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

Authors: Wei Gu
Journal: Abstract and Applied Analysis
 (2014)

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## 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.


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## Body

## 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:(1)r(x,t)ut-duxx=f(u(x,t),u(x,t-s),x,t),hhhhhhhhhhhhhhhhhh(x,t)∈(0,1)×(0,T],(2)u(x,t)=ϕ(x,t),x∈[0,1],t∈[-s,0],(3)u(0,t)=α(t),u(1,t)=β(t),t∈(0,T],
where d>0 is a constant and s>0 is the delay term, r(x,t)∈C((0,1)×(0,T]), 0<c0≤r(x,t)≤c1. In the special case of r(x,t)=1, 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 f=f(u(x,t-s)). 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 Section2, 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)
Letm be an integer satisfying ms≤T<(m+1)s, denote Il=(ls,ls+s), l=-1,0,…,m-1, Im=(ms,T), and I=∪p=-1mIp, assume that (1)–(3) has a unique solution u∈C6,4(I×(0,T]) and that u and its partial derivatives are all bounded by a constant c2;(H2)
f(μ,ν,x,t) has bounded first-order continuous partial derivatives, and we denote
(4)c3=max⁡|ε1|≤ε0,|ε2|≤ε0⁡|fμ(u(x,t)+ε1,u(x,t-s)+ε2,x,t)|,c4=max⁡|ε1|≤ε0,|ε2|≤ε0⁡|fν(u(x,t)+ε1,u(x,t-s)+ε2,x,t)|,
where ε0>0, c3, and c4 are constants, (x,t)∈(0,1)×(0,T].First letM and j be two positive integers; then, we take h=1/M, τ=s/j, xi=ih, tk=kτ, tk+1/2=(tk+tk+1)/2. Define Ωhτ=Ωh×Ωτ, where Ωh={xi∣0≤i≤M}, Ωτ={tk∣-j≤k≤N}, N=[T/τ]. Denote Uik=u(xi,tk), 0≤i≤M, -j≤k≤N, throughout this paper. Let
(5)W={vik∣0≤i≤M,-j≤k≤N}
be the grid function space defined on Ωhτ. The following notations are made:
(6)vik+(1/2)=vik+vik+12,δtvik+(1/2)=vik+1-vikτ,δxvi+(1/2)k=vi+1k-vikh,δx2vik=vi+1k-2vik+vi-1kh2,Avik=112(vi-1k+10vkk+vi+1k).
Considering (1) at the point (xi,tk+(1/2)), we have
(7)r(xi,tk+(1/2))∂u∂t(xi,tk+(1/2))-d∂2u∂x2(xi,tk+(1/2))=f(u(xi,tk+(1/2)),u(xi,tk+(1/2)-j),xi,tk+(1/2)),0≤i≤M,0≤k≤N-1.
From Taylor expansion, we have
(8)∂u∂t(xi,tk+(1/2))=δtUik+(1/2)-τ224∂3u∂t3(xi,ηik),ηik∈(tk,tk+1),∂2u∂x2(xi,tk+(1/2))=12[∂2u∂x2(xi,tk)+∂2u∂x2(xi,tk+1)]-τ28∂4u(∂x2∂t2)(xi,γik)=12(δx2Uik+δx2Uik+1)-h224[∂4u∂x4(ξik,tk)+∂4u∂x4(ξik+1,tk+1)]-τ28∂4u∂x2∂t2(xi,γik),ξik,ξik+1∈(xi-1,xi+1),γik∈(tk,tk+1),f(u(xi,tk+(1/2)),u(xi,tk+(1/2)-j),xi,tk+(1/2))=f(32Uik-12Uik-1,12Uik+1-j+12Uik-j,xi,tk+(1/2))+3τ28∂2u(xi,ρk)∂t2fμ(ζik,ςik,xi,tk+(1/2))-τ28∂2u(xi,ϱk)∂t2fν(ζik,ςik,xi,tk+(1/2)),
where ρk∈(tk-1,tk+1/2), ϱk∈(tk-j,tk+1-j), ζik is between u(xi,tk+(1/2)) and (3/2)Uik-(1/2)Uik-1, and ςik is between u(xi,tk+(1/2)-j) and (1/2)Uik+1-j+(1/2)Uik-j. Substituting (8) into (7), denote rik+(1/2)=r(xi,tk+(1/2)); we obtain
(9)rik+(1/2)δtUik+(1/2)-d2[∂2u∂x2(xi,tk)+∂2u∂x2(xi,tk+1)]=f(32Uik-12Uik-1,12Uik+1-j+12Uik-j,xi,tk+(1/2))+τ2R¯ik,1≤i≤M,0≤k≤N-1,
where
(10)R¯ik=rik+(1/2)24∂3u∂t3(xi,ηik)-d8∂4u∂x2∂t2(xi,γik)+38∂2u(xi,ρk)∂t2fμ(ζik,ςik,xi,tk+(1/2))-18∂2u(xi,ϱk)∂t2fν(ζik,ςik,xi,tk+(1/2)).
