Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.

基于蒙特卡洛法的Euler-Bernoulli梁基频和振型求解方法

将Rayleigh法和蒙特卡洛法相结合,在Euler-Bernoulli梁理论假设下求解了均匀梁、变截面梁和附带集中质量的变截面梁自由振动问题.对原本连续的梁结构模型进行离散化处理,利用蒙特卡洛法给出梁结构的假设振型.将假设得到的梁结构振型函数代入Rayleigh法,多次计算过程中,将历次基频所得值与计算所得最小值进行比较,根据其相对误差判断是否满足收敛条件,进而求得基频及对应的振型.结果表明,不同计算模型中基频最大误差不超过10%,能够满足工程需求,且精度和时间的控制参数调整灵活,使用者可根据自身需要自行调节.该方法理论简明,适用范围广泛,能够快速准确地求解诸多类型的梁结构基频和振型.

能量泛函及Euler-Lagrange方程在图像降噪中的应用研究

研究了自适应分数阶偏微分方程修正模型的能量泛函及Euler-Lagrange方程。首先,定义了自适应分数阶偏微分方程修正模型的能量泛函,其中包含未知函数和拉格朗日乘子的集合。然后,通过求解能量泛函的极值方程,推导出了Euler-Lagrange方程。最后,讨论了Euler-Lagrange方程在自适应分数阶偏微分方程修正模型中的应用。

扩散系数Holder连续的随机微分方程的截断Euler-Maruyama方法

本文研究漂移系数超线性增长和扩散系数Holder连续的随机微分方程的截断Euler-Maruyama方法的强收敛性.研究结果显示强收敛率依赖于Holder指数.本文给出一个例子验证所得的结果.

随机年龄结构固定资产系统倒向Euler法的p阶矩耗散性

在单边Lipschitz条件下,研究了一类随机年龄结构固定资产系统倒向Euler法数值解的p阶矩耗散性.当0<p<1时,步长满足一定条件可以得到系统的p阶矩耗散性;而p=2时,在对步长没有任何限制条件的情况下,得到了系统的均方耗散性.最后,通过数值例子验证了理论结果的可行性和有效性.

Bound-Preserving Discontinuous Galerkin Methods with Modified Patankar Time Integrations for Chemical Reacting Flows

In this paper,we develop bound-preserving discontinuous Galerkin(DG)methods for chemical reactive flows.There are several difficulties in constructing suitable numerical schemes.First of all,the density and internal energy are positive,and the mass fraction of each species is between 0 and 1.Second,due to the rapid reaction rate,the system may contain stiff sources,and the strong-stability-preserving explicit Runge-Kutta method may result in limited time-step sizes.To obtain physically relevant numerical approximations,we apply the bound-preserving technique to the DG methods.Though traditional positivity-preserving techniques can successfully yield positive density,internal energy,and mass fractions,they may not enforce the upper bound 1 of the mass fractions.To solve this problem,we need to(i)make sure the numerical fluxes in the equations of the mass fractions are consistent with that in the equation of the density;(ii)choose conservative time integrations,such that the summation of the mass fractions is preserved.With the above two conditions,the positive mass fractions have summation 1,and then,they are all between 0 and 1.For time discretization,we apply the modified Runge-Kutta/multi-step Patankar methods,which are explicit for the flux while implicit for the source.Such methods can handle stiff sources with relatively large time steps,preserve the positivity of the target variables,and keep the summation of the mass fractions to be 1.Finally,it is not straightforward to combine the bound-preserving DG methods and the Patankar time integrations.The positivity-preserving technique for DG methods requires positive numerical approximations at the cell interfaces,while Patankar methods can keep the positivity of the pre-selected point values of the target variables.To match the degree of freedom,we use polynomials on rectangular meshes for problems in two space dimensions.To evolve in time,we first read the polynomials at the Gaussian points.Then,suitable slope limiters can be applied to enforce the positivity of the solutions at those points,which can be preserved by the Patankar methods,leading to positive updated numerical cell averages.In addition,we use another slope limiter to get positive solutions used for the bound-preserving technique for the flux.Numerical examples are given to demonstrate the good performance of the proposed schemes.

