Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation

This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.

Numerical Stability and Oscillations of Runge-Kutta Methods for Differential Equations with Piecewise Constant Arguments of Advanced Type

For differential equations with piecewise constant arguments of advanced type, numerical stability and oscillations of Runge-Kutta methods are investigated. The necessary and sufficient conditions under which the numerical stability region contains the analytic stability region are given. The conditions of oscillations for the Runge-Kutta methods are obtained also. We prove that the Runge-Kutta methods preserve the oscillations of the analytic solution. Moreover, the relationship between stability and oscillations is discussed. Several numerical examples which confirm the results of our analysis are presented.

H-stability of the Runge-Kutta methods with general variable stepsize for system of pantograph equations with two delay terms

This paper deals with H-stability of the Runge-Kutta methods with a general variable stepsize for the system of pantograph equations with two delay terms. It is shown that the Runge-Kutta methods with a regular matrix A are H-stable if and only if the modulus of the stability function at infinity is less than 1.

Algebraic Stability of Multistep Runge-Kutta Methods

A series of sufficient and necessary conditions for the algebraic stability of multistepRunge-Kutta methods is obtained, most of which can be regarded as extension of the relevant results available for Runge-Kutta methods, especially, for Radau Ⅰ A, Radau Ⅱ A and Gaussian Runge-Kutta methods.

Enforcing Strong Stability of Explicit Runge-Kutta Methods with Superviscosity

A time discretization method is called strongly stable(or monotone),if the norm of its numerical solution is nonincreasing.Although this property is desirable in various of contexts,many explicit Runge-Kutta(RK)methods may fail to preserve it.In this paper,we enforce strong stability by modifying the method with superviscosity,which is a numerical technique commonly used in spectral methods.Our main focus is on strong stability under the inner-product norm for linear problems with possibly non-normal operators.We propose two approaches for stabilization:the modified method and the filtering method.The modified method is achieved by modifying the semi-negative operator with a high order superviscosity term;the filtering method is to post-process the solution by solving a diffusive or dispersive problem with small superviscosity.For linear problems,most explicit RK methods can be stabilized with either approach without accuracy degeneration.Furthermore,we prove a sharp bound(up to an equal sign)on diffusive superviscosity for ensuring strong stability.For nonlinear problems,a filtering method is investigated.Numerical examples with linear non-normal ordinary differential equation systems and for discontinuous Galerkin approximations of conservation laws are performed to validate our analysis and to test the performance.

Numerical Stability of the Runge-Kutta Methods for Equations u′(t) = au(t)+bu([K/N*t]) in Science Computation

Differential equation has widely applied in science and engineering calculation. Runge Kutta method is a main method for solving differential equations. In this paper, the numerical properties of Runge-Kutta methods for the equation u′(t) = au(t)+bu([K/N* t]) is dealed with, where K and N is relatively prime and K < N,K,N∈ Z+. The conditions are obtained under which the numerical solutions preserve the analytical stability properties of the analytic ones and some numerical experiments are given.

Stability of Stochastic Logistic Model with Ornstein-Uhlenbeck Process for Cell Growth of Microorganism in Fermentation Process

Current research is concerned with the stability of stochastic logistic equation with Ornstein-Uhlenbeck process. First, this research proves that the stochastic logistic model with Ornstein-Uhlenbeck process has a positive solution. After that, it also introduces the sufficient conditions for stochastically stability of stochastic logistic model for cell growth of microorganism in fermentation process for positive equilibrium point by using Lyapunov function. In addition, this research establishes the sufficient conditions for zero solution as mentioned in Appendix A due to the cell growth of microorganism &mu;max, which cannot be negative in fermentation process. Furthermore, for numerical simulation, current research uses the 4-stage stochastic Runge-Kutta (SRK4) method to show the reality of the results.

The Stability of Runge－Kutta Methods for Systems of Delay Differential Equations

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《针织家居服》行业标准修订内容解读

为了帮助标准使用人员更加准确地理解和运用《针织家居服》新标准,从而引导产品向高质量的方向发展,促进行业健康前进,文章详细介绍《针织家居服》新标准的主要修订内容和依据,包括家居服术语和定义的增加,超出标准范围的号型标注方法的增加,部分内在质量考核指标如水洗尺寸变化率、水洗后扭曲率、耐湿摩擦色牢度、洗后外观质量等的调整,洗液沾色程度考核指标和试验方法的增加,以及部分试验方法的调整等,对满足生产企业、消费者、检测部门需求以及正确理解和使用标准起到帮助。

Stable Vertical Takeoff of an Insect-Mimicking Flapping-Wing System Without Guide Implementing Inherent Pitching Stability

We briefly summarized how to design and fabricate an insect-mimicking flapping-wing system and demonstrate how to implement inherent pitching stability for stable vertical takeoff. The effect of relative locations of the Center of Gravity （CG） and the mean Aerodynamic Center （AC） on vertical flight was theoretically examined through static force balance considera- tion. We conducted a series of vertical takeoff tests in which the location of the mean AC was determined using an unsteady Blade Element Theory （BET） previously developed by the authors. Sequential images were captured during the takeoff tests using a high-speed camera. The results demonstrated that inherent pitching stability for vertical takeoff can be achieved by controlling the relative position between the CG and the mean AC of the flapping system.

