolic equations, and the computational algorithm is based on the method of monotone iterations using upper and lower solutions as the initial iterations. The monotone iterative method yields improved upper and lower bounds of the solution in each iteration, and the sequence of iterations converges monotonically to a solution for both the time-dependent and the steady-state problems. An important consequence of this method is that it leads to a bifurcation point that determines the dynamic behavior of the time-dependent problem in relation to the corresponding steady-state problem. This bifurcation point also determines whether the steady-state problem has one or two non-negative solutions, and is explicitly given in terms of the physical parameters of the system and the type of boundary conditions. Numerical results are presented for both the time-dependent and the steady-state problems under various boundary conditions, including a test problem with known analytical solution. These numerical results exhibit the predicted dynamic behavior of the time-dependent solution given by the theoretical analysis. Also discussed are the numerical stability of the computational algorithm and the convergence of the finite difference solution to the corresponding continuous solution of the reaction–diffusion problem. © 1993 John Wiley & Sons, Inc.     

A study of mixed discrete bilevel programs using semidefinite and semi-infinite programming

In this paper, we study the bilevel programming problem with discrete polynomial lower level problem. We start by transforming the problem into a bilevel problem comprising a semidefinite program (SDP for short) in the lower level problem. Then, we are able to deduce some conditions of existence of solutions for the original problem. After that, we again change the bilevel problem with SDP in the lower level problem into a semi-infinite program. With the aid of the exchange technique, for simple bilevel programs, an algorithm for computing a global optimal solution is suggested, the convergence is shown, and a numerical example is given.     

Analyzing the performance of simultaneous generalized hill climbing algorithms

Simultaneous generalized hill climbing (SGHC) algorithms provide a framework for using heuristics to simultaneously address sets of intractable discrete optimization problems where information is shared between the problems during the algorithm execution. Many well-known heuristics can be embedded within the SGHC algorithm framework. This paper shows that the solutions generated by an SGHC algorithm are a stochastic process that satisfies the Markov property. This allows problem probability mass functions to be formulated for particular sets of problems based on the long-term behavior of the algorithm. Such results can be used to determine the proportion of iterations that an SGHC algorithm will spend optimizing over each discrete optimization problem. Sufficient conditions that guarantee that the algorithm spends an equal number of iterations in each discrete optimization problem are provided. SGHC algorithms can also be formulated such that the overall performance of the algorithm is independent of the initial discrete optimization problem chosen. Sufficient conditions are obtained guaranteeing that an SGHC algorithm will visit the globally optimal solution for each discrete optimization problem. Lastly, rates of convergence for SGHC algorithms are reported that show that given a rate of convergence for the embedded GHC algorithm, the SGHC algorithm can be designed to preserve this rate.     

Numerical approximations for the nonlinear time fractional reaction–diffusion equation

In this article, we propose an implicit pseudospectral scheme for nonlinear time fractional reaction–diffusion equations with Neumann boundary conditions, which is based upon Gauss–Lobatto–Legendre–Birkhoff pseudospectral method in space and finite difference method in time. A priori estimate of numerical solution is given firstly. Then the existence of numerical solution is proved by Brouwer fixed point theorem and the uniqueness is obtained. It is proved rigorously that the fully discrete scheme is unconditionally stable and convergent. Furthermore, we develop a modified scheme by adding correction terms for the problem with nonsmooth solutions. Numerical examples are given to verify the theoretical analysis.     

离散扰动下向量均衡问题的稳定性

利用集合序列P—K收敛的概念,讨论了离散扰动下的向量均衡问题弱有效解的稳定性．提出了一个新的向量均衡问题的极小化序列的概念．给出了各种充分条件以确保集合的包含关系,并举例阐述相应的结论．     

Numerical Treatment of Homogeneous Semi-Markov Processes in Transient Case–a Straightforward Approach

This paper presents the numerical solution of the process evolution equation of a homogeneous semi-Markov process (HSMP) with a general quadrature method. Furthermore, results that justify this approach proving that the numerical solution tends to the evolution equation of the continuous time HSMP are given. The results obtained generalize classical results on integral equation numerical solutions applying them to particular kinds of integral equation systems. A method for obtaining the discrete time HSMP is shown by applying a very particular quadrature formula for the discretization. Following that, the problem of obtaining the continuous time HSMP from the discrete one is considered. In addition, the discrete time HSMP in matrix form is presented and the fact that the solution of the evolution equation of this process always exists is proved. Afterwards, an algorithm for solving the discrete time HSMP is given. Finally, a simple application of the HSMP is given for a real data social security example.     

