Stability Of Linear Multistep Methods Research Articles

Delay-Dependent Stability of Linear Multistep Methods for Neutral Systems with Distributed Delays

Delay-Dependent Stability of Linear Multistep Methods for Neutral Systems with Distributed Delays

Stable linear multistep methods with off-step points for the solution of ordinary differential equations

Of recent, stability has become an important concept and a qualitative property in any numerical integration scheme. In this work, we propose two stable linear multistep methods with off-step points for the numerical integration of ordinary differential equations whose development is collocation and interpolation based. The boundary locus techniques show that the proposed schemes are zero-stable, A-stable and -stable for some step number and are found suitable for stiff differential equations. Numerical results obtained compare favourably with some existing methods in literature.

Delay-dependent stability of linear multi-step methods for linear neutral systems

Delay-dependent stability of linear multi-step methods for linear neutral systems

Energy stable higher-order linear ETD multi-step methods for gradient flows: application to thin film epitaxy

We discuss how to combine exponential time differencing technique with multi-step method to develop higher order in time linear numerical scheme that are energy stable for certain gradient flows with the aid of a generalized viscous damping term. As an example, a stabilized third order in time accurate linear exponential time differencing (ETD) scheme for the epitaxial thin film growth model without slope selection is proposed and analyzed. An artificial stabilizing term $$A\tau ^3\frac{\partial \Delta ^3 u}{\partial t}$$ is added to ensure energy stability, with ETD-based multi-step approximations and Fourier pseudo-spectral method applied in the time integral and spatial discretization of the evolution equation, respectively. Long-time energy stability and an $$\ell ^{\infty }(0,T; \ell ^2)$$ error analysis are provided, based on the energy method. In addition, a few numerical experiments are presented to demonstrate the energy decay and convergence rate.

Discrete C1 convergence of linear multistep methods

We prove that strongly stable linear multistep methods are convergent in the discrete C 1 norm.

Delay-dependent stability of linear multistep methods for differential systems with distributed delays

This paper deals with the stability of linear multistep methods for multi-dimensional differential systems with distributed delays. The delay-dependent stability of linear multistep methods with compound quadrature rules is studied. Several new sufficient criteria of delay-dependent stability are obtained by means of the argument principle. An algorithm is provided to check delay-dependent stability. An example that illustrates the effectiveness of the derived theoretical results is given.

Delay-dependent stability of linear multistep methods for DAEs with multiple delays

This paper aims to investigate the asymptotic stability of linear multistep (LM) methods for linear differential-algebraic equations (DAEs) with multiple delays. Based on the argument principle, we first establish the delay-dependent stability criteria of analytic solutions; then, we propose some practically checkable conditions for weak delay-dependent stability of numerical solutions derived by implicit LM methods. Lagrange interpolations are used to compute the delayed terms. Several numerical examples are given to illustrate the theoretical results.

Linear multistep methods for impulsive delay differential equations

This paper deals with the convergence and stability of linear multistep methods for a class of linear impulsive delay differential equations. Numerical experiments show that the Simpson’s Rule and two-step BDF method are of order p=0 when applied to impulsive delay differential equations. An improved linear multistep numerical process is proposed. Convergence and stability conditions of the numerical solutions are given in the paper. Numerical experiments are given in the end to illustrate the conclusion.

Modified Broström procedure in patients with generalized joint laxity

This paper deals with the convergence and stability of linear multistep methods for a class of linear impulsive delay differential equations. Numerical experiments show that the Simpson’s Rule and two-step BDF method are of order p=0 when applied to impulsive delay differential equations. An improved linear multistep numerical process is proposed. Convergence and stability conditions of the numerical solutions are given in the paper. Numerical experiments are given in the end to illustrate the conclusion.

Stiffly stable second derivative linear multistep methods with two hybrid points

This paper presents a family of hybrid linear multistep methods (LMMs) with second derivative term for the numerical solution of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The methods are stiffly stable for step number k ≤ 7.

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.

