Uniformly exponentially stable approximation for the transmission line with variable coefficients and its application

We analyze an ideal transmission line, which is defined by the telegraph equation with variable coefficients, from the perspectives of numerical analysis and control theory in this note. Because the spatially semi-discrete scheme of the original system is insufficient for discussing uniform exponential stability, we apply a similar transform to the continuous system and produce an intermediate system that may be easily analyzed. To begin, we discuss uniform exponential stability for the intermediate system using an so called average central-difference semi-discrete scheme and the direct Lyapunov function approach. The proof is the same as in the continuous case. The Trotter-Kato Theorem is used to demonstrate the stability and consistency of numerical approximation scheme. Finally, we propose a semi-discrete strategy for the original system through an inverse transform. All results on intermediate system are then translated into the original system. The numerical state reconstruction problem is addressed as an essential application of the main results. Furthermore, several numerical simulations are used to validate the effectiveness of the numerical approximating algorithms.     

STABILIZED NUMERICAL APPROXIMATIONS OF THE BACKWARD PROBLEM OF A PARABOLIC EQUATION

1　IntroductionLetΩ be a bounded domain in Rn and Ω be its boundary.ThenΣ =Ω× ( 0 ,1 ) is abounded domain in Rn+1 .We consider the following backwad problem of a prabolic equa-tion: u t= ni,j=1 xiaij( x) u xj -c( x) u,　　 ( x,t)∈Σ,( 1 )u| Ω× [0 ,1 ] =0 , ( 2 )u| t=1 =g( x) . ( 3 )　　 Where { aij( x) } are smooth functions given onΩ satisfyingaij( x) =aji( x) ,　　 1≤ i,j≤ n, ( 4)α0 ni=1ζ2i ≤ ni,j=1aij( x)ζiζj≤α1 ni=1ζ2i,　　 ζ∈ Rn,x∈Ω. ( 5)　　Where0 <α…     

Computing Stability of Differential Equations with Bounded Distributed Delays

This paper deals with the stability analysis of scalar delay integro-differential equations (DIDEs). We propose a numerical scheme for computing the stability determining characteristic roots of DIDEs which involves a linear multistep method as time integration scheme and a quadrature method based on Lagrange interpolation and a Gauss–Legendre quadrature rule. We investigate to which extent the proposed scheme preserves the stability properties of the original equation. We derive and prove a sufficient condition for (asymptotic) stability of a DIDE (with a constant kernel) which we call RHP-stability. Conditions are obtained under which the proposed scheme preserves RHP-stability. We compare the obtained results with corresponding ones using Newton–Cotes formulas. Results of numerical experiments on computing the stability of DIDEs with constant and nonconstant kernel functions are presented.     

Two-parameter bifurcation in a two-dimensional simplified Hodgkin–Huxley model

The dynamical behaviors of a two-dimensional simplified Hodgkin–Huxley model exposed to external electric fields are investigated based on the qualitative analysis and numerical simulation. A necessary and sufficient condition is given for the existence of the Hopf bifurcation. The stability of equilibrium points and limit cycles is also studied. Moreover, the canards and bifurcation are discussed in the simplified model and original model. The dynamical behaviors of the simplified model are consistent with the original model. It would be a great help to further investigations of the original model.     

A Dynamical Method for Solving the Obstacle Problem

In this paper, we consider the unilateral obstacle problem, trying to find the numerical solution and coincidence set. We construct an equivalent format of the original problem and propose a method with a second-order in time dissipative system for solving the equivalent format. Several numerical examples are given to illustrate the effectiveness and stability of the proposed algorithm. Convergence speed comparisons with existent numerical algorithm are also provided and our algorithm is fast.     

