非线性Galerkin算法的收敛性和复杂性

本文给出了数值求解非线性发展方程的全离散非线性Ｇａｌｅｒｋｉｎ算法,即将空间离散时的谱非线性Ｇａｌｅｒｋｉｎ算法和时间离散的Ｅｕｌｅｒ差分格式相结合,得到了显式和隐式两种全离散数值格式,相应地也考虑了显式和隐式的Ｇａｌｅｒｋｉｎ全离散格式,并分别分析了上述四种全离散格式的收敛性和复杂性,经过比较得出结论;在某些约束条件下,非线性Ｇａｌｅｒｋｉｎ算法和Ｇａｌｅｒｋｉｎ算法具有相同阶的收敛速度,然而前     

Almost sure exponential stability of numerical solutions for stochastic delay differential equations with jumps

This paper deals with the almost sure exponential stability of the Euler-type methods for nonlinear stochastic delay differential equations with jumps by using the discrete semimartingale convergence theorem. It is shown that the explicit Euler method reproduces the almost sure exponential stability under an additional linear growth condition. By replacing the linear growth condition with the one-sided Lipschitz condition, the backward Euler method is able to reproduce the stability property.     

两类变时间步长的非线性Galerkin算法的稳定性

1．引言近年来,随着计算机的飞速发展,人们越来越关心非线性发展方程解的渐进行为．为了较精确地描述解在时间t→∞时的渐进行为,人们发展了一类惯性算法,即非线性Galerkin算法．该算法是将来解空间分解为低维部分和高维部分,相应的方程可以分别投影到它们上面,它的解也相应地分解为两部分,大涡分量和小涡分量;然后核算法给出大涡分量和小涡分量之间依赖关系的一种近似,以便容易求出相应的近似解．许多研究表明,非线性Galerkin算法比通常的Galerkin算法节省可观的计算量．当数值求解微分方程时,计算机只能对已知数据进行有限位…     

Dissipativity of the linearly implicit Euler scheme for Navier‐Stokes equations with delay

In this article, we study the dissipativity of the linearly implicit Euler scheme for the 2D Navier‐Stokes equations with time delay volume forces (NSD). This scheme can be viewed as an application of the implicit Euler scheme to linearized NSD. Therefore, only a linear system is needed to solve at each time step. The main results we obtain are that this scheme is L² dissipative for any time step size and H¹ dissipative under a time‐step constraint. As a consequence, the existence of a numerical attractor of the discrete dynamical system is established. A by‐product of the dissipativity analysis of the linearly implicit Euler scheme for NSD is that the dissipativity of an implicit‐explicit scheme for the celebrated Navier‐Stokes equations that treats the volume forces term explicitly is obtained.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2114–2140, 2017     

两个带变时间步的两重网格算法的稳定性分析

对用于求解非线性发展方程的两个带变时间步的两重网格算法，对空间变量用有限元离散，对时间变量分别用一阶精度Euler显式和二阶精度半隐式差分格式离散，然后构造两重网格算法，通过深入的稳定性分析，得出本文的算法优于标准全离散有限元算法。     

On the deformations of the incompressible Euler equations

We consider systems of deformed system of equations, which are obtained by some transformations from the system of incompressible Euler equations. These have similar properties to the original Euler equations including the scaling invariance. For one form of deformed system we prove that finite time blow-up actually occurs for ‘generic’ initial data, while for the other form of the deformed system we prove the global in time regularity for smooth initial data. Moreover, using the explicit functional relations between the solutions of those deformed systems and that of the original Euler system, we derive the condition of finite time blow-up of the Euler system in terms of solutions of one of its deformed systems. As another application of those relations we deduce a lower estimate of the possible blow-up time of the 3D Euler equations. This research was supported partially by the KOSEF Grant no. R01-2005-000-10077-0     

Adaptive techniques for Landau–Lifshitz–Gilbert equation with magnetostriction

In this paper we propose a time–space adaptive method for micromagnetic problems with magnetostriction. The considered model consists of coupled Maxwell's, Landau–Lifshitz–Gilbert (LLG) and elastodynamic equations. The time discretization of Maxwell's equations and the elastodynamic equation is done by backward Euler method, the space discretization is based on Whitney edge elements and linear finite elements, respectively. The fully discrete LLG equation reduces to an ordinary differential equation, which is solved by an explicit method, that conserves the norm of the magnetization.     