Three-stage Stiffly Accurate Runge-Kutta Methods for Stiff Stochastic Differential Equations

In this paper we discuss diagonally implicit and semi-implicit methods based on the three-stage stiffly accurate Runge-Kutta methods for solving Stratonovich stochastic differential equations（SDEs）.Two methods,a three-stage stiffly accurate semi-implicit（SASI3） method and a three-stage stiffly accurate diagonally implicit （SADI3） method,are constructed in this paper.In particular,the truncated random variable is used in the implicit method.The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of stiff SDEs.

Stepsize Selection in Explicit Runge-Kutta Methods for Moderately Stiff Problems

We present an algorithm for determining the stepsize in an explicit Runge-Kutta method that is suitable when solving moderately stiff differential equations. The algorithm has a geometric character, and is based on a pair of semicircles that enclose the boundary of the stability region in the left half of the complex plane. The algorithm includes an error control device. We describe a vectorized form of the algorithm, and present a corresponding MATLAB code. Numerical examples for Runge-Kutta methods of third and fourth order demonstrate the properties and capabilities of the algorithm.

Design and Analysis of Some Third Order Explicit Almost Runge-Kutta Methods

In this paper, we propose two new explicit Almost Runge-Kutta (ARK) methods, ARK3 (a three stage third order method, i.e., s = p = 3) and ARK34 (a four-stage third-order method, i.e., s = 4, p = 3), for the numerical solution of initial value problems (IVPs). The methods are derived through the application of order and stability conditions normally associated with Runge-Kutta methods;the derived methods are further tested for consistency and stability, a necessary requirement for convergence of any numerical scheme;they are shown to satisfy the criteria for both consistency and stability;hence their convergence is guaranteed. Numerical experiments carried out further justified the efficiency of the methods.

Dissipativity of Multistep Runge-Kutta Methods for Nonlinear Neutral Delay-Integro-Differential Equations with Constrained Grid

This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear neutral delay-integro-differential equations.We investigate the dissipativity properties of-algebraically stable multistep Runge-Kutta methods with constrained grid.The finite-dimensional and infinite-dimensional dissipativity results of-algebraically stable multistep Runge-Kutta methods are obtained.

Stability analysis of Runge-Kutta methods for nonlinear neutral delay integro-differential equations

The sufficient conditions for the stability and asymptotic stability of Runge-Kutta methods for nonlinear neutral delay integro-differential equations are derived. A numerical test that confirms the theoretical results is given in the end.

Stability Analysis of Runge-Kutta Methods for Delay Integro-Differential Equations

Considering a linear system of delay integro-differential equations with a constant delay whose zero solution is asympototically stable, this paper discusses the stability of numerical methods for the sys-tem. The adaptation of Runge-Kutta methods with a Lagrange interpolation procedure was focused on in-heriting the asymptotic stability of underlying linear systems. The results show that an A-stable Runge-Kutta method preserves the asympototic stability of underlying linear systems whenever an unconstrained grid is used.

A CLASS OF TWO-STEP CONTINUITY RUNGE-KUTTA METHODS FOR SOLVING SINGULAR DELAY DIFFERENTIAL EQUATIONS AND ITS STABILITY ANALYSIS

In this paper, a class of two-step continuity Runge-Kutta（TSCRK） methods for solving singular delay differential equations（DDEs） is presented. Analysis of numerical stability of this methods is given. We consider the two distinct cases： （i）τ≥ h, （ii）τ 〈 h, where the delay τ and step size h of the two-step continuity Runge-Kutta methods are both constant. The absolute stability regions of some methods are plotted and numerical examples show the efficiency of the method.

STABILITY ANALYSIS OF RUNGE-KUTTA METHODS FOR NONLINEAR SYSTEMS OF PANTOGRAPH EQUATIONS

This paper is concerned with numerical stability of nonlinear systems of pantograph equations. Numerical methods based on （k, l）-algebraically stable Runge-Kutta methods are suggested. Global and asymptotic stability conditions for the presented methods are derived.

Stability analysis and a priori error estimate of explicit Runge-Kutta discontinuous Galerkin methods for correlated random walk with density-dependent turning rates

In this paper,we analyze the explicit Runge-Kutta discontinuous Galerkin(RKDG)methods for the semilinear hyperbolic system of a correlated random walk model describing movement of animals and cells in biology.The RKDG methods use a third order explicit total-variation-diminishing Runge-Kutta(TVDRK3)time discretization and upwinding numerical fluxes.By using the energy method,under a standard CourantFriedrichs-Lewy(CFL)condition,we obtain L2stability for general solutions and a priori error estimates when the solutions are smooth enough.The theoretical results are proved for piecewise polynomials with any degree k 1.Finally,since the solutions to this system are non-negative,we discuss a positivity-preserving limiter to preserve positivity without compromising accuracy.Numerical results are provided to demonstrate these RKDG methods.

Stability Analysis of Runge-Kutta Methods for Nonlinear Neutral Volterra Delay-Integro-Differential Equations

This paper is concerned with the numerical stability of implicit Runge-Kutta methods for nonlinear neutral Volterra delay-integro-differential equations with constant delay.Using a Halanay inequality generalized by Liz and Trofimchuk,we give two sufficient conditions for the stability of the true solution to this class of equations.Runge-Kutta methods with compound quadrature rule are considered.Nonlinear stability conditions for the proposed methods are derived.As an illustration of the application of these investigations,the asymptotic stability of the presented methods for Volterra delay-integro-differential equations are proved under some weaker conditions than those in the literature.An extension of the stability results to such equations with weakly singular kernel is also discussed.