Finite difference methods for second order in space,first order in time hyperbolic systems and the linear shifted wave equation as a model problem in numerical relativity

Motivated by the problem of solving the Einstein equations, we discuss high order finite difference discretizations of first order in time, second order in space hyperbolic systems. Particular attention is paid to the case when first order derivatives that can be identified with advection terms are approximated with non-centered finite difference operators. We first derive general properties of these discrete operators, then we extend a known result on numerical stability for such systems to general order of accuracy. As an application we analyze the shifted wave equation, including the behavior of the numerical phase and group speeds at different orders of approximations. Special attention is paid to when the use of off-centered schemes improves the accuracy over the centered schemes.     

High order finite volume approximations of differential operators on nonuniform grids

When computing numerical solutions to partial differential equations, difference operators that mimic the crucial properties of the differential operators are usually more accurate than those that do not. Properties such as symmetry, conservation, stability, and the duality relationships and identities between the gradient, curl, and divergence operators are all important. Using the finite volume method, we have derived local, accurate, reliable and efficient difference methods t divergence, gradient, and curl operators are defined using a discrete versions of the divergence theorem and Stokes' theorem. These methods are especially powerful on coarse nonuniform grids and in calculations where the mesh moves to track interfaces or shocks. Numerical examples comparing local second and fourth-order finite volume approximations to conservation laws on very rough grids are used to demonstrate the advantages of the higher order methods.     

On the validity of the variational approximation in discrete nonlinear Schrödinger equations

The variational approximation is a well known tool to approximate localized states in nonlinear systems. In the context of a discrete nonlinear Schrödinger equation with a small coupling constant, we prove error estimates for the variational approximations of site-symmetric, bond-symmetric, and twisted discrete solitons. This is shown for various trial configurations, which become increasingly more accurate as more parameters are taken. It is also shown that the variational approximation yields the correct spectral stability result and controls the oscillatory dynamics of stable discrete solitons for long but finite time intervals.     

Development of arbitrary-order multioperators-based schemes for parallel calculations.: Part 2: Families of compact approximations with two-diagonal inversions and related multioperators

One-parameter families of compact approximations to grid functionals with inverses of two-point operators and their properties are described. As particular examples, interpolation/extrapolations operators, quadratures formulas and approximations to derivatives are presented. Using operators from the families with fixed parameters values as basis operators, their linear combinations providing formally arbitrary-order approximations (multioperators) are constructed. Numerical illustrations are presented. Special emphasis is placed on first derivatives discretizations in the context of conservation laws. As an example, a highly accurate tenth-order scheme is outlined and tested against the Burgers’ equation. It is shown how extrapolation multioperators can be used to create boundary closures.     

Elliptic PDE formulation and boundary conditions of the spherical harmonics method of arbitrary order for general three-dimensional geometries

The inherent complexity of the radiative transfer equation makes the exact treatment of radiative heat transfer impossible even for idealized situations and simple boundary conditions. Therefore, a wide variety of efficient solution methods have been developed for the RTE. Among these solution methods the spherical harmonics method, the moment method, and the discrete ordinates method provide means to obtain higher-order approximate solutions to the equation of radiative transfer. Although the assembly of the governing equations for the spherical harmonics method requires tedious algebra, their final form promises great accuracy for any given order, since it is a spectral method (rather than finite difference/finite volume in the case of discrete ordinates). In this study, a new methodology outlined in a previous paper on the spherical harmonics method (P_N) is further developed. The new methodology employs successive elimination of spherical harmonic tensors, thus reducing the number of first-order partial differential equations needed to be solved simultaneously by previous P_N approximations (=(N+1)²). The result is a relatively small set (=N(N+1)/2) of second-order, elliptic partial differential equations, which can be solved with standard PDE solution packages. General boundary conditions and supplementary conditions using rotation of spherical harmonics in terms of local coordinates are formulated for the general P_N approximation for arbitrary three-dimensional geometries. Accuracy of the P_N approximation can be further improved by applying the “modified differential approximation” approach first developed for the P₁-approximation. Numerical computations are carried out with the P₃ approximation for several new two-dimensional problems with emitting, absorbing, and scattering media. Results are compared to Monte Carlo solutions and discrete ordinates simulations and a discussion of ray effects and false scattering is provided.     

