Discrete Dirichlet boundary value problems with upper semicontinuous right-hand sides

This paper deals with the relationship between solutions of Dirichlet boundary value problems (BVPs) for second order systems of differential inclusions with upper semicontinuous right-hand sides and associated numerical discrete Dirichlet BVPs of second order difference inclusions. First, the existence and estimate of solutions to the discrete BVP is discussed uniformly with respect to the discrete step size. Then convergence of solutions of the numerical discrete BVP and the corresponding semicontinous BVP is studied. Related results are also mentioned which motivated our study of this problem.     

Discrete non-linear inequalities and applications to boundary value problems

In this paper, we generalize some existing discrete Gronwall-Bellman-Ou-Iang-type inequalities to more general situations. These are in turn applied to study the boundedness, uniqueness, and continuous dependence of solutions of certain discrete boundary value problem for difference equations.     

Continuity correction for discrete barrier options with two barriers

Discrete barrier options are the options whose payoffs are determined by underlying prices at a finite set of times. We consider the discrete barrier option with two barriers. Broadie et al. (1997) [16] proposed a continuity correction for the discretely monitored barrier option. We extend this idea to barrier option with two barriers. The proof for discrete chained barrier option is provided and numerical results show the continuity correction approximation is remarkably accurate.     

Discrete maximum principle for the finite element solution of linear non-stationary diffusion–reaction problems

The maximum principle is one of the basic characteristic properties of solutions of second order partial differential equations of parabolic (and elliptic) types. The preservation of this property for solutions of corresponding discretized problems is a very natural requirement in reliable and meaningful numerical modelling of various real-life phenomena (heat conduction, air pollution, etc.). In the present paper we analyse a full discretization of a quite general class of linear parabolic equations and present sufficient conditions for the validity of a discrete analogue of the maximum principle in the case when bilinear finite elements are used for discretization in space.     

A Finite Volume Method Preserving Maximum Principle for the Conjugate Heat Transfer Problems with General Interface Conditions

In this paper, we present a unified finite volume method preserving discrete maximum principle (DMP) for the conjugate heat transfer problems with general interface conditions. We prove the existence of the numerical solution and the DMP-preserving property. Numerical experiments show that the nonlinear iteration numbers of the scheme in [24] increase rapidly when the interfacial coefficients decrease to zero. In contrast, the nonlinear iteration numbers of the unified scheme do not increase when the interfacial coefficients decrease to zero, which reveals that the unified scheme is more robust than the scheme in [24]. The accuracy and DMP-preserving property of the scheme are also veried in the numerical experiments.     

Incipient sediment transport for non-cohesive landforms by the discrete element method (DEM)

We introduce a numerical method for incipient sediment transport past bedforms. The approach is based on the discrete element method (DEM) [1], simulating the micro-mechanics of the landform as an aggregate of rigid spheres interacting by contact and friction. A continuous finite element approximation [2] predicts the boundary shear stress field due to the fluid flow, resulting in drag and lift forces acting over the particles. Numerical experiments verify the method by reproducing results by Shields [3] and other authors for the initiation of motion of a single grain. A series of experiments for sediments with varying compacity and constituting piles yields enhanced relationships between threshold shear stress and friction Reynolds number, to define incipient sediment transport criterion for flows over small-scale bed morphologies.     

Discrete multi-projection methods for eigen-problems of compact integral operators

In this paper, a discrete multi-projection method is developed for solving the eigenvalue problem of a compact integral operator with a smooth kernel. We propose a theoretical framework for analysis of the convergence of these methods. The theory is then applied to establish super-convergence results of the corresponding discrete Galerkin method, collocation method and their iterated solutions. Numerical examples are presented to illustrate the theoretical estimates for the error of these methods.     

On the competition of elastic energy and surface energy in discrete numerical schemes

The Γ-limit of certain discrete free energy functionals related to the numerical approximation of Ginzburg–Landau models is analysed when the distance h between neighbouring points tends to zero. The main focus lies on cases where there is competition between surface energy and elastic energy. Two discrete approximation schemes are compared, one of them shows a surface energy in the Γ-limit. Finally, numerical solutions for the sharp interface Cahn–Hilliard model with linear elasticity are investigated. It is demonstrated how the viscosity of the numerical scheme introduces an artificial surface energy that leads to unphysical solutions.      

A new discrete filled function algorithm for discrete global optimization

A definition of the discrete filled function is given in this paper. Based on the definition, a discrete filled function is proposed. Theoretical properties of the proposed discrete filled function are investigated, and an algorithm for discrete global optimization is developed from the new discrete filled function. The implementation of the algorithms on several test problems is reported with satisfactory numerical results.     

一类完全离散的经典风险模型的破产概率

本文研究引入利率的完全离散经典风险模型,得到一个有限时间内的破产概率的递推公式,最后给出了一个数值算例.     

Regularization Tools version 4.0 for Matlab 7.3

This communication describes version 4.0 of Regularization Tools, a Matlab package for analysis and solution of discrete ill-posed problems. The new version allows for under-determined problems, and it is expanded with several new iterative methods, as well as new test problems and new parameter-choice methods.      