Convergence and stability of finite element nonlinear Galerkin method for the Navier-Stokes equations

In this article, we present a new fully discrete finite element nonlinear Galerkin method, which are well suited to the long time integration of the Navier-Stokes equations. Spatial discretization is based on two-grid finite element technique; time discretization is based on Euler explicit scheme with variable time step size. Moreover, we analyse the boundedness, convergence and stability condition of the finite element nonlinear Galerkin method. Our discussion shows that the time step constraints of the method depend only on the coarse grid parameter and the time step constraints of the finite element Galerkin method depend on the fine grid parameter under the same convergence accuracy. Received February 2, 1994 / Revised version received December 6, 1996     

Numerical analysis of the hyperbolic two-temperature model

The system of hyperbolic heat conduction problems is solved numerically. The explicit and fully implicit Euler type schemes for the time integration of the nonstationary problem are proposed and investigated. Space derivatives are approximated by using the finite volume method, resulting in conservative and monotonous discrete approximations of the second order of accuracy. The stability analysis is done in the L ₂ and energy norms for a simplified one-temperature equation and the system of two equations, describing the temperature and the flux. Results of numerical experiments are presented. This work was supported by the Lithuanian State Science and Studies Foundation within the projects B-03/2007, B-09/2007 and by the Agency for International Science and Technology Development Programmes in Lithuania within the EUREKA projects E!3691 OPTCABLES and E!3483 EULASNET LASCAN.     

Banach空间中带Poisson跳的随机种群方程的数值分析

本文研究Banach空间中带Poisson跳的随机种群方程,通过离散使之成为随机微分方程,进而运用显式Euler公式来分析其数值解与解析解的误差.     

Convergence of the Point Vortex Methods for Euler Equation on Half Plane

1.IntroductionThevortexmethodsareefficientnumericalmethodofsimulatingincompressibleflowathighReynold'snumber-TheconvergenceofthevortexmethodsfortheinitialvalueproblemsofEulerequationwasfirstobtainedbyHald[41,thentheresultswereimprovedbyseveralauthors[1'2'3'5].Butinfact,manypracticalproblemsareconsideredinaboundeddomainoranexteriordomain,andthenumericalboundaryconditionhasanimportanteffectonnumericalresult.Theparticletrajectoriesofexactsolutionwillnotgooutfromthedomain,butitisnotthecaseinprat…     

A nonlinear stochastic growth model on discrete time domains

In this paper, we introduce nonlinear stochastic dynamic problems on discrete time domains where events may occur at unevenly spaced time points. We define Euler equation and transversality condition for the problem. We prove that the Euler equation and the transversality condition are sufficient for the existence of the optimal solution. Next we generalize discrete time Cagan type rational expectation model to multivariate case. As an application of the main results, we obtain an explicit solution to a log-linearized nonlinear stochastic growth model.     

用线方法求解带混合边值条件的抛物方程

研制了分别用显式Euler法、隐式Euler法、Crank-Nicolson格式(梯形方法)求解带第一、第二及混合边值条件的抛物问题的应用软件,通过求解若干抛物问题对该软件作了测试,获得了预期的数值结果,讨论了时间和空间步长的变化对格式计算结果的影响,得到了三种方法的稳定性、收敛精度和计算量.     

Note for “Some Exact Blowup Solutions to the pressureless Euler equations in R” [Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 2993-2998]

In this supplementary note, we can generalize the exact solutions for the pressureless Euler equations in [Yuen MW. Some exact blowup solutions to the pressureless Euler equations in R^N, Commun. Nonlinear Sci. Numer. Simulat. 2011;16:2993-8]. Here, the solutions are constructed in implicit or explicit forms.     

