Applied Mathematics Letters 23 21 457 461 Contents lists available at ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml Nonlinear stability of discontinuous Galerkin methods for delay differential equations Dongfang Li, Chengjian Zhang School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 4374, China a r c l e i n f o a b s t r a c t Article history: Received 28 March 29 Received in revised form 18 November 29 Accepted 4 December 29 Keywords: Nonlinear stability Discontinuous Galerkin methods Delay differential equations The present paper is devoted to a study of nonlinear stability of discontinuous Galerkin methods for delay differential equations. Some concepts, such as global and analogously asymptotical stability are introduced. We derive that discontinuous Galerkin methods lead to global and analogously asymptotical stability for delay differential equations. And these nonlinear stability properties reveal to the reader the relation between the perturbations of the numerical solution and that of the initial value or the systems. 29 Elsevier Ltd. All rights reserved. 1. Introduction The past several decades have witnessed a large developmenn numerical analysis for various delay differential equations DDEs [1 3]. Most researchers focused their attention on finite difference methods, such as one-leg methods, Runge Kutta methods and so on [4 8]. Besides the above methods, is well known that the discontinuous Galerkin DG schemes are also a class of locally conservative, stable, and superconvergence methods, which are widely used in scientific fields such as computational fluids, gas dynamics, chemical transport and so on see e.g. [9 15]. As regards many excellent properties, it will be interesting to apply these methods to DDEs. Recently, Zhang and Li derived optimal superconvergence results for DDEs [16]. When applied m-degree DG method to the equation, they obtained the error estimates u U Oh 2m+1, m 1 at the integer nodal points and u U Oh m+2, m 2 at some points in every interval. And numerical tests confirmed the methods theoretical results. To our knowledge, these were almost the only results for this topic up until now. It would seem, therefore, that further investigation on stability analysis is needed. As an important part, nonlinear stability plays an important role in computational implementation and intrigues many researchers in numerical analysis of DDEs. For example, P-stability and GP-stability were firsntroduced to describe the nonlinear stability of such problems [4,6,7]. Later, Torelli [17] introduced the concepts of RN- and GRN-stability and proved that the backward Euler method is GRN-stable for nonautonomous nonlinear problems. Bellen and Zennaro [18] further pointed out that the two-stage Lobatto 3C method is GRN-stable. Next, Huang et al. introduced the concepts of R- and GR- stability and AR- and GAR- stability, which are analogues of P- and RN- and GP- and GRN- stability. One-leg methods, Runge Kutta methods and more general linear methods are used to reveal these stability properties [5,19]. In the present paper, we focus on nonlinear stability of DG methods for a class of DDEs. And we show the reader that the perturbations of the numerical solution are controlled by the initial perturbations from the system and the method. This projecs supported by NSFC nsfc187178. Corresponding author. E-mail addresses: lidongfang1983@gmail.com D. Li, cjzhang@mail.hust.edu.cn C. Zhang. 893-9659/$ see front matter 29 Elsevier Ltd. All rights reserved. doi:1.116/j.aml.29.12.3