Acting operator A on both sides of (9), we have
(11)Arik+(1/2)δtUik+(1/2)-d2[A∂2u∂x2(xi,tk)+A∂2u∂x2(xi,tk+1)]=Af(32Uik-12Uik-1,12Uik+1-j+12Uik-j,xi,tk+(1/2))+τ2AR¯ik,1≤i≤M-1,0≤k≤N-1.
Resorting to the following Lemma, we can obtain the estimation of the operator A.Lemma 1 (see [19, 20]).
Suppose thatq(x)∈C6[xi-1,xi+1]; then, we have
(12)112[q′′(xi-1)+10q′′(xi)+q′′(xi+1)]-1h2[q(xi-1)-2q(xi)+q(xi+1)]=h4240q(6)(ωi),
where ωi∈(xi-1,xi+1).From Lemma1 and Taylor expansion, we obtain
(13)A∂2u∂x2(xi,tk)=δx2Uik+h4240∂6u∂x6(θik,tk),hhhhhhhhhhhhhhhhhθik∈(xi-1,xi+1),A∂2u∂x2(xi,tk+1)=δx2Uik+1+h4240∂6u∂x6(θik+1,tk+1),hhhhhhhhhhhhhhhhhhhhhhθik+1∈(xi-1,xi+1).
Inserting (13) into (11), we have
(14)Arik+(1/2)δtUik+(1/2)-dδx2Uik+1/2=Af(32Uik-12Uik-1,12Uik+1-j+12Uik-j,xi,tk+(1/2))+Rik,1≤i≤M-1,0≤k≤N-1,
where
(15)Rik=τ2AR¯ik+dh4480[∂6u∂x6(θik,tk)+∂6u∂x6(θik+1,tk+1)].From0<c0≤rik+1/2≤c1 and assumptions (H1) and (H2), we have
(16)|R¯ik|≤c5,1≤i≤M-1,0≤k≤N-1,
such that
(17)|Rik|≤c6(τ2+h4),1≤i≤M-1,0≤k≤N-1.Discretizing the initial and boundary conditions of (2) and (3), we obtain
(18)Uik=ϕ(xi,tk),0≤i≤M,-j≤k≤0,(19)U0k=α(tk),UMk=β(tk),1≤k≤N.
Replacing Uik by uik in (14) and omitting Rik, we obtain the following compact difference scheme:
(20)Arik+(1/2)δtuik+(1/2)-dδx2uik+1/2=Af(32uik-12uik-1,12uik+1-j+12uik-j,xi,tk+(1/2)),1≤i≤M-1,0≤k≤N-1,(21)uik=ϕ(xi,tk),0≤i≤M,-j≤k≤0,(22)u0k=α(tk),uMk=β(tk),1≤k≤N.


## 3. The Solvability, Convergence, and Stability of the Compact Difference Scheme

Define the following grid function space onΩh:
(23)V={v∣v=(v0,v1,…,vM),v0=vM=0}.
