A note on the steady-state advection-diffusion equation

In this note we show that the numerical solution of the advection-diffusion equation can be improved by considering the asymptotic behaviour of its analytical solution. This is accomplished by including a correction term based on the numerical differentiation of the asymptotic (Pe » 1, Pe being the Peclet number) solution. This correction forces the usual oscillations associated with centred schemes to disappear.     

An investigation of an Emden-Fowler equation from thin film flow

A third-order ordinary differential equation (ODE) for thin film flow with both Neumann and Dirichlet boundary conditions is transformed into a second-order nonlinear ODE with Dirichlet boundary conditions.Numerical solutions of the nonlinear second-order ODE are investigated using finite difference schemes.A finite difference formulation to an Emden-Fowler representation of the second-order nonlinear ODE is shown to converge faster than a finite difference formulation of the standard form of the second-order nonlinear ODE.Both finite difference schemes satisfy the von Neumann stability criteria.When mapping the numerical solution of the second-order ODE back to the variables of the original third-order ODE we recover the position of the contact line.A nonlinear relationship between the position of the contact line and physical parameters is obtained.     

A note on the stability of solutions of the Hill equation with a square-wave coefficient

The results of Van der Pol and Strutt [Phil. Mag.5, 18 (1928)] on the stability of solutions to the Hill equation for a square-wave coefficient have been re-examined. More accurate and extensive results for the stability boundaries are presented, with emphasis on the closed unstable regions which are imbedded in the stable regions.     

A note on the delayed heat equation: Instability with respect to initial data

Working in the context of a simple, one-dimensional, initial-boundary value problem involving homogeneous Dirichlet boundary data, we show that the time delayed heat equation can exhibit a type of instability with respect to the initial condition (IC); specifically, we show that a slight (in the L² sense) change in the IC can change a well-posed problem to an ill-posed one. We also establish that a physically realistic solution is possible only if the IC is of a (very) specific form. The main implication of this study is that the single and dual phase lag models, which have been put forward as possible alternatives to Fourier’s law, are not valid constitutive relations for the thermal flux vector.     

A note on the exact travelling wave solution to the KdV–Burgers equation

In the present note, by use of the hyperbolic tangent method, a progressive wave solution to the Korteweg–de Vries–Burgers (KdVB) equation is presented. The solution we introduced here is less restrictive and comprises some solutions existing in the current literature (see [Wave Motion 11 (1989) 559; Wave Motion 14 (1991) 369]).     

时滞抛物型方程的高精度精细积分法

对一类时滞抛物型方程初边值问题,提出了关于空间步长是四阶精度的高精度无条件稳定的精细积分法.数值算例表明,本文提出的精细积分法具有很高的精度,因而是一种有效的数值方法.     

用状态方程直接积分法求解饱和土Biot固结方程

试采用状态方程直接积分法求解饱和土Biot固结问题,首次得到以位移未知量表示的状态方程、位移和孔隙流体压力的计算公式。算例结果表明,本方法的计算效率较高,同时精确可靠。     

关于虚拟激励法的一个注记

从具有随机频率与随机相位的随机谐和函数出发,证明了当随机频率与相位均为均匀分布而随机幅值与功率谱密度平方根成正比时,该随机过程的功率谱即精确地等于目标功率谱.进而表明:只需遍历频率区间,即可由单一谐和函数激励下的响应幅值给出响应的功率谱密度,从而揭示了虚拟激励法的物理意义.研究还表明:为了给出结构响应的功率谱密度,实际...     

Improved precise integration method for differential Riccati equation

An improved precise integration method(IPIM) for solving the differential Riccati equation(DRE) is presented.The solution to the DRE is connected with the exponential of a Hamiltonian matrix,and the precise integration method(PIM) for solving the DRE is connected with the scaling and squaring method for computing the exponential of a matrix.The error analysis of the scaling and squaring method for the exponential of a matrix is applied to the PIM of the DRE.Based on the error analysis,the criterion for choosing two parameters of the PIM is given.Three kinds of IPIMs for solving the DRE are proposed.The numerical examples show that the IPIM is stable and gives the machine accuracy solutions.     

A note on the Cauchy relations

The Cauchy relations originated in the context of the molecular theory of elasticity. Assuming them implies a reduction of the independent elastic constants which characterize the constitutive tensor in the linear anisotropic case. This paper brings to light a property of the elastic tensor through its decomposition into a symmetric part and into a skew part in the 2nd and 3rd indices. The skew part, which vanishes if and only if the Cauchy relations hold, is connected with a self-equilibrated stress field. The isotropic case is examined as a particular example.

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Sommario Le relazioni di Cauchy ebbero origine nell'ambito della teoria molecolare dell'elasticità; la loro assunzione implica una riduzione delle costanti elastiche indipendenti che caratterizzano il tensore costitutivo nel caso anisotropo lineare. In questa breve nota si mette in luce una proprietà del tensore elastico attraverso la sua decomposizione in una parte simmetrica ed in una antisimmetrica nel 2° e 3° indice: la parte antisimmetrica, che risulta nulla se e solo se valgono le relazioni di Cauchy, è associata ad un campo di tensioni autoequilibrate. Si analizza, come caso particolare, il caso isotropo.

     

动量守恒札记

通过对理论力学教材中某一常见习题的分析,阐明判断系统是否存在动量守恒,不仅要看表面现象,更要注意隐藏的动量守恒关系.     

A note on the first Korn's inequality

The first Korn's inequality is extended to unbounded domains and to classes of functions having a singular point.     

解对流方程的子域精细积分并行算法

基于子域精细积分的思想,针对对流方程初边值问题,首先提出了含参数a＞0的一族三层显格式和一族二层隐格式,它们的局部截断误差分别为O(a△t+△t2+△x2)和O(α△t+△t+△x2).当参数a≥(In△t-ln△x)/2△t时三层显格式是稳定的,而二层隐格式则对所有的参数α＞0都是无条件稳定的.然后,以二层隐格式为基础,设计了一种交替分组显武迭代(AGEI)方法,并证明了该迭代过程的收敛性.由于三层显格式和AGEI方法的整个计算过程都是显式的,所以非常适合于并行计算.文末的数值算例表明,上述方法具有很高的精确度和良好的实用性.     

A note on the reservoir routing problem

The reservoir routing problem has been recognized and managed for quite some time, but this paper presents a consistent mathematical approach by applying only the mass conservation principle (instead of various ‘ad hoc’ approaches). Application of the mass conservation principle facilitates the presentation of integral and differential formulations based on either the water volume or the water level of an arbitrarily-shaped reservoir. For the solution, new functions with a clear physical meaning are introduced. An error analysis is performed using a new approach based on spectral operators (traditionally, Richardson extrapolation has been used). It becomes evident that the integral formulations are more advantageous in terms of stability; however, they take more time to program and to calculate. The calculations in the examples are based on actual data from an artificial water reservoir (lake) Botonega in Croatia using the authors’ computer program that was created in the Mathcad environment.