A characterisation of oscillations in the discrete two-dimensional convection-diffusion equation

It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation. However, except for one-dimensional problems, there is little analysis of this phenomenon. In this paper, we present an analysis of the two-dimensional problem with constant flow aligned with the grid, based on a Fourier decomposition of the discrete solution. For Galerkin bilinear finite element discretisations, we derive closed form expressions for the Fourier coefficients, showing them to be weighted sums of certain functions which are oscillatory when the mesh Péclet number is large. The oscillatory functions are determined as solutions to a set of three-term recurrences, and the weights are determined by the boundary conditions. These expressions are then used to characterise the oscillations of the discrete solution in terms of the mesh Péclet number and boundary conditions of the problem.

     

Matrix correction of a dual pair of improper linear programming problems

A matrix is sought that solves a given dual pair of systems of linear algebraic equations. Necessary and sufficient conditions for the existence of solutions to this problem are obtained, and the form of the solutions is found. The form of the solution with the minimal Euclidean norm is indicated. Conditions for this solution to be a rank one matrix are examined. On the basis of these results, an analysis is performed for the following two problems: modifying the coefficient matrix for a dual pair of linear programs (which can be improper) to ensure the existence of given solutions for these programs, and modifying the coefficient matrix for a dual pair of improper linear programs to minimize its Euclidean norm. Necessary and sufficient conditions for the solvability of the first problem are given, and the form of its solutions is described. For the second problem, a method for the reduction to a nonlinear constrained minimization problem is indicated, necessary conditions for the existence of solutions are found, and the form of solutions is described. Numerical results are presented.     

Numerical methods for the linear Boltzmann transport equation in slab geometry

An iterative method for computing numerical solutions of a finite-difference system corresponding to the linear Boltzmann equation in slab geometry is presented. This iterative scheme gives a straightforward marching process starting from the given boundary and initial conditions. It is shown that with a suitable initial iteration the sequence of iterations converges monotonically to a unique solution of the finite-difference system. This monotone convergence leads to improved upper and lower bounds of the solution in each iteration, and to the well-posedness of the discrete system in the sense of Hadamard. It also leads to the convergence of the discrete system to the continuous system as the mesh size of the space–velocity–time variables approaches to zero. Under a mild restriction on the time-increment the discrete system is numerically stable, independent of the mesh-size of the space and velocity. An error estimate for the computed solution due to simultaneous initial and iteration error is obtained. Also given are some numerical results for the time-dependent and the steady-state solutions.     

Global attractor of a coupled finite difference reaction diffusion system with delays

In the study of asymptotic behavior of solutions for reaction diffusion systems, an important concern is to determine whether and when the system has a global attractor which attracts all positive time-dependent solutions. The aim of this paper is to investigate the global attraction problem for a finite difference system which is a discrete approximation of a coupled system of two reaction diffusion equations with time delays. Sufficient conditions are obtained to ensure the existence and global attraction of a positive solution of the corresponding steady-state system. Applications are given to three types of Lotka-Volterra reaction diffusion models, where time-delays may appear in the opposing species.     

Exact Solutions of Discrete Kinetic Models and Stationary Problems for the Plane Broadwell Model

A review of exact solutions of discrete velocity models is given. Different methods of constructing the solutions are discussed. New methods are proposed for the stationary Broadwell model, and new class of solutions is obtained. Two-dimensional flows in channel are studied on the basis of exact solutions and analogy with the Carleman model. Explicit example of non-unique solution to a boundary value problem is given. Exact hydrodynamic equations are derived for the stationary Broadwell model. A role of the Navier–Stokes approximation is discussed in detail by comparison with these exact equations.     