If v∈V, we introduce the following notations:
(24)∥v∥=h∑i=1M-1‍(vi)2,|v|1=h∑i=1M‍(vi-vi-1h)2,∥v∥∞=max⁡0≤i≤M|vi|.
By [16, 17, 19], we have the following two inequalities:
(25)∥v∥∞≤12|v|1,(26)∥v∥≤16|v|1.
For the analysis of the difference scheme, the following Lemma is introduced.Lemma 2 (see [16, 17, 19]).
Assume that{Fk∣k≥0} to be nonnegative sequence and satisfies
(27)Fk+1≤A+Bτ∑i=1k‍Fl,k=0,1,…;
then
(28)Fk+1≤Aexp⁡(Bkτ),k=0,1,2,…,
where A and B are nonnegative constants.Theorem 3.
Under the condition that5c0-c1>0, the compact difference scheme (20)–(22) has a unique solution.Proof.
Denote thatλ=dτ/h2; then, difference scheme (20)–(22) can be reformed as
(29)(112ri-1k+1/2-λ2)ui-1k+1+(1012rik+1/2+λ)uik+1+(112ri+1k+1/2-λ2)ui+1k+1=(112ri-1k+1/2+λ2)ui-1k+(1012rik+1/2-λ)uik+(112ri+1k+1/2+λ2)ui+1k+τ12f(32ui-1k-12ui-1k-1,12ui-1k+1-j+12ui-1k-j,xi-1,tk+(1/2))+10τ12f(32uik-12uik-1,12uik+1-j+12uik-j,xi,tk+(1/2))+τ12f(32ui+1k-12ui+1k-1,12ui+1k+1-j+12ui+1k-j,xi+1,tk+(1/2)).
The mathematical induction method will be used in the proof of this theorem. Denote(30)uk=(u0k,u1k,…,uMk).
Notice that uk(-j≤k≤0) is determined by the initial condition (21). Suppose that ul has been determined.
Letk=l in (20); the linear algebraic equations with respect to ul+1 can be obtained. Under the condition that 5c0-c1>0, we have
(31)|1012ril+1/2+λ|-|112ri-1l+1/2-λ2|-|112ri+1l+1/2-λ2|≥1012ril+1/2-112ri-1l+1/2-112ri+1l+1/2≥10c0-2c112>0.
Thus, the coefficient matrix of the linear algebraic system is strictly diagonally dominant and then there exists a unique solution ul+1. By the inductive principle, the proof ends.Denoteeik=Uik-uik, 0≤i≤M, -j≤k≤N; subtracting (20)–(22) from (14), (18), and (19), respectively, the following error equations can be obtained:
(32)Arik+(1/2)δteik+(1/2)-dδx2eik+1/2=A[f(32Uik-12Uik-1,12Uik+1-j+12Uik-j,xi,tk+(1/2))-f(32uik-12uik-1,12uik+1-j+12uik-j,xi,tk+(1/2))]+Rik,1≤i≤M-1,0≤k≤N-1,(33)eik=0,0≤i≤M,-j≤k≤0,(34)e0k=0,eMk=0,1≤k≤N.Theorem 4.
Denote(35)C=c63Td(9c0-c1)exp⁡(2(5c32+c42)d(9c0-c1)T).
If the following conditions are satisfied:
(36)τ≤(ϵ04C)1/2,h≤(ϵ04C)1/4,
then we have
(37)∥ek∥∞≤C(τ2+h4),0≤k≤N,
where ϵ0>0 is a constant.Proof.
Actinghδteik+(1/2) on (32) and summing up for i from 1 to M-1, we obtain
(38)h∑i=1M-1‍(Arik+(1/2)δteik+(1/2))δteik+(1/2)-dh∑i=1M-1‍(δx2eik+1/2)δteik+(1/2)=h∑i=1M-1‍{A[f(32Uik-12Uik-1,12Uik+1-j+12Uik-j,xi,tk+(1/2))-f(32uik-12uik-1,12uik+1-j+12uik-j,xi,tk+(1/2))]}δteik+(1/2)+h∑i=1M-1‍(Rik)δteik+(1/2),1≤i≤M-1,0≤k≤N-1.