Limitations of frequency domain approximation for detecting chaos in fractional order systems

In this paper, we analytically study the influences of using frequency domain approximation in numerical simulations of fractional order systems. The number and location of equilibria, and also the stability of these points, are compared between the original system and its frequency based approximated counterpart. It is shown that the original system and its approximation are not necessarily equivalent according to the number, location and stability of the fixed points. This problem can cause erroneous results in special cases. For instance, to prove the existence of chaos in fractional order systems, numerical simulations have been largely based on frequency domain approximations, but in this paper we show that this method is not always reliable for detecting chaos. This approximation can numerically demonstrate chaos in the non-chaotic fractional order systems, or eliminate chaotic behavior from a chaotic fractional order system.     

一类含有稳定参数的Adams型隐式方法及其新算法

§1.引言 数值积分Stiff常微分方程初值问题,其积分过程的稳定性相当重要.用传统的数值方法,如Adams方法等,为保证计算稳定性,积分步长受到相当的限制.在stiff常微分方程初值问题的数值解法中,Gear方法是目前最通用的方法之一.但是,当阶p大     

Stabilization of Interior-Point Methods for Linear Programming

The paper studies numerical stability problems arising in the application of interior-point methods to primal degenerate linear programs. A stabilization procedure based on Gaussian elimination is proposed and it is shown that it stabilizes all path following methods, original and modified Dikin's method, Karmarkar's method, etc.     

On Mayfield’s stability proof for the discretized transparent boundary condition for the parabolic equation

Mayfield’s numerical implementation of transparent boundary condition for the Schrödinger-type parabolic equations is revisited. An inaccuracy in the original proof of the conditional stability for the resulting scheme is pointed out. The highly unusual and impressive original result is reestablished and a new proof is presented. Some further remarks and estimates on the instability which occurs when the Mayfield condition is violated are given.     

The High Order Exponentially Fitted Nonequidistant Extrapolation Methods for Stiff Systems

A class of exponentially fitted nonequidistant extrapolation methods based on an L-stable linear single-step formula are studied. Theorems about the extrapolation coefficients are given. These methods keep good numerical stability "quasi-L-stability", while raising the accuracy of the original formula.     

Numerical analysis of a population model of marine invertebrates with different life stages

We analyze an age-structured population model of marine invertebrates with life stages composed of sessile adults and pelagic larvae contained in a local habitat. In the model, space is the principal limiting resource. The long time simulation of this kind of coupled systems is difficult. Here, we propose a numerical method in order to investigate the asymptotic behavior of the solutions. Our numerical analysis provides an original knowledge about the mechanisms that govern the stability of the system as, for example, the different performance depending on the vital functions.     

Stability preserving non-standard finite difference scheme for a harvesting Leslie–Gower predator–prey model

A non-standard finite difference scheme for a harvesting Leslie–Gower equations is constructed. It is shown that the obtained difference system has the same dynamics as the original continuous system, such as positivity of solutions, equilibria and their local stability properties, irrespective of the size of numerical time step. To illustrate the analytical results, we present some numerical simulations.     

求解非定常对流扩散方程的流函数松弛方法

提出了一种求解线性和非线性对流扩散方程的流函数松弛方法.方法的主要思想是利用流函数松弛近似将原始的方程转化成等价的松弛方程组,新的松弛方程组是带源项的双曲系统.通过稳定性分析可以知道新系统的耗散系数可由松弛系数调整.数值实现亦证明这个方法可以快速有效地描述对流扩散方程的解.     

Auxiliary Equations Approach for the Stochastic Unsteady Navier-Stokes Equations with Additive Random Noise

This paper presents a Martingale regularization method for the stochastic Navier-Stokes equations with additive noise. The original system is split into two equivalent parts, the linear stochastic Stokes equations with Martingale solution and the stochastic modified Navier-Stokes equations with relatively-higher regularities. Meanwhile, a fractional Laplace operator is introduced to regularize the noise term. The stability and convergence of numerical scheme for the pathwise modified Navier-Stokes equations are proved.The comparisons of non-regularized and regularized noises for the Navier-Stokes system are numerically presented to further demonstrate the efficiency of our numerical scheme.     

