非线性延迟积分微分方程线性多步法的渐近稳定性

本文研究了非线性延迟积分微分方程线性多步法的渐近稳定性.证明了在约束网格下,带有复合求积公式A-稳定的线性多步法能够保持解析解的渐近稳定性.文章最后,数值试验验证了本文的结论.     

Asymptotic stability of general linear methods for systems of linear neutral delay differential-algebraic equations

This paper is concerned with numerical stability of general linear methods (GLMs) for a system of linear neutral delay differential-algebraic equations. A sufficient and necessary condition for asymptotic stability of GLMs solving such system is derived. Based on this main result, we further investigate the asymptotic stability of linear multistep methods, Runge–Kutta methods, and block θ-methods, respectively. Numerical experiments confirm our theoretical result.     

Stability of numerical methods for volterra integro-differential equations

A theory of weak stability for linear multistep methods for the numerical solution of Volterra integro-differential equations is developed, and a connection between this theory and the corresponding theory for ordinary differential equations is established. In addition, the order of such methods is discussed, and a new starting procedure is proposed and analyzed.     

Discretization of parabolic inequalities

We consider the time discretization with linear multistep methods of an abstract parabolic variational inequality. For such a discretization, we prove stability and convergence under suitable regularity assumptions. Moreover, we prove that, under suitable assumptions, the methods considered converge with order at least one. We performed numerical essays which lead to the conjecture that the order is higher.     

Nonlinear contractivity of a class of semi-implicit multistep methods

The stability and contractivity of generalized linear multistep methods are studied for a large class of nonlinear stiff initial value problems. These methods are characterized by the fact that the coefficients of the integration formulas are matrices depending on the Jacobian or on an approximation to the Jacobian. Conditions for the parameters of such a multistep method are given which ensure that the method gives contractive numerical solutions over a large class of nonlinear dissipative systems for sufficiently small stepsizesh, where the restriction onh is not due to the stiffness of the problem. Stability and contractive properties of special methods of this class are reported.     

Convergence Results of One-Leg and Linear Multistep Methods for Multiply Stiff Singular Perturbation Problems

One-leg methods and linear multistep methods are two class of important numerical methods applied to stiff initial value problems of ordinary differential equations. The purpose of this paper is to present some convergence results of A-stable one-leg and linear multistep methods for one-parameter multiply stiff singular perturbation problems and their corresponding reduced problems which are a class of stiff differential-algebraic equations. Received April 14, 2000; revised June 30, 2000     

A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation

Many simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge-Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic order     

一类求解比例延迟积分微分方程线性多步法的散逸性

考虑了比例延迟积分微分方程的数值方法的散逸性。首先,通过变换将原方程变为常延迟积分微分方程,然后把一类线性多步法应用到以上问题中,用线性插值程序和复合梯形公式分别逼近延迟项和积分项,证明了在一定条件下,该数值方法具有散逸性。     

Approximate factorization for a viscous wave equation

A general procedure to construct ADI methods for multidimensional problems was originated by Beam and Warming using the method of approximate factorization. In this paper, we extend the method of approximate factorization to solve a viscous wave equation. The method can be combined with any implicit linear multistep method for the time integration of the wave equation. The stability of the factored schemes and their underlying schemes is analyzed based on a discrete Fourier analysis and the energy method. Convergence proofs are presented and numerical results supporting our analysis are provided.     

The direct numerical solution of a volterra integral equation arising out of viscoelastic stress in materials

In this paper the convergence of a class of linear multistep methods for a Volterra integral equation, arising out of viscoelastic stress in materials, is analysed and the theoretical orders of convergence are verified by numerical results.     

General linear methods for ordinary differential equations

General linear methods were introduced as the natural generalizations of the classical Runge–Kutta and linear multistep methods. They have potential applications, especially for stiff problems. This paper discusses stiffness and emphasises the need for efficient implicit methods for the solution of stiff problems. In this context, a survey of general linear methods is presented, including recent results on methods with the inherent RK stability property.     