Euler函数方程φ(xy)=28(φ(x)+φ(y))的正整数解

利用初等方法研究了不定方程φ(xy)=28(φ(x)+φ(y))的可解性问题,并给出了该方程的全部正整数解,其中φ(n)是Euler函数。

The discontinuous Petrov-Galerkin method for one-dimensional compressible Euler equations in the Lagrangian coordinate

In this paper, a Petrov-Galerkin scheme named the Runge-Kutta control volume （RKCV） discontinuous finite ele- ment method is constructed to solve the one-dimensional compressible Euler equations in the Lagrangian coordinate. Its advantages include preservation of the local conservation and a high resolution. Compared with the Runge-Kutta discon- tinuous Galerkin （RKDG） method, the RKCV method is easier to implement. Moreover, the advantages of the RKCV and the Lagrangian methods are combined in the new method. Several numerical examples are given to illustrate the accuracy and the reliability of the algorithm.

Adaptive split-step Fourier method for simulating ultrashort laser pulse propagation in photonic crystal fibres

In this paper, the generalized nonlinear Schrodinger equation （GNLSE） is solved by an adaptive split-step Fourier method （ASSFM）. It is found that ASSFM must be used to solve GNLSE to ensure precision when the soliton selffrequency shift is remarkable and the photonic crystal fibre （PCF） parameters vary with the frequency considerably. The precision of numerical simulation by using ASSFM is higher than that by using split-step Fourier method in the process of laser pulse propagation in PCFs due to the fact that the variation of fibre parameters with the peak frequency in the pulse spectrum can be taken into account fully.

Geophysical Study: Estimation of Deposit Depth Using Gravimetric Data and Euler Method (Jalalabad Iron Mine, Kerman Province of IRAN)

Mineral exploration is done by different methods. Geophysical and geochemical studies are two powerful tools in this field. In integrated studies, the results of each study are used to determine the location of the drilling boreholes. The purpose of this study is to use field geophysics to calculate the depth of mineral reserve. The study area is located 38 km from Zarand city called Jalalabad iron mine. In this study, gravimetric data were measured and mineral depth was calculated using the Euler method. 1314 readings have been performed in this area. The rocks of the region include volcanic and sedimentary. The source of the mineralization in the area is hydrothermal processes. After gravity measuring in the region, the data were corrected, then various methods such as anomalous map remaining in levels one and two, upward expansion, first and second-degree vertical derivatives, analytical method, and analytical signal were drawn, and finally, the depth of the deposit was estimated by Euler method. As a result, the depth of the mineral deposit was calculated to be between 20 and 30 meters on average.

General Modified Split-Step Balanced Methods for Stiff Stochastic Differential Equations

A class of general modified split-step balanced methods proposed in the paper can be applied to solve stiff stochastic differential systems with m-dimensional multiplicative noise. Compared to some other already reported split-step balanced methods, the drift increment function of the methods can be taken from any chosen ane-step ordinary differential equations （ODEs） solver. The schemes is proved to be strong convergent with order one. For the mean-square stability analysis, the investigation is confined to two cases. Some numerical experiments are reported to testify the performance and the effectiveness of the methods.

The Semi-implicit Euler Method for Stochastic Pantograph Equations with Jumps

In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square sense under the Local Lipschitz condition.