Stability of Galerkin and Inertial Algorithms with variable time step size

This paper provides Galerkin and Inertial Algorithms for solving a class of nonlinear evolution equations. Spatial discretization can be performed by either spectral or finite element methods; time discretization is done by Euler explicit or Euler semi-implicit difference schemes with variable time step size. Moreover, the boundedness and stability of these algorithms are studied. By comparison, we find that the boundedness and stability of Inertial Algorithm are superior to the ones of Galerkin Algorithm in the case of explicit scheme and the boundedness and stability of two algorithms are same in the case of semi-implicit scheme.     

Explicit periodic travelling waves for a discrete lambda–omega reaction–diffusion system

In 1973, Kopell and Howard introduced a λ–ω reaction–diffusion system and found an explicit family of periodic travelling wave solutions lying on circles with radius less than 1. Since λ–ω systems represent universal models for studying chemical processes, and onset of turbulent behaviour, etc., explicit solutions of λ–ω systems with delays or discrete λ–ω systems can be of further help when the only method for obtaining other solutions is through numerical computation. There are now much investigations of various λ–ω systems. However, it is of interest to note that none attempts to find explicit travelling wave solutions. In this paper, we investigate the existence of explicit solutions for the simplest Euler scheme of a λ–ω system with delays or advancements which is described as a coupled pair of partial difference equations. We are able to provide necessary as well as sufficient conditions for the existence of numerical periodic travelling wave solutions. Additionally, we also provide some examples to show that our explicit solutions are qualitatively different from those found by Kopell and Howard and hence they may be of interests for specialists in the area of reaction–diffusion systems.     

Convergence of the Tamed Euler Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Non-global Lipschitz Continuous Coefficients

In this paper, we deal with the strong convergence of numerical methods for stochastic differential equations with piecewise continuous arguments (SEPCAs) with at most polynomially growing drift coefficients and global Lipschitz continuous diffusion coefficients. An explicit and time-saving tamed Euler method is used to solve this type of SEPCAs. We show that the tamed Euler method is bounded in pth moment. And then the convergence of the tamed Euler method is proved. Moreover, the convergence order is one-half. Several numerical simulations are shown to verify the convergence of this method.     

带跳的时滞随机微分方程近似解的收敛性

本文研究了一类具有Possion跳的时滞随机微分方程（SDDEwJPs）．在一般情况下SDDEwJPs没有解析解．因此合适的数值逼近法，例如欧拉法，就是在研究它们性质时所采用的重要工具．本文在局部李普希兹条件下证明了欧拉近似解强收敛于SDDEwJPs的精确解（分析解）．     

Multiple grid and Osher's scheme for the efficient solution of the steady Euler equations

An iterative method is developed for the solution of the steady Euler equations for inviscid flow. The system of hyperbolic conservation laws is discretized by a finite-volume Osher-discretization. The iterative method is a multiple grid (FAS) iteration with symmetric Gauss-Seidel (SGS) as a relaxation method. Initial estimates are obtained by full multigrid (FMG). In the pointwise relaxation the equations are kept in block-coupled form and local linearization of the equations and the boundary conditions is considered. The efficient formulation of Osher's discretization of the 2-D non-isentropic steady Euler equations and its linearization is presented. The efficiency of FAS-SGS iteration is shown for a transonic model problem. It appears that, for the problem considered, the rate of convergence is almost independent of the gridsize and that for all meshsizes the discrete system is solved up to truncation error accuracy in only a few (2 or 3) iteration cycles.     

On the randomized Euler schemes for ODEs under inexact information

Numerical Algorithms - We analyse errors of randomized explicit and implicit Euler schemes for approximate solving of ordinary differential equations (ODEs). We consider classes of ODEs for which...