Über A (α)-stabile Verfahren hoher Konsistenzordnung

For each α ε (0, π/2), the existence ofA (α)-stable linear multistep methods with arbitrary order of consistency is shown by an explicit construction. Some characteristic data of the methods and numerical results are given.     

Stability of difference Volterra equations: Direct Liapunov method and numerical procedure

A procedure to construct Liapunov functionals for discrete Volterra equations is proposed. Using this procedure stability conditions are derived for general Volterra difference equations. Some applications of the proposed procedure for obtaining stability conditions for linear multistep methods for Volterra integro-differential equations are presented.     

Block methods for parabolic equations

Block methods for the finite difference solution of linear one dimensional parabolic partial differential equations are considered. These schemes use two linear multistep formulae which, when applied simultaneously, advance the numerical solution by two time steps. No special starting procedure is required for their implementation. By careful choice of the coefficients in these formulae, all of the block methods derived in this paper are unconditionally stable and have high order accuracy. In addition, some of these schemes are suitable for problems involving a discontinuity between the initial and boundary conditions. The results of numerical experiments on two test problems are presented.     

Discretizzazioni A(Θ)-stabili di equazioni differenziali astratte

We study a semi-discretization in time of linear parabolic problems by using A(Θ)-stable linear multistep methods of arbitrary order. The original problems are reduced to sequences of elliptic equations, which can be approximated by Galerkin methods. The stability and error estimates are uniform with respect to these space-discretizations.      

High order Bessel fitting methods for the numerical integration of the Schrödinger equation

In this paper we show how to construct explicit multistep algorithms for an accurate and efficient numerical integration of the radial Schr?dinger equation. The proposed methods are Bessel fitting, that is to say, they integrate exactly any linear combination of Bessel and Newman functions and ordinary polynomials. They are the first of the like methods that can achieve any order.     

P-stable linear symmetric multistep methods for periodic initial-value problems

In this paper we present a new kind of P-stable multistep methods for periodic initial-value problems. From the numerical results obtained by the new method to well-known periodic problems, show the superior efficiency, accuracy, stability of the method presented in this paper.     

High Order Strong Stability Preserving Time Discretizations

Strong stability preserving (SSP) high order time discretizations were developed to ensure nonlinear stability properties necessary in the numerical solution of hyperbolic partial differential equations with discontinuous solutions. SSP methods preserve the strong stability properties—in any norm, seminorm or convex functional—of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the connections between the timestep restrictions for strong stability preservation and contractivity. Numerical examples demonstrate that common linearly stable but not strong stability preserving time discretizations may lead to violation of important boundedness properties, whereas SSP methods guarantee the desired properties provided only that these properties are satisfied with forward Euler timestepping. We review optimal explicit and implicit SSP Runge–Kutta and multistep methods, for linear and nonlinear problems. We also discuss the SSP properties of spectral deferred correction methods. The work of S. Gottlieb was supported by AFOSR grant number FA9550-06-1-0255. The work of D.I. Ketcheson was supported by a US Dept. of Energy Computational Science Graduate Fellowship under grant DE-FG02-97ER25308. The research of C.-W. Shu is supported in part by NSF grants DMS-0510345 and DMS-0809086.     

Stability analysis of explicit multirate methods

If stiff differential systems can be divided into stiff and nonstiff subsystems, each subsystem can be integrated with different step sizes. These methods are called multirate methods, and they have been successfully used in several practical problems, especially in real-time simulations.In this paper a new way to perform the stability analysis of multirate versions of linear multistep methods is presented. This stability analysis makes use of the multirate z-transform method.     

Conditionally Stable Multidimensional Schemes for Advective Equations

It is shown that the isotropic wave-like multidimensional spatial stencils combined with linear multistep and Runge-Kutta time marching schemes provide more favorable stability restrictions for advective initial-value problems. Under certain conditions the maximum allowable time step can be doubled compared to using conventional spatial stencils. Consequently, this paper shows that the multidimensional optimizations of spatial schemes, involving more grid points, are not inherently less efficient in terms of the processing time. Three numerical tests solving the two and three dimensional advection equations are carried out to experiment the stability restrictions found in the previous sections.