ON DISCRETE PROJECTION AND NUMERICAL BOUNDARY CONDITIONS FOR THE NUMERICAL SOLUTION OF THE UNSTEADY INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

1. IntroductionLet us consider the unsteady incompressible Navier--Stokes equations (INSE)on a two--dimensional rectangular region fl with boundary 0fl. Here w = (u, v)" is tl1e velocityvector, p is the pressure, and f a known vector function of x) y, and…     

The difference between a class of discrete fractional and integer order boundary value problems

In this paper, we point out the differences between a class of fractional difference equations and the integer-order ones. We show that under the same boundary conditions, the problem of the fractional order is nonresonant, while the integer-order one is resonant. Then we analyse the discrete fractional boundary value problem in detail. Then the uniqueness and multiplicity of the solutions for the discrete fractional boundary value problem are obtained by two new tools established in 2012, respectively.     

Error inequalities for quintic and biquintic discrete Hermite interpolation

In this paper we shall develop a class of discrete Hermite interpolates in one and two independent variables. Further, we offer explicit error bounds in ?_∞ norm for the quintic and biquintic discrete Hermite interpolates. Some numerical examples are included to illustrate the results obtained.     

Discrete Gradient Method: Derivative-Free Method for Nonsmooth Optimization

A new derivative-free method is developed for solving unconstrained nonsmooth optimization problems. This method is based on the notion of a discrete gradient. It is demonstrated that the discrete gradients can be used to approximate subgradients of a broad class of nonsmooth functions. It is also shown that the discrete gradients can be applied to find descent directions of nonsmooth functions. The preliminary results of numerical experiments with unconstrained nonsmooth optimization problems as well as the comparison of the proposed method with the nonsmooth optimization solver DNLP from CONOPT-GAMS and the derivative-free optimization solver CONDOR are presented.     

Discrete and continuous maximum principles for parabolic and elliptic operators

The major qualitative properties of linear parabolic and elliptic operators/PDEs are the different maximum principles (MPs). Another important property is the stabilization property (SP), which connects these two types of operators/PDEs. This means that under some assumptions the solution of the parabolic PDE tends to an equilibrium state when t→∞, which is the solution of the corresponding elliptic PDE. To solve PDEs we need to use some numerical methods, and it is a natural requirement that these qualitative properties are preserved on the discrete level. In this work we investigate this question when a two-level discrete mesh operator is used as the discrete model of the parabolic operator (which is a one-step numerical procedure for solving the parabolic PDE) and a matrix as a discrete elliptic operator (which is a linear algebraic system of equations for solving the elliptic PDE). We clarify the relation between the discrete parabolic maximum principle (DPMP), the discrete elliptic maximum principle (DEMP) and the discrete stabilization property (DSP). The main result is that the DPMP implies the DSP and the DEMP.     

Boundary value problems on weighted networks

We present here a systematic study of general boundary value problems on weighted networks that includes the variational formulation of such problems. In particular, we obtain the discrete version of the Dirichlet Principle and we apply it to the analysis of the inverse problem of identifying the conductivities of the network in a very general framework. Our approach is based on the development of an efficient vector calculus on weighted networks which mimetizes the calculus in the smooth case. The key tool is an adequate construction of the tangent space at each vertex. This allows us to consider discrete vector fields, inner products and general metrics. Then, we obtain discrete versions of derivative, gradient, divergence and Laplace-Beltrami operators, satisfying analogous properties to those verified by their continuous counterparts. On the other hand we develop the corresponding integral calculus that includes the discrete versions of the Integration by Parts technique and Green’s Identities. Finally, we apply our discrete vector calculus to analyze the consistency of difference schemes used to solve numerically a Robin boundary value problem in a square.     

Necessary and sufficient conditions for discrete and differential inclusions of elliptic type

This paper deals for the first time with the Dirichlet problem for discrete (PD), discrete approximation problem on a uniform grid and differential (PC) inclusions of elliptic type. In the form of Euler-Lagrange inclusion necessary and sufficient conditions for optimality are derived for the problems under consideration on the basis of new concepts of locally adjoint mappings. The results obtained are generalized to the multidimensional case with a second order elliptic operator.     

Discrete weighted least-squares method for the Poisson and biharmonic problems on domains with smooth boundary

In this article a discrete weighted least-squares method for the numerical solution of elliptic partial differential equations exhibiting smooth solution is presented. It is shown how to create well-conditioned matrices of the resulting system of linear equations using algebraic polynomials, carefully selected matching points and weight factors. Two simple algorithms generating suitable matching points, the Chebyshev matching points for standard two-dimensional domains and the approximate Fekete points of Sommariva and Vianello for general domains, are described. The efficiency of the presented method is demonstrated by solving the Poisson and biharmonic problems with the homogeneous Dirichlet boundary conditions defined on circular and annular domains using basis functions in the form satisfying and in the form not satisfying the prescribed boundary conditions.     

The uniqueness of solutions to discrete, vector, two-point boundary value problems

Difference equations which may arise as discrete approximations to two-point boundary value problems for systems of second-order, ordinary differential equations are investigated and conditions are formulated under which solutions to the discrete problem are unique. Some existence, uniqueness implies existence, and convergence theorems for solutions to the discrete problem are also presented.