The unconditional stable and convergent difference methods with intrinsic parallelism for quasilinear parabolic systems

A kind of the general finite difference schemes with intrinsic parallelism forthe boundary value problem of the quasilinear parabolic system is studied without assum-ing heuristically that the original boundary value problem has the unique smooth vectorsolution. By the method of a priori estimation of the discrete solutions of the nonlineardifference systems, and the interpolation formulas of the various norms of the discretefunctions and the fixed-point technique in finite dimensional Euclidean space, the exis-tence and uniqueness of the discrete vector solutions of the nonlinear difference systemwith intrinsic parallelism are proved. Moreover the unconditional stability of the generalfinite difference schemes with intrinsic parallelism is justified in the sense of the continu-ous dependence of the discrete vector solution of the difference schemes on the discretedata of the original problems in the discrete w_2~(2,1) norms. Finally the convergence of thediscrete vector solutions of the certain differe     

Stressed state of elastic plane with cavities

An algorithm for the numerical solution of the problem of defining the stressed state of a loaded plane with cavities of arbitrary configuration is given. A system of singular integral equations is given, which is obtained by using displacement potentials, is quantizied by the method of discrete singularities and finite differences. Existence and uniqueness conditions for solutions are formulated. The algorithm is debugged on a test example and is applied to a study of interrelation of elliptic cavities when the distance between them decreases.Translated from Dinamicheskie Sistemy, No. 7, pp. 8–13, 1988.     

Numerical methods for a coupled system of differential equations arising from a thermal ignition problem

This article is concerned with monotone iterative methods for numerical solutions of a coupled system of a first‐order partial differential equation and an ordinary differential equation which arises from fast‐igniting catalytic converters in automobile engineering. The monotone iterative scheme yields a straightforward marching process for the corresponding discrete system by the finite‐difference method, and it gives not only a computational algorithm for numerical solutions of the problem but also the existence and uniqueness of a finite‐difference solution. Particular attention is given to the “finite‐time” blow‐up property of the solution. In terms of minimal sequence of the monotone iterations, some necessary and sufficient conditions for the blow‐up solution are obtained. Also given is the convergence of the finite‐difference solution to the continuous solution as the mesh size tends to zero. Numerical results of the problem, including a case where the continuous solution is explicitly known, are presented and are compared with the known solution. Special attention is devoted to the computation of the blow‐up time and the critical value of a physical parameter which determines the global existence and the blow‐up property of the solution. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

A discrete multivariate mean value theorem with applications

We establish a discrete multivariate mean value theorem for the class of positive maximum component sign preserving functions. A constructive and combinatorial proof is given based upon a simplicial algorithm and vector labeling. Moreover, we apply this theorem to a discrete nonlinear complementarity problem and an economic equilibrium problem with indivisibilities and show the existence of solutions in both problems under certain mild conditions.     

子矩阵约束下的Hermite-Hamilton矩阵反问题

该文讨论了子矩阵约束下矩阵反问题$AX=B$的Hermite-Hamilton矩阵解.给出了解存在的充要条件和通解的一般表达式.且对任一给定矩阵,在解集合中求出了其最佳逼近解.     

Monotone iterative methods for finite difference system of reaction-diffusion equations

Summary This paper presents an existence-comparison theorem and an iterative method for a nonlinear finite difference system which corresponds to a class of semilinear parabolic and elliptic boundary-value problems. The basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences from a suitable linear discrete system. Using upper and lower solutions as two distinct initial iterations, two monotone sequences from a suitable linear system are constructed. It is shown that these two sequences converge monotonically from above and below, respectively, to a unique solution of the nonlinear discrete equations. This formulation leads to a well-posed problem for the nonlinear discrete system. Applications are given to several models arising from physical, chemical and biological systems. Numerical results are given to some of these models including a discussion on the rate of convergence of the monotone sequences.     

一类带有交叉扩散的捕食模型的共存问题

研究带有齐次Dirichlet边界条件的捕食-食饵模型,得到了平凡解存在的条件,并给出半平凡解存在的充分条件以及解的先验估计,最后利用Shauder不动点定理,得到问题至少有一个正解存在的充分条件.该结果说明只要捕获率足够小,物种的交叉扩散相对弱,问题就至少存在一个正解.