Constructive Approximations of Markov Operators

We construct piecewise linear Markov finite approximations of Markov operators defined on L ¹([0, 1] ^N ) and we study various properties, such as consistency, stability, and convergence, for the purpose of numerical analysis of Markov operators.     

Unified compact ADI methods for solving nonlinear viscous and nonviscous wave equations

In this paper, two unified alternating direction implicit (ADI) methods, based on the combination of fourth-order compact difference for the approximations of the second spatial derivatives with approximation factorization of difference operators, are presented for solving a two-dimensional (2D) and three-dimensional (3D) nonlinear viscous and nonviscous wave equations, respectively. By the discrete energy method, it is shown that their solutions converge to exact solutions with an order of two in time and four in space in L²- and H¹-norms. Finally, numerical findings testify the computational efficiency of the algorithms.     

Modification of Multiple Knot $B$-Spline Wavelet for Solving (Partially) Dirichlet Boundary Value Problem

A construction of multiple knot B-spline wavelets has been given in [C. K. Chui and E. Quak, Wavelet on a bounded interval, In: D. Braess and L. L. Schumaker, editors. Numerical methods of approximation theory. Basel: Birkhauser Verlag; (1992), pp. 57-76]. In this work, we first modify these wavelets to solve the elliptic (partially) Dirichlet boundary value problems by Galerkin and Petrov Galerkin methods. We generalize this construction to two dimensional case by Tensor product space. In addition, the solution of the system discretized by Galerkin method with modified multiple knot B-spline wavelets is discussed. We also consider a nonlinear partial differential equation for unsteady flows in an open channel called Saint-Venant. Since the solving of this problem by some methods such as finite difference and finite element produce unsuitable approximations specially in the ends of channel, it is solved by multiple knot B-spline wavelet method that yields a very well approximation. Finally, some numerical examples are given to support our theoretical results.     

The finite element simulation for the shallow impurity in quantum dots

The finite element method (FEM) has been implemented in this paper to investigate the electronic state of shallow hydrogenic impurities in spherical GaAs (Al,As) quantum dots (SQDs) by taking into account the finite value of realistic potential barrier height. The nonlinear partial differential equations have been discretized by means of Galerkin's weighted residue method assuming a uniform partition and using a quadratic Lagrange basis function for each finite element of the physical domain considered. The impurity binding energies have been calculated numerically by solving the governing equations and compared with the variational method. The results we have obtained are in excellent agreement with those published in the literature.     

一种非线性滤波器的生成与应用

研究一类非线性微分变换算子及其离散格式,它们在信号和图象处理中可视为一种非线性滤波器,并具有线性滤波器所没有的一些特点.     

Lump Solutions for Two Mixed Calogero-Bogoyavlenskii-Schiff and Bogoyavlensky-Konopelchenko

Based on the Hirota bilinear operators and their generalized bilinear derivatives, we formulate two new (2+1)-dimensional nonlinear partial differential equations, which possess lumps. One of the new nonlinear differential equations includes the generalized Calogero-Bogoyavlenskii-Schiff equation and the generalized Bogoyavlensky-Konopelchenko equation as particular examples, and the other has the same bilinear form with different $D_p$-operators. A class explicit lump solutions of the new nonlinear differential equation is constructed by using the Hirota bilinear approaches. A specific case of the presented lump solution is plotted to shed light on the charateristics of the lump.     

Quantum phase properties associated to solvable quantum systems using the nonlinear coherent states approach

In this paper we study the quantum phase properties of “nonlinear coherent states” and “solvable quantum systems with discrete spectra” using the Pegg-Barnett formalism in a unified approach. The presented procedure will then be applied to few special solvable quantum systems with known discrete spectrum as well as to some new classes of nonlinear oscillators with particular nonlinearity functions. Finally the associated phase distributions and their nonclasscial properties such as the squeezing in number and phase operators have been investigated, numerically.     

Feynman path integral and ordering rules on discrete finite space

The Feynman path integral is constructed for systems whose configuration space is a discrete finite set. The construction is based on the operator formulation of quantum mechanics on a finite discrete space. We derive connections between operators and introduce the analogue of the^*-multiplication for discrete symbols.     