458 D. Li, C. Zhang / Applied Mathematics Letters 23 21 457 461 The paper is structured as follows. In Section 2, we present DG methods for DDEs, some concepts of stability are also collected. In Section 3, we show that our DG methods lead to global and analogously asymptotical stability for DDEs. Finally, in Section 4, we end with some extensive conclusions. 2. DG methods for DDEs and Consider the following nonlinear DDEs: y t f t, y, yt τ, t > yt ψt, t z t f t, z, zt τ, t > zt ϕt, t. Here τ is a positive delay term, ψt and ϕt are continuous, f : [t, + ] X X X, such that 2.1 and 2.2 own a unique solution, respectively, where X is a real or complex Hilbert space. Moreover, we assume there exist some inner product, and the induced norm such that Re u 1 u 2, f t, u 1, v f t, u 2, v α u 1 u 2 2 2.3 f t, u, v 1 f t, u, v 2 β v 1 v 2, where α and β are constants. The conditions 2.3 were widely used in literature with respect to nonlinear stability of numerical methods for DDEs see. for example [1,2,5]. For the discretization of system 2.1 by a class of DG methods, we denote the interval I i, +1 for i,..., n, and define m-degree discontinuous finite element space as follows: S h υ : υ Ii P m I i, i 1, 2,...}, where P m I i denotes the set of all polynomials of degree m on I i. An m-degree discontinuous finite element Yt S h, which is approximations to y in 2.1, can be defined as follows: +1 Ytv dt + Ŷtv f t, Yt, Yt τvdt, v S h. 2.4 I i I i Here i n and Ŷt is defined by cf. [1] ψt, t Ŷt Yt} + C n [Yt] < t < t n Yt n t t n where C n is a positive real number and Y } 1 2 Yt+ i + Yt i [Y ] Yt i Yt + i Yt ± i lim ɛ Y ± ɛ. Remark 2.1. When τ, the literature [9,13] reveal us superconvergence results on each interval for C n 1 2, respectively, and the order of the method at the points t n is 2m + 1. Similarly, the adaptation of the same DG method for the problem 2.2 leads to the approximations Zt to z. Now we introduce some nonlinear stability concepts, which reveal to the reader the relation between the perturbations of the numerical solution and that of the initial value or the system. And these concepts are nonlinear analogues of that existed in the literature [2]. Definition 2.1. A numerical method is called globally stable if Yt n Zt n C max ψt ϕt, t n t holds under some assumptions, where C is a positive constant, Yt and Zt are numerical approximations to 2.1 and 2.2, respectively. Definition 2.2. A numerical method is said to be analogously asymptotically stable if the numerical solutions Yt and Zt satisfy lim T + T under some assumptions. Yt Zt 2 dt < 2.1 2.2 2.5
D. Li, C. Zhang / Applied Mathematics Letters 23 21 457 461 459 3. Nonlinear stability for DDEs In this section, we shall show DG methods lead to the global and analogously asymptotical stability for the DDEs. Theorem 3.1. Assume that the conditions 2.3 hold and α + β. Then, the DG method for DDE is globally stable. Proof. Let Yt and Zt be two sequences of approximations to problems 2.1 and 2.2, respectively, and write et Yt Zt êt Ŷt Ẑt F f t, Yt, Yt τ f t, Zt, Zt τ. With the notation, the DG methods with the same stepsize h for 2.1 and 2.2 yields: +1 etv dt + êtv Fvdt. I i I i Setting v et in the formulation 3.1, and integrating by parts gives: 1 +1 2 e2 t + êtet Fvdt. I i Summing up from to n 1, we find 1 2 e2 t + êtet i1 As in Cockburn [1], we have Here +1 3.1 Fetdt. 3.2 1 2 e2 t + êtet +1 1 2 e2 t + n Θt n 1 2 e2. 3.3 i1 Θt n 1 2 e2 t + n 1 2 e2 t + n êt net n + 1 2 [e2 ] + ê [e ] 1 2 e2 + + êe + + 1 2 e2 êt n et n et + n ê e }[e ] where we used the fact that [e 2 t n ] 2et n }[et n ]. i1 According to the definition of êt in 2.5, we get i1 ê e e + + 1 2 [e]2, 3.4 Θt n C i [e ] 2 + 1 2 [e]2. i1 On the other hand, noting that β, we have Fetdt f t, Yt, Yt τ f t, Zt, Yt τ etdt + α f t, Zt, Yt τ f t, Zt, Zt τ etdt et 2 dt + β et τ et dt α + 1 2 β et 2 dt + 1 2 β et τ 2 dt 3.5