Construction of algebraically stable DIMSIMs

The class of general linear methods for ordinary differential equations combines the advantages of linear multistep methods (high efficiency) and Runge–Kutta methods (good stability properties such as A-, L-, or algebraic stability), while at the same time avoiding the disadvantages of these methods (poor stability of linear multistep methods, high cost for Runge–Kutta methods). In this paper we describe the construction of algebraically stable general linear methods based on the criteria proposed recently by Hewitt and Hill. We also introduce the new concept of ϵ-algebraic stability and investigate its consequences. Examples of ϵ-algebraically stable methods are given up to order p=4.

On the construction of high order A(α)-stable hybrid linear multistep methods for stiff IVPs in ODEs

In this paper, we present a class of A(α)-stable hybrid linear multistep methods for numerical solving stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The method considered uses a second derivative like the Enright’s second derivative linear multistep methods for stiff IVPs in ODEs.

A Continuous Formulation of A(α)-Stable Second Derivative Linear Multistep Methods for Stiff IVPs in ODEs

This paper considers family of A(α)-stable second derivative linear multistep methods of order p = k + 3 for step number k ≥ 15 for the solution of stiff IVPs in ODEs. The methods are demonstrated to be A(α)-stable for k ≥ 13. At k = 14, the method is stable but not A(α)-stable. The instability of the new methods sets in when k > 15. Numerical examples are given to demonstrate the application of the methods.

Linear Multistep Methods for Impulsive Differential Equations

This paper deals with the convergence and stability of linear multistep methods for impulsive differential equations. Numerical experiments demonstrate that both the mid‐point rule and two‐step BDF method are of order p = 0 when applied to impulsive differential equations. An improved linear multistep method is proposed. Convergence and stability conditions of the improved methods are given in the paper. Numerical experiments are given in the end to illustrate the conclusion.

Stability analysis of linear multistep and Runge–Kutta methods for neutral multidelay-differential-algebraic systems

This paper is concerned with the stability analysis of linear multistep and Runge–Kutta methods for neutral multidelay-differential-algebraic systems. We focus on the system with no restrictive assumption on the structure of the coefficient matrices. It is shown that under some conditions, A -stable linear multistep methods and A -stable Runge–Kutta methods can preserve the delay-independent stability of the underlying linear systems. Numerical experiments further confirm the theoretical results.

Some stiffly stable second derivative continuous linear multistep methods with a hybrid point for stiff IVPS in ODEs

Based on Geari¦s fixed step size backward differentiation methods, Gear (1968), second derivative continuous linear multistep methods with an off-step point are presented. This type of methods provides a means of bypassing the order barriers imposed by Dahlquist (1963) and also provides continuous solutions of IVPs in ODEs. The stiff stability of these methods is determined by using the boundary locus. Instability sets in at k ƒ­10 . Numerical results of the methods solving a non-linear and a linear stiff initial value problems in ordinary differential equations are compared to that of the state -of -the-art code, ODE 15s in MATLAB.

Stability of linear multistep methods for nonlinear neutral delay differential equations in Banach space

This paper is devoted to investigating the nonlinear stability properties of linear multistep methods for the solution to neutral delay differential equations in Banach space. Two approaches to numerically treating the “neutral term” are considered, which allow us to prove several results on numerical stability of linear multistep methods. These results provide some criteria for choosing the step size such that the numerical method is stable. Some examples of application and a numerical experiment, which further confirms the main results, are given.

A class of stiffly stable hybrid second derivative continuous linear multistep methods for stiff IVPs in ODEs

A class of stiffly stable hybrid second derivative continuous linear multistep methods for stiff IVPs in ODEs

Errors of linear multistep methods for singularly perturbed Volterra delay-integro-differential equations

This paper is concerned with the error behaviour of linear multistep methods applied to singularly perturbed Volterra delay-integro-differential equations. We derive global error estimates of A ( α ) -stable linear multistep methods with convergent quadrature rule. Numerical experiments confirm our theoretical analysis.