有限点方法研究

在二维散乱离散点集上研究一类无网格方法——有限点方法(Finite Point Method,简称FPM),建立方法的基础.采用方向微商和方向差商讨论有限点方法,建立各阶各方向微商间的关系式.利用这些关系式,根据被逼近点的邻点数目差异,分别建立数值方向微商的五点公式及少点(两点、三点、四点)公式;研究五点公式的可解性条件与可允许邻点集;获得典型微分算子的数值方向微商公式等.理论分析和数值试验表明,随着邻点数目的增加,相应数值公式的逼近精度随之提高.这类近似公式不仅为在散乱离散点集上构造各类偏微分方程的格式奠定了基础,同时,也可应用于偏微分方程非结构网格计算方法,提高方法的精度.     

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators

In this paper some generalized operators of Fractional Calculus (FC) are investigated that are useful in modeling various phenomena and systems in the natural and human sciences, including physics, engineering, chemistry, control theory, etc., by means of fractional order (FO) differential equations. We start, as a background, with an overview of the Riemann-Liouville and Caputo derivatives and the Erdélyi-Kober operators. Then the multiple Erdélyi-Kober fractional integrals and derivatives of R-L type of multi-order (δ ₁,…,δ _m ) are introduced as their generalizations. Further, we define and investigate in detail the Caputotype multiple Erdélyi-Kober derivatives. Several examples and both known and new applications of the FC operators introduced in this paper are discussed. In particular, the hyper-Bessel differential operators of arbitrary order m > 1 are shown as their cases of integer multi-order. The role of the so-called special functions of FC is emphasized both as kernel-functions and solutions of related FO differential equations.     

Inverse problems for matrix Sturm-Liouville operators

Inverse spectral problems for nonselfadjoint matrix Sturm-Liouville differential operators on a finite interval and on the half-line are studied. As a main spectral characteristic, we introduce the so-called Weyl matrix and prove that the specification of the Weyl matrix uniquely determines the matrix potential and the coefficients of the boundary conditions. Moreover, for a finite interval, we also study the inverse problems of recovering matrix Sturm-Liouville operators from discrete spectral data (eigenvalues and “weight” numbers) and from a system of spectra. The results thus obtained are natural generalizations of the classical results in inverse problem theory for scalar Sturm-Liouville operators. Dedicated to the memory of B. M. Levitan     

First variation formula and conservation laws in several independent discrete variables

This paper sets the scene for discrete variational problems on an abstract cellular complex that serves as discrete model of R^p and for the discrete theory of partial differential operators that are common in the Calculus of Variations. A central result is the construction of a unique decomposition of certain partial difference operators into two components, one that is a vector bundle morphism and other one that leads to boundary terms. Application of this result to the differential of the discrete Lagrangian leads to unique discrete Euler and momentum forms not depending either on the choice of reference on the base lattice or on the choice of coordinates on the configuration manifold, and satisfying the corresponding discrete first variation formula. This formula leads to discrete Euler equations for critical points and to exact discrete conservation laws for infinitesimal symmetries of the Lagrangian density, with a clear physical interpretation.     

On Nonlinear Differential Equations That Describe Localized Processes

Abstract

In this paper we want to characterize nonlinear differential equations that describe processes allowing a localization operation in each subdomain of domain in which we consider the process. We formulate this localization condition by means of visual representations and give this operation a mathematical sense. Then we obtain a general form for such equation as well as put in it certain general physical contents, taking into account the fact that nonlinear operators from physically intelligent equations satisfy this condition.     

Numerical Solution of Fractional Partial Differential Equations by Discrete Adomian Decomposition Method

In this paper we find the solution of linear as well as nonlinear fractional partial differential equations using discrete Adomian decomposition method. Here we develop the discrete Adomian decomposition method to find the solution of fractional discrete diffusion equation, nonlinear fractional discrete Schrodinger equation, fractional discrete Ablowitz-Ladik equation and nonlinear fractional discrete Burger's equation. The obtained solution is verified by comparison with exact solution when $\alpha=1$.