System reduction using factor division algorithm and eigen spectrum analysis

A mixed method is proposed which combines the factor division algorithm with the eigen spectrum analysis for deriving reduced order models of high-order linear time invariant systems. Pole centroid and system stiffness of both original and reduced order systems remain same in this method. The proposed method guarantees stability of the reduced model if the original high-order system is stable and is comparable in quality with the other well known existing methods of order reduction. The method is illustrated by four numerical examples including one example of a multivariable system.     

Numerical solutions of random mean square Fisher-KPP models with advection

This paper deals with the construction of numerical stable solutions of random mean square Fisher-Kolmogorov-Petrosky-Piskunov (Fisher-KPP) models with advection. The construction of the numerical scheme is performed in two stages. Firstly, a semidiscretization technique transforms the original continuous problem into a nonlinear inhomogeneous system of random differential equations. Then, by extending to the random framework, the ideas of the exponential time differencing method, a full vector discretization of the problem addresses to a random vector difference scheme. A sample approach of the random vector difference scheme, the use of properties of Metzler matrices and the logarithmic norm allow the proof of stability of the numerical solutions in the mean square sense. In spite of the computational complexity, the results are illustrated by comparing the results with a test problem where the exact solution is known.     

Incremental displacement-correction schemes for incompressible fluid-structure interaction

In this paper we introduce a class of incremental displacement-correction schemes for the explicit coupling of a thin-structure with an incompressible fluid. These methods enforce a specific Robin–Neumann explicit treatment of the interface coupling. We provide a general stability and convergence analysis that covers both the incremental and the non-incremental variants. Their stability properties are independent of the added-mass effect. The superior accuracy of the incremental schemes (with respect to the original non-incremental variant) is highlighted by the error estimates, and then confirmed in a benchmark by numerical experiments.     

Multiphysics mixed finite element method with Nitsche's technique for Stokes-poroelasticity problem

In this article, we propose a multiphysics mixed finite element method with Nitsche's technique for Stokes-poroelasticity problem. Firstly, we reformulate the poroelasticity part of the original problem by introducing two pseudo-pressures to into a “fluid–fluid” coupled problem so that we can use the classical stable finite element pairs to deal with this problem conveniently. Then, we prove the existence and uniqueness of weak solution of the reformulated problem. And we use Nitsche's technique to approximate the coupling condition at the interface to propose a loosely-coupled time-stepping method to solve three subproblems at each time step–a Stokes problem, a generalized Stokes problem and a mixed diffusion problem. And the proposed method does not require any restriction on the choice of the discrete approximation spaces on each side of the interface provided that appropriate quadrature methods are adopted. Also, we give the stability analysis and error estimates of the loosely-coupled time-stepping method. Finally, we give the numerical tests to show that the proposed numerical method has a good stability and no “locking” phenomenon.     

Errata for "Hybrid Methods in the Numerical Solution of Volterra Integro--differential Equations"

In the above paper [IMA J. num. Analysis (1982) 2, 21–35]the theory of hybrid methods in ordinary differential equationswas extended to deal with the numerical solution of Volterraintegro—differential equations. The stability regionsfor two particular methods were given. The region given in Fig.1(b) (ibid., p. 34) is incorrect. Our purpose is to providethe correct stability region and to correct some associatederrors. We employ the notation of the original paper.     

非Hermite线性方程组的若干预处理迭代算法

非Hermite线性方程组在科学和工程计算中有着重要的理论研究意义和使用价值,因此如何高效求解该类线性方程组,一直是研究者所探索的方向.通过提出一种预处理方法,对非Hermite线性方程组和具有多个右端项的复线性方程组求解的若干迭代算法进行预处理,旨在提高原算法的收敛速度.最后通过数值试验表明,所提出的若干预处理迭代算法与原算法相比较,预处理算法迭代次数大大降低,且收敛速度明显优于原算法.除此之外,广义共轭A-正交残量平方法(GCORS2)的预处理算法与其他算法相比,具有良好的收敛性行为和较好的稳定性.