46 D. Li, C. Zhang / Applied Mathematics Letters 23 21 457 461 α + 1 2 β et 2 dt + 1 2 β et 2 dt 1 2 β et 2 dt + 1 2 β et 2 dt 1 2 β et 2 dt 1 βτ max ψt ϕt 2. 3.6 2 t Now, together with 3.3, 3.5 and 3.6, we get e 2 t n e2 + βτ max ψt ϕt 2 1 + βτ max ψt ϕt 2 3.7 t t therefore, the method is globally stable. Theorem 3.2. Assume that the conditions 2.3 hold and α +β <. Then, the DG method for DDEs is analogously asymptotically stable. Proof. Let σ α + β <. Like the Eq. 3.6 in proof 3.1, we obtain Fetdt α + 1 2 β et 2 dt + 1 2 β et τ 2 dt α + 1 2 β σ 1 2 β 1 2 β et 2 dt + σ et 2 dt + 1 2 β et 2 dt et 2 dt + 1 2 β et 2 dt et 2 dt 1 βτ max ψt ϕt 2 + σ 2 t Now, together with 3.3, 3.5 and 3.8, we derive e 2 t n e2 + βτ max ψt ϕt 2 + 2σ t Since σ <, we find that there exists a constant C satisfying et 2 dt. 3.8 et 2 dt. 3.9 lim T + T et 2 dt C, which completes the proof. 3.1 4. Equations with several delays Consider the following equation with several delays: y t f t, y, yt τ 1,..., yt τ m, t yt ψt, t where τ i >, i 1,..., m. In fact, there are no additional difficulties in modifying the given results to this more general case. However, we do not list them here for the sake of brevity. References [1] A. Bellen, M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 23. [2] H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, 24.
D. Li, C. Zhang / Applied Mathematics Letters 23 21 457 461 461 [3] J. Kuang, Y. Cong, Stability of Numerical Methods for Delay Differential Equations, Science Press, Beijing, 25. [4] V.K. Barwell, Special stability problems for functional equations, BIT 15 1975 13 135. [5] C. Huang, H. Fu, S. Li, G. Chen, Stability analysis of Runge Kutta methods for non-linear delay differential equations, BIT 39 1999 27 28. [6] K.J. in t Hout, A new interpolation procedure for adapting Runge Kutta methods to delay differential equations, BIT 32 1992 634 649. [7] M. Zennaro, P-stability of Runge Kutta methods for delay differential equations, Numer. Math. 49 1986 35 318. [8] M. Zennaro, Asymptotic stability analysis of Runge Kutta methods for nonlinear systems of delay differential equations, Numer. Math. 77 1997 549 563. [9] S. Adjerid, K.D. Devine, J.E. Flaherty, L. Krivodonova, A posteriori error estimation for discontinuous Glerkin solutions of hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 191 22 197 1112. [1] B. Cockburn, Discontinuous Galerkin methods, ZAMM. Z. Angew. Math. Mech. 83 11 23 731 754. [11] M. Delfour, W. Hager, F. Trochu, Discontinuous Galerkin methods for ordinary differential equations, Math. Comput. 36 1981 455 473. [12] A. Ern, J. Proft, A posteriori discontinuous Galerkin error estimates for transient convectioncdiffusion equations, Appl. Math. Lett. 18 25 833 841. [13] D. Estep, A posteriori error bounds and global error control for approximations of ordinary differential equations, SIMA J. Numer. Anal. 32 1995 1 48. [14] A. Romkes, S. Prudhomme, J.T. Oden, A priori error analyses of a stabilized Discontinuous Galerkin method, Comput. Math. Appl. 46 23 1289 1211. [15] T. Zhang, J. Li, S. Zhang, Superconvergence of discontinuous Galerkin methods for hyperbolic systems, J. Comput. Appl. Math. 223 29 725 734. [16] C. Zhang, D. Li, The DGFE methods for delay differential equations submitted for publication. [17] L. Torelli, Stability of numerical methods for delay differential equations, J. Comput. Appl. Math. 25 1989 15 26. [18] A. Bellen, M. Zennaro, Strong contractivity properties of numerical methods for ordinary and delay differential equations, Appl. Numer. Math. 9 1992 321 346. [19] C. Huang, H. Fu, S. Li, H. Fu, G. Chen, Nonlinear stability of general linear methods for delay differential equations, BIT 42 22 38 392. [2] C. Zhang, S. Vandewalle, Stability analysis of Volterra delay-integro-differential equations and their backward differentiation time discretization, J. Comput. Appl. Math. 164 24 797 814.