M.Z. Liu

Stability of Runge-Kutta methods in the numerical solution of linear impulsive differential equations

This paper deals with the stability analysis of the analytic and numerical solutions of linear impulsive differential equations. The numerical method with variable stepsize is defined, the conditions that the numerical solutions preserve the stability property of the analytic ones are obtained and some numerical experiments are given.

Stability analysis of Runge–Kutta methods for unbounded retarded differential equations with piecewise continuous arguments ☆

Abstract This paper deals with the asymptotical stability of the analytic solutions for the unbounded retarded differential equations with piecewise continuous arguments (EPCA) u ′ ( t ) = f ( t , u ( t ) , u ( [ t N ] ) ) , t ⩾ 0 , u ( 0 ) = u 0 , where N ∈ Z + . A sufficient condition that the differential equations are asymptotically stable is derived. This paper is also concerned with the stability analysis of the Runge–Kutta methods for equation u ′ ( t ) = au ( t ) + bu ( [ t N ] ) , t ⩾ 0 , u ( 0 ) = u 0 , where N ∈ Z + . The conditions that the numerical solutions preserve the stability of the analytic solutions are obtained and some numerical experiments are given.

Numerical Hopf bifurcation of linear multistep methods for a class of delay differential equations

Abstract In this paper, we consider the discretization of parameter-dependent delay differential equation of the form y ′ ( t ) = f ( y ( t ) , y ( t - 1 ) , τ ) , τ ⩾ 0 , y ∈ R d . It is shown that if the delay differential equation undergoes a Hopf bifurcation at τ = τ ∗ , then the discrete scheme undergoes a Hopf bifurcation at τ ( h ) = τ ∗ + O ( h p ) for sufficiently small step size h, where p ⩾ 1 is the order of the strictly stable linear multistep method. The direction of numerical Hopf bifurcation and stability of bifurcating invariant curve are the same as that of the corresponding delay differential equation.

Stability of the Euler–Maclaurin methods for neutral differential equations with piecewise continuous arguments

Abstract This paper deals with the stability analysis of the Euler–Maclaurin methods for neutral differential equations with piecewise continuous arguments u ′ ( t ) = au ( t ) + ∑ i = 0 N a i u ( i ) ( [ t ] ) . The stability regions of the Euler–Maclaurin methods are determined. The conditions under which the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given.

Stability of a class of Runge-Kutta methods for a family of pantograph equations of neutral type

This paper deals with the stability of Runge-Kutta methods for a class of neutral infinite delay-differential equations with different proportional delays. Under suitable conditions, the asymptotic stability of some Runge-Kutta methods with variable stepsize are considered by the stability function at infinity. It is proved that the even-stage Gauss-Legendre methods are not asymptotically stable, but the Radau IA methods, Radau IIA methods and Lobatto IIIC methods are all asymptotically stable. Furthermore, some numerical experiments are given to demonstrate the main conclusions.

Numerical stability analysis of differential equations with piecewise constant arguments with complex coefficients

Abstract In this paper we consider the analytical and numerical stability regions of Runge–Kutta methods for differential equations with piecewise continuous arguments with complex coefficients. It is shown that the analytical stability region contained in the numerical one is violated for a ∉ R by the geometric technique. And we give the conditions under which the analytical stability region is contained in the union of the numerical stability regions of two Runge–Kutta methods. At last, some experiments are given.

Exponential stability of the exact solutions and the numerical solutions for a class of linear impulsive delay differential equations

This paper is concerned with exponential stability of a class of linear impulsive delay differential equations (IDDEs). Exponential stability of this kind of equations is studied by the properties of delay differential equations (DDEs) without impulsive perturbations. When different delay differential equations (DDEs) without impulsive perturbations are chosen, different sufficient conditions for exponential stability of the linear impulsive delay differential equations (IDDEs) are provided. Numerical methods for this kind of equations are constructed. The convergence and exponential stability of the numerical solutions are studied and some experiments are given.

Stability of the Rosenbrock methods for the neutral delay differential-algebraic equations

This paper develops the Rosenbrock methods for the neutral delay differential-algebraic equations (NDDAEs) and proves that the Rosenbrock methods equipped with suitable interpolation are GP-stable under proper assumption for the linear neutral delay differential-algebraic equations with constant coefficients. Furthermore, the GP-stability of the Runge-Kutta methods is also considered. The discussions are supported by the numerical experiments.

Exponential stability of Euler-Maruyama solutions for impulsive stochastic differential equations with delay

This paper establishes a method to study the exponential stability of Euler-Maruyama (EM) method for impulsive stochastic differential equations with delay. By using the properties of M-matrix and stochastic analysis technique, some conditions under which the EM solution is exponentially mean-square stable are obtained. Some examples are provided to illustrate the results.

Convergence and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments

Abstract In this paper, we consider the stochastic differential equations with piecewise continuous arguments (SDEPCAs) in which both the drift and the diffusion coefficients do not satisfy the global Lipschitz and linear growth conditions, especially the diffusion coefficients are highly non-linear growing. It is proved that the split-step theta (SST) method with θ ∈ [ 1 2 , 1 ] is strongly convergent to SDEPCAs under the local Lipschitz, monotone and one-sided Lipschitz conditions. It is also obtained that the SST method with θ ∈ ( 1 2 , 1 ] preserves the exponential mean square stability of SDEPCAs under the monotone condition and some condition on the step-size. Without any restriction on the step-size, there exists θ ∗ ∈ ( 1 2 , 1 ] such that the SST method with θ ∈ ( θ ∗ , 1 ] is exponentially stable in mean square. Moreover, for sufficiently small step-size, the rate constant can be reproduced. Some numerical simulations are presented to illustrate the analytical theory.