Mathematical induction will be used to prove this theorem. Notice that∥ek∥∞=0(-j≤k≤0) and suppose that (37) is true for 0≤k≤l; we will show that (37) is also true for k=l+1.
In the following, each term of (38) will be estimated:
(39)h∑i=1M-1‍(Arik+(1/2)δteik+(1/2))δteik+(1/2)≥h12∑i=1M-1‍[9rik+(1/2)-(ri-1k+(1/2)+ri+1k+(1/2))2](δteik+(1/2))2≥9c0-c112∥δtek+(1/2)∥2,-dh∑i=1M-1‍(δx2eik+(1/2))δteik+(1/2)=d2τ(|ek+1|12-|ek|12),h∑i=1M-1‍(Rik)δteik+(1/2)≤69c0-c1h∑i=1M-1‍(Rik)2+9c0-c124h∑i=1M-1‍(δteik+(1/2))2≤69c0-c1c62(τ2+h4)2+9c0-c124∥δtek+(1/2)∥2.
From the inductive assumption and (36), we have
(40)∥ek∥∞≤C(τ2+h4)≤ϵ02,0≤k≤l.
From (H2), we have
(41)|f(32Uik-12Uik-1,12Uik+1-j+12Uik-j,xi,tk+(1/2))-f(32uik-12uik-1,12uik+1-j+12uik-j,xi,tk+(1/2))|≤c3|32eik-12eik-1|+c4|12eik+1-j+12eik-j|,1≤i≤M,0≤k≤l.
It then follows that
(42)|A[f(32Uik-12Uik-1,12Uik+1-j+12Uik-j,xi,tk+(1/2))-f(32uik-12uik-1,12uik+1-j+12uik-j,xi,tk+(1/2))]|≤A(c3|32eik-12eik-1|+c4|12eik+1-j+12eik-j|),1≤i≤M-1,0≤k≤l.
From the inequality above, we obtain
(43)h∑i=1M-1‍{A[f(32Uik-12Uik-1,12Uik+1-j+12Uik-j,xi,tk+(1/2))-f(32uik-12uik-1,12uik+1-j+12uik-j,xi,tk+(1/2))]}×δteik+(1/2)≤h∑i=1M-1‍{A(c3|32eik-12eik-1|+c4|12eik+1-j+12eik-j|)}×δteik+(1/2)≤69c0-c1h×∑i=1M-1‍{A(c3|32eik-12eik-1|+c4|12eik+1-j+12eik-j|)}2+9c0-c124∥δteik+(1/2)∥2≤69c0-c1h×∑i=1M-1‍(c3|32eik-12eik-1|+c4|12eik+1-j+12eik-j|)2+9c0-c124∥δtek+(1/2)∥2≤129c0-c1×[c32h∑i=1M-1‍(32eik-12eik-1)2+c42h∑i=1M-1‍(12eik+1-j+12eik-j)2]+9c0-c124∥δtek+(1/2)∥2≤129c0-c1×[52c32h∑i=1M-1‍((eik)2+(eik-1)2)+12c42h∑i=1M-1‍((eik+1-j)2+(eik-j)2)]+9c0-c124∥δtek+(1/2)∥2=309c0-c1c32(∥ek∥2+∥ek-1∥2)+69c0-c1c42(∥ek+1-j∥2+∥ek-j∥2)+9c0-c124∥δtek+(1/2)∥2,0≤k≤l.
Inserting (39)–(43) into (38), we obtain
(44)d2τ(|ek+1|12-|ek|12)≤309c0-c1c32(∥ek∥2+∥ek-1∥2)+69c0-c1c42(∥ek+1-j∥2+∥ek-j∥2)+69c0-c1c62(τ2+h4)2,0≤k≤l.
The above inequality has the following form:
(45)|ek+1|12≤|ek|12+2τd309c0-c1c32(∥ek∥2+∥ek-1∥2)+2τd69c0-c1c42(∥ek+1-j∥2+∥ek-j∥2)+2τd69c0-c1c62(τ2+h4)2,0≤k≤l.
