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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 810352, 8 pages
A Compact Difference Scheme for a Class of Variable Coefficient Quasilinear Parabolic Equations with Delay
School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan, Hubei 430073, China
Received 19 February 2014; Accepted 15 May 2014; Published 5 June 2014
Academic Editor: Zhongxiao Jia
Copyright © 2014 Wei Gu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A linearized compact difference scheme is provided for a class of variable coefficient parabolic systems with delay. The unique solvability, unconditional stability, and convergence of the difference scheme are proved, where the convergence order is four in space and two in time. A numerical test is presented to illustrate the theoretical results.
From the twentieth century, more and more scholars have been attracted into the research on the theory of delay differential equations (DDEs) [1–4]. As we know, most DDEs have no analytical solutions; efficient numerical methods solving for DDEs and delay partial differential equations (DPDEs) need to be considered deeply. Recently, many scholars consider the numerical investigation on DPDEs. For instance, Marzban and Tabrizidooz  considered a hybrid approximation method for solving Hutchinson’s equation; Jackiewicz and Zubik-Kowal  considered Chebyshev spectral collocation and waveform relaxation methods for nonlinear DPDEs and finite difference methods were considered to solve delay parabolic partial differential equations in [7–9]; Li et al. [10–12] constructed finite element methods to solve reaction-diffusion equations with delay. The numerical research of DPDEs focused on stability analysis can be referred to in .
The following variable coefficient parabolic systems with delay are considered in this paper: where is a constant and is the delay term, , . In the special case of , numerical solutions of (1)–(3) have been considered in [14–17]. Ferreira and da Silva considered a backward Euler scheme and proved the stability and convergence by the energy method in . A Crank-Nicolson scheme and a linearized compact difference scheme were proposed by Zhang and Sun in  and Sun and Zhang in , respectively. Q. Zhang and C. Zhang considered a new linearized compact multisplitting scheme in . Gu and Wang constructed a Crank-Nicolson scheme in  to solve a special case of (1), where . In this paper, a linearized compact difference scheme solving for (1)–(3) will be constructed. The unique solvability, unconditional stability, and convergence of the difference scheme are proved, where the convergence order is four in space and two in time. A numerical test is presented to illustrate the theoretical results.
The paper is organized as follows. In Section 2, a linearized compact difference scheme is constructed to solve (1)–(3). Section 3 considers the solvability, stability, and convergence of the provided difference scheme. In Section 4, a numerical test is presented to illustrate the theoretical results. Section 5 gives a brief discussion of this paper.
2. The Compact Difference Scheme and Local Truncation Error
Throughout this paper, the following assumptions are assumed to be true.(H1)Let be an integer satisfying , denote , , , and , assume that (1)–(3) has a unique solution and that and its partial derivatives are all bounded by a constant ;(H2) has bounded first-order continuous partial derivatives, and we denote where , , and are constants, .
First let and be two positive integers; then, we take , , , , . Define , where , , . Denote , , , throughout this paper. Let be the grid function space defined on . The following notations are made: Considering (1) at the point , we have From Taylor expansion, we have where , , is between and , and is between and . Substituting (8) into (7), denote ; we obtain where Acting operator on both sides of (9), we have Resorting to the following Lemma, we can obtain the estimation of the operator .
From and assumptions (H1) and (H2), we have such that
3. The Solvability, Convergence, and Stability of the Compact Difference Scheme
Define the following grid function space on : If , we introduce the following notations: By [16, 17, 19], we have the following two inequalities: For the analysis of the difference scheme, the following Lemma is introduced.
Proof. Denote that ; then, difference scheme (20)–(22) can be reformed as
The mathematical induction method will be used in the proof of this theorem. Denote Notice that is determined by the initial condition (21). Suppose that has been determined.
Let in (20); the linear algebraic equations with respect to can be obtained. Under the condition that , we have Thus, the coefficient matrix of the linear algebraic system is strictly diagonally dominant and then there exists a unique solution . By the inductive principle, the proof ends.
Theorem 4. Denote If the following conditions are satisfied: then we have where is a constant.
Proof. Acting on (32) and summing up for from 1 to , we obtain
Mathematical induction will be used to prove this theorem. Notice that and suppose that (37) is true for ; we will show that (37) is also true for .
In the following, each term of (38) will be estimated: From the inductive assumption and (36), we have From (H2), we have It then follows that From the inequality above, we obtain
Inserting (39)–(43) into (38), we obtain The above inequality has the following form: Summing up (45) for , noticing (33), and exploiting (26), we have By Lemma 2, we have From (25), we obtain By the inductive principle, this completes the proof.
Similar to the proof of Theorem 4, the following stability result can be obtained
Theorem 5. Denote Then, there exist constants and such that under the condition that and are small enough and .
4. Numerical Test
Example 1. Consider the following problem: where . The exact solution of (54) is .
Table 1 provides some numerical results of difference scheme (20)–(22) solving for (54) with step-sizes and . Table 2 gives the maximum absolute errors between numerical solutions and exact solutions with different step-sizes. From Table 2, we can see that when the space step-size and the time step-size are reduced by a factor of 1/2 and 1/4, respectively, then the maximum absolute errors are reduced by a factor of approximately 1/16.
Figure 1 provides us the error curves of numerical solutions for (54) at by using scheme (20)–(22). Figures 2 and 3 give the error surface of the numerical solutions with step-sizes , , and , , respectively.
Generally speaking, from the results of the tables and the figures provided, we can see that the numerical results are coincident with the theoretical results.
In this paper, a compact difference scheme is constructed to solve a type of variable coefficient delay partial differential equations, and the difference scheme is proved to be unconditionally stable and convergent. Finally, a numerical test is presented to illustrate the theoretical results.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This work is supported by the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (no. 2013693) and the National Natural Science Foundation of China (nos. 71301166, 11301544, and 11201487).
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