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

doi:10.1016/j.amc.2004.10.008

Applied Mathematics and Computation

Volume 168, Issue 2, 15 September 2005, Pages 1128-1144

https://doi.org/10.1016/j.amc.2004.10.008 Get rights and content

Abstract

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.

Introduction

In recent years, much research has been focused on the stability of numerical methods for the delay differential equations (DDEs). This system can be found in wide variety of scientific and engineering fields such as biology, physics, ecology and so on (see also [3], [12], [14]). Many papers paid attention to the research on the numerical methods. For example, the numerical stability of the θ-methods can be referred to [11], [13] and that of the Runge–Kutta methods can be referred to [9], [10]. Especially, a comprehensive list of the numerical methods is displayed in [2]. The GP-stability and the GPL-stability of Rosenbrock methods for one-dimensional delay differential equations are considered. It is shown that the Rosenbrock method for DDEs is GP-stable if and only if the corresponding Rosenbrock method for the ordinary differential equations (ODEs) is A-stable in [5]. Moreover, it is proved that the method is GPL-stable if and only if it is L-stable for ODEs in [6].

During the same period, some work has also been done in the field of differential-algebraic equations (DAEs), see also [4], [7], [15], and delay differential-algebraic equations (DDAEs), see also [1], [8], [16].

However, most references mentioned above are concerned with the stability analysis of the Runge–Kutta methods and the θ-methods. Especially, in Ref. [16], the author has mainly discussed the stability of the linear multistep methods, the Runge–Kutta methods and the θ-methods.

In this paper, the Rosenbrock methods are extended to NDDAEs and the asymptotic stability of Rosenbrock methods for the linear NDDAEs with constant coefficients is considered. The stability of the Rosenbrock methods for the linear constant coefficient DAEs is studied, which makes the analysis for the NDDAEs’ case understood better. Furthermore, the conditions for the Rosenbrock methods to be stable are given for the NDDAEs, in the procedure of which we have made use of the interpolation technique that is referred to Ref. [9].

Section snippets

Brief introduction of Rosenbrock methods

Consider the nonautonomous system $x^{'} (t) = f (t, x (t)),$ where x, f are d-dimensional vectors. The s-stage Rosenbrock methods take form as the following: $\begin{matrix} (I - h γ_{ii} J) K_{n, i} = hf (t_{n} + α_{i} h, x_{n} + \sum_{j = 1}^{i - 1} α_{ij} K_{n, j}) \\ + h^{2} γ_{i} f_{t} (t_{n}, x_{n}) + hJ \sum_{j = 1}^{i - 1} γ_{ij} K_{n, j}, \\ x_{n + 1} = x_{n} + \sum_{i = 1}^{s} b_{i} K_{n, i}, \end{matrix}$ where t_n = nh and h is the stepsize, α_ij, γ_ij, α_i, γ_i are real coefficients, $α_{i} = \sum_{j = 1}^{i - 1} α_{ij}$ and $γ_{i} = \sum_{j = 1}^{i} γ_{ij}$ for i = 1, 2, … , s, I denotes the identity matrix and J denotes the Jacobi matrix f_x(t_n, x_n), x_n is the approximation of x(t_n) and K_n,i is the approximation of the

Linear constant coefficient DAEs

Consider the simplest kind of linear differential-algebraic equations ${Ax}^{'} (t) + Bx (t) = 0,$ where $A, B \in R^{d \times d}$ are constant matrices and A is singular. The solvability of (3), which is essentially the existence and uniqueness of the solution, is given by the following theorem in [4].

Lemma 3.1

The system (3) is solvable if and only if the matrix pencil λA + B is regular, i.e., not identically singular for any λ.

Definition 3.1

The solution x(t) of system (3) is said to be asymptotically stable, if there exists a constant b, for any

The asymptotic stability of Rosenbrock methods for NDDAEs

In this section, we will consider NDDAEs ${Ax}^{'} (t) + Bx (t) + {Cx}^{'} (t - τ) + Dx (t - τ) = 0,$ where $A, B, C, D \in R^{d \times d}$ is constant matrices and A is singular. We are going to construct the Rosenbrock methods for the system (10) and analyze the asymptotic stability of the numerical methods.

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^☆: This paper is supported by the National Natural Science Foundation of China (10271036).

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

Abstract

Introduction

Section snippets

Brief introduction of Rosenbrock methods

Linear constant coefficient DAEs

The asymptotic stability of Rosenbrock methods for NDDAEs

Appl. Math. Comput.

Appl. Numer. Math.

The numerical solution of delay-differential-algebraic equations of retarded and neutral type

SIAM J. Numer. Anal.

Numerical Methods for Delay Differential Equations

Differential-Differernce Equations

Numerical Solution of Initial-Value Problem in Differential-Algebra Equations, Equations

Asymptotic stability of Rosenbrock methods for delay differential equations

J. Syst. Simul.

Solving Ordinary Differential Equations