Abstract and Applied Analysis

Volume 2014 (2014), Article ID 810352, 8 pages

http://dx.doi.org/10.1155/2014/810352

Research Article

A Compact Difference Scheme for a Class of Variable Coefficient Quasilinear Parabolic Equations with Delay

Wei Gu

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.

Abstract.

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.

1. Introduction

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 [5] considered a hybrid approximation method for solving Hutchinson’s equation; Jackiewicz and Zubik-Kowal [6] 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 [13].

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 [14]. A Crank-Nicolson scheme and a linearized compact difference scheme were proposed by Zhang and Sun in [15] and Sun and Zhang in [16], respectively. Q. Zhang and C. Zhang considered a new linearized compact multisplitting scheme in [17]. Gu and Wang constructed a Crank-Nicolson scheme in [18] 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 .

Lemma 1 (see [19, 20]). Suppose that ; then, we have

where .

From Lemma 1 and Taylor expansion, we obtain

Inserting (13) into (11), we have

where

From and assumptions (H1) and (H2), we have

such that

Discretizing the initial and boundary conditions of (2) and (3), we obtain

Replacing by in (14) and omitting , we obtain the following compact difference scheme:

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.

Lemma 2 (see [16, 17, 19]). Assume that to be nonnegative sequence and satisfies

then

where and are nonnegative constants.

Theorem 3. Under the condition that , the compact difference scheme (20)–(22) has a unique solution.

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.

Denote , , ; subtracting (20)–(22) from (14), (18), and (19), respectively, the following error equations can be obtained:

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.

To discuss the stability of the difference scheme (20)–(22), we consider the following problem:

where is the perturbation caused by . The following difference scheme solving for (49) can be obtained:

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

In this section, a numerical test is considered to validate the algorithms provided in this paper, and the numerical solutions of the example are obtained by exploiting scheme (20)–(22).

Define

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.

Table 1: Numerical results of (54) when , .

, )
Numerical solution
Exact solution

(0.5, 0.1)
0.667184
0.667184

(0.5, 0.2)
0.727837
0.727837

(0.5, 0.3)
0.788490
0.788490

(0.5, 0.4)
0.849143
0.849143

(0.5, 0.5)
0.909796
0.909796

(0.5, 0.6)
0.970449
0.970449

(0.5, 0.7)
1.031102
1.031102

(0.5, 0.8)
1.091755
1.091755

(0.5, 0.9)
1.152408
1.152408

(0.5, 1.0)
1.213061
1.213061

Table 2: Maximum norm errors of (54) with different step-sizes.

1/10
1/100

*

1/20
1/400

15.842

1/40
1/1600

15.999

1/80
1/6400

16.046

1/160
1/25600

19.546

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.

Figure 1: Error curves of difference scheme (20)–(22) solving for problem (54) with different step-sizes when .

Figure 2: Error surface maps of difference scheme (20)–(22) solving for problem (54) with step-size ; .

Figure 3: Error surface maps of difference scheme (20)–(22) solving for problem (54) with step-size ; .

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.

5. Conclusion

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.

Acknowledgments

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).

References

J. H. Wu, Theory and Application of Partial Functional Differential Equation, Springer, New York, NY, USA, 1996.

A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, UK, 2003.

H. Brunner, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge, UK, 2004.

D. Li and C. Zhang, “ error estimates of discontinuous Galerkin methods for delay differential equations,” Applied Numerical Mathematics, vol. 82, pp. 1–10, 2014.

H. R. Marzban and H. R. Tabrizidooz, “A hybrid approximation method for solving Hutchinson's equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 100–109, 2012.

Z. Jackiewicz and B. Zubik-Kowal, “Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations,” Applied Numerical Mathematics, vol. 56, no. 3-4, pp. 433–443, 2006.

A. R. Ansari, S. A. Bakr, and G. I. Shishkin, “A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations,” Journal of Computational and Applied Mathematics, vol. 205, no. 1, pp. 552–566, 2007.

D. Li, C. Zhang, and W. Wang, “Long time behavior of non-Fickian delay reaction-diffusion equations,” Nonlinear Analysis: Real World Applications, vol. 13, no. 3, pp. 1401–1415, 2012.

F. A. Rihan, “Computational methods for delay parabolic and time-fractional partial differential equations,” Numerical Methods for Partial Differential Equations, vol. 26, no. 6, pp. 1556–1571, 2010.

D. Li, C. Zhang, and H. Qin, “LDG method for reaction-diffusion dynamical systems with time delay,” Applied Mathematics and Computation, vol. 217, no. 22, pp. 9173–9181, 2011.

D. Li and C. Zhang, “On the long time simulation of reaction-diffusion equations with delay,” The Scientific World Journal, vol. 2014, Article ID 186802, 5 pages, 2014.

D. Li, C. Tong, and J. Wen, “Stability of exact and discrete energy for non-fickian reaction-diffusion equations with a variable delay,” Abstract and Applied Analysis, vol. 2014, Article ID 840573, 9 pages, 2014.

H. Tian, “Asymptotic stability of numerical methods for linear delay parabolic differential equations,” Computers & Mathematics with Applications, vol. 56, no. 7, pp. 1758–1765, 2008.

J. A. Ferreira and P. M. da Silva, “Energy estimates for delay diffusion-reaction equations,” Journal of Computational Mathematics, vol. 26, no. 4, pp. 536–553, 2008.

Z. B. Zhang and Z. Z. Sun, “A Crank-Nicolson scheme for a class of delay nonlinear parabolic differential equations,” Journal on Numerical Methods and Computer Applications, vol. 31, no. 2, pp. 131–140, 2010.

Z.-Z. Sun and Z.-B. Zhang, “A linearized compact difference scheme for a class of nonlinear delay partial differential equations,” Applied Mathematical Modelling, vol. 37, no. 3, pp. 742–752, 2013.

Q. Zhang and C. Zhang, “A new linearized compact multisplitting scheme for the nonlinear convection-reaction-diffusion equations with delay,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 12, pp. 3278–3288, 2013.

W. Gu and P. Wang, “A Crank-Nicolson difference scheme for solving a type of variable coefficient delay partial differential equations,” Journal of Applied Mathematics, vol. 2014, Article ID 560567, 6 pages, 2014.

Z.-Z. Sun, The Numerical Methods for Partial Equations, Science Press, Beijing, China, 2005, Chinese.

Z.-Z. Sun, “Compact difference schemes for heat equation with Neumann boundary conditions,” Numerical Methods for Partial Differential Equations, vol. 25, no. 6, pp. 1320–1341, 2009.