Euler-Bernoulli海洋立管涡致强迫振动响应研究

针对海洋立管(Pipe-in-pipe,PIP)系统在海水作用下发生的振动问题,开展了对PIP系统在涡致强迫振动下的动力学响应研究,分析了在涡致强迫振动下海洋立管外管直径、轴向拉力、外激力频率对海洋立管位移响应的影响规律。基于Euler-Bernoulli双梁模型,采用Lamb-Oseen涡模型,建立了动力学模型,利用格林函数法求得该强迫振动的稳态响应。结果表明,随着管道直径增加,外激力增加,产生最大力幅值的位置离管道越远;轴向拉力对外部管道的影响较大,对内部管道的影响较小;无因次频率取0.4时,外部管道位移超出允许变形极限,内外管壁发生周期碰撞,易对海洋立管造成损伤。

Analysis of regular and chaotic dynamics of the Euler-Bernoulli beams using finite difference and finite element methods

Chaotic vibrations of flexible non-linear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions（symmetric and non-symmetric）are studied in this work.Reliability of the obtained results is verified by the finite difference method（FDM）and the finite element method（FEM）with the Bubnov-Galerkin approximation for various boundary conditions and various dynamic regimes（regular and non-regular）.The influence of boundary conditions on the Euler-Bernoulli beams dynamics is studied mainly,dynamic behavior vs.control parameters { ωp,q0 } is reported,and scenarios of the system transition into chaos are illustrated.

A Preconditioned Gridless Method for Solving Euler Equations at Low Mach Numbers

A preconditioned gridless method is developed for solving the Euler equations at low Mach numbers.The preconditioned system in a conservation form is obtained by multiplying apreconditioning matrix of the type of Weiss and Smith to the time derivative of the Euler equations,which are discretized using agridless technique wherein the physical domain is distributed by clouds of points.The implementation of the preconditioned gridless method is mainly based on the frame of the traditional gridless method without preconditioning,which may fail to converge for low Mach number simulations.Therefore,the modifications corresponding to the affected terms of preconditioning are mainly addressed.The numerical results show that the preconditioned gridless method still functions for compressible transonic flow simulations and additionally,for nearly incompressible flow simulations at low Mach numbers as well.The paper ends with the nearly incompressible flow over a multi-element airfoil,which demonstrates the ability of the method presented for treating flows over complicated geometries.

Sloshing simulation of standing wave with time-independent finite difference method for Euler equations

The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional （2D） tank are studied. The irregular tank is mapped onto a fixed square domain through proper mapping functions. A staggered mesh system is employed in a 2D tank to calculate the elevation of the transient fluid. A time-independent finite difference method, which is developed by Bang- fuh Chen, is used to solve the Euler equations for incompressible and inviscid fluids. The numerical results agree well with the analytic solutions and previously published results. The sloshing profiles of surge and heave motion with initial standing waves are presented. The results show very clear nonlinear and beating phenomena.

Variational iteration method for solving compressible Euler equations

This paper applies the variational iteration method to obtain approximate analytic solutions of compressible Euler equations in gas dynamics. This method is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Using this method, a rapid convergent sequence is produced which converges to the exact solutions of the problem. Numerical results and comparison with other two numerical solutions verify that this method is very convenient and efficient.

Input-to-state stability of Euler-Maruyama method for stochastic delay control systems

This paper develops the mean-square exponential input-to-state stability（exp-ISS） of the Euler-Maruyama（EM） method for stochastic delay control systems（SDCSs）.The definition of mean-square exp-ISS of numerical methods is established.The conditions of the exact and EM method for an SDCS with the property of mean-square exp-ISS are obtained without involving control Lyapunov functions or functional.Under the global Lipschitz coefficients and mean-square continuous measurable inputs,it is proved that the mean-square exp-ISS of an SDCS holds if and only if that of the EM method is preserved for a sufficiently small step size.The proposed results are evaluated by using numerical experiments to show their effectiveness.

Global Existence of Smooth Solutions for the One-Dimensional Full Euler System for a Dusty Gas

We study the existence of global-in-time classical solutions for the one-dimensional nonisentropic compressible Euler system for a dusty gas with large initial data.Using the characteristic decomposition method proposed by Li et al.(Commun Math Phys 267:1–12,2006),we derive a group of characteristic decompositions for the system.Using these characteristic decompositions,we find a sufficient condition on the initial data to ensure the existence of global-in-time classical solutions.