LDG method for reaction–diffusion dynamical systems with time delay☆
Introduction
In this paper, a local discontinuous Galerkin method (LDG) is introduced to investigate the following reaction dynamical systems with time delaywith periodic boundary conditions and the initial condition
These kinds of delay equations are widely proposed as models for population ecology, cell biology and control theory. For example, the diffusive Mackey–Glass equationis used to describe a single-species population with age-structure and diffusion [23]. And the following diffusive Hematopoiesis modelis presented to investigate the dynamics of blood cell production [17]. For a more detailed description of such models, we refer the readers to the papers [9], [16], [18], [20], [23], [24] and the references therein. Most of the results in the literature indicate that a delay term could change the dynamic properties of a system such as stability, oscillation, bifurcations, chaos etc. That is the reason that such equations became the focus of researchers in numerical analysis and simulation.
The LDG method we discussed in this paper is a particular version of the discontinuous Galerkin (DG) method, which uses a completely discontinuous piecewise polynomial space for the numerical solution and the test functions in space, coupled with time discretization using one-leg method with an interpolation procedure for the delay term. The first LDG method was developed by Cockburn and Shu for solving convection–diffusion problems in [7]. After that, extensive work was done to develop LDG methods for various kinds of time-dependent partial differential equations, such as a series of nonlinear wave equations [25], KDV type equations [26], [27], nonlinear dispersive equations [13], Sobolev equations [8] etc. The main idea of this method is to rewrite the partial differential equation into a first-order system, and only then apply the DG method on the system with the help of carefully chosen interface numerical fluxes. There are several excellent properties (e.g. parallelization, adaptivity, and simple treatment of boundary conditions) that makes the method attractive to researchers in numerical computation and analysis. So far, the LDG method is proved to be a locally conservative, stable, and high-order accurate method (see e.g. [4], [5], [14], [30]).
Though application of the LDG methods in the filed of PDEs has proliferated in recent years, effort in applying the method to delay systems is still lacking. Nonlinear term f(t, u(x, t), u(x, t − τ)) on the RHS of Eq. (1.1) appears on the right side of equal sign. How does the nonlinear term affect stability and convergence of the method for the system? To the authors’ knowledge, little research has been conducted in studying this topic up to now. Hence, in this paper, we develop the fundamental concepts and test the LDG method for nonlinear delay systems of the form (1.1).
The rest of the paper is organized as follows. In Section 2, we describe in detail the implementation of the numerical methods for delay systems. Section 3 explains how the nonlinear part affects the stability of the underlying systems. Section 4 is devoted to convergence analysis of the LDG method. Section 5 shows experimental studies for verifying the effectiveness of the proposed methods. Finally, conclusions for this paper are summarized in Section 6.
Section snippets
Algorithm
In this section, we will describe in detail the implementation of the numerical method for the delay system.
Stability analysis of LDG methods
Stability constitutes an important characteristic of the behavior of a reaction dynamical systems with delay. However, stability analysis for these systems is very complicated. One of the reasons is that the nonlinear terms are difficult to control. In the present paper, we assume that there exist some inner product 〈·, ·〉 and the induced norm ∣·∣ such thatwhere α ⩽ 0 and β ⩾ 0 are constants. The conditions (3.1) were widely used
Convergence analysis
In this section, we shall discuss the error analysis of the semi-discrete LDG scheme for the reaction–diffusion dynamical systems with time delay. The given error estimate is of optimal order, provided that the initial approximations are chosen as the L2 projection of the initial function defined in Section 2.
First, we introduce some lemmas, which will assist in the proof of our main result. Lemma 4.1 For a given smooth function ω, the projection and are the unique functions in Sh which satisfy, for see [6]
Numerical results
In this section, we present several numerical experiments to illustrate the effectiveness of the LDG method. In what follows, the time discretization is done by the two-step explicit one-leg method (the formula (2.8)). We take a small time stepsize (e.g. Δt = h3, h = (xb − xa)/N) to avoid the influence of the time discretication in our numerical experiments.
Conclusions
In this paper a LDG method is introduced for solving nonlinear reaction–diffusion dynamical systems with time delay. Global stability of the LDG method is derived. The stability result implies that the perturbations of the numerical solutions are controlled by the initial perturbations from the system and the method. Moreover, we show that if polynomials of degree k are used, the methods are (k + 1)th order accurate in space. These two results indicate that the algorithm is a good candidate to
Acknowledgments
The authors thank prof. Chi-wang Shu for his wonderful classes in Beijing summer school. They are also grateful to the anonymous referees, whose valuable suggestions and comments largely improve the quality of the paper.
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This work is supported by NSFC (No. 10871078), 863 Program of China (No. 2009AA044501) and Graduates’ innovation fund of HUST (No. HF-08-02-2011-011).