Summing up (45) for k, noticing (33), and exploiting (26), we have
(46)|ek+1|12≤|e0|12+2τd309c0-c1c32×∑m=0k‍(∥em∥2+∥em-1∥2)+2τd69c0-c1c42×∑m=0k‍(∥em+1-j∥2+∥em-j∥2)+2τd69c0-c1c62∑m=0l‍(τ2+h4)2≤24(5c32+c42)d(9c0-c1)τ∑m=1k‍∥em∥2+12c62(l+1)τd(9c0-c1)(τ2+h4)2≤4(5c32+c42)d(9c0-c1)τ×∑m=1k‍|em|2+12c62Td(9c0-c1)(τ2+h4)2,0≤k≤l.
By Lemma 2, we have
(47)|el+1|12≤12c62Td(9c0-c1)exp⁡(4(5c32+c42)d(9c0-c1)T)(τ2+h4)2.
From (25), we obtain
(48)∥el+1∥∞≤c63Td(9c0-c1)exp⁡(2(5c32+c42)d(9c0-c1)T)(τ2+h4).
By the inductive principle, this completes the proof.To discuss the stability of the difference scheme (20)–(22), we consider the following problem:
(49)r(x,t)vt-dvxx=f(v(x,t),v(x,t-s),x,t),hhhhhhhhhhhhhhhhh(x,t)∈(0,1)×(0,T],v(x,t)=ϕ(x,t)+ψ(x,t),x∈[0,1],t∈[-s,0],v(0,t)=α(t),v(1,t)=β(t),t∈(0,T],
where ψ(x,t) is the perturbation caused by ϕ(x,t). The following difference scheme solving for (49) can be obtained:
(50)Arik+(1/2)δtvik+(1/2)-dδx2vik+1/2=Af(32vik-12vik-1,12vik+1-j+12vik-j,xi,tk+(1/2)),1≤i≤M-1,0≤k≤N-1,vik=ϕ(xi,tk)+ψik,0≤i≤M,-j≤k≤0,v0k=α(tk),vMk=β(tk),1≤k≤N.Similar to the proof of Theorem4, the following stability result can be obtainedTheorem 5.
Denote(51)ηik=vik-uik,0≤i≤M,-j≤k≤N.
Then, there exist constants c7 and c8 such that
(52)∥ηk∥∞≤c7τh∑m=-j0‍∑i=lM-1‍(ψik)2
under the condition that h and τ are small enough and max⁡-j≤k≤0,0≤i≤M|ψik|≤c8.


## 4. Numerical Test

In this section, a numerical test is considered to validate the algorithms provided in this paper, and the numerical solutionsuik of the example are obtained by exploiting scheme (20)–(22).Define(53)E∞(h,τ)=max⁡0≤i≤M,0≤k≤N|u(xi,tk)-uik|.Example 1.
Consider the following problem:(54)r(x,t)ut-2uxx=u(x,t)+u(x,t-0.1)+2e-x,hhhhhhhhhhhhhhمhhhhhhx∈(0,1),t∈(0,1],u(x,t)=e-x(1+t),x∈(0,1),t∈[-0.1,0],u(0,t)=1+t,u(1,t)=e-1(1+t),t∈(0,1],
where r(x,t)=4t+5.9. The exact solution of (54) is u(x,t)=e-x(1+t).Table1 provides some numerical results of difference scheme (20)–(22) solving for (54) with step-sizes h=1/10 and τ=1/100. 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 h=1/10, τ=1/100.
(
x, t)
Numerical solution
Exact solution
|
u
(
x
i
,
t
k
)
-
u
i
k
|
(0.5, 0.1)
0.667184
0.667184
8.652
e
-
009
(0.5, 0.2)
0.727837
0.727837
1.623
e
-
008
(0.5, 0.3)
0.788490
0.788490
2.258
e
-
008
(0.5, 0.4)
0.849143
0.849143
2.805
e
-
008
(0.5, 0.5)
0.909796
0.909796
3.288
e
-
008
(0.5, 0.6)
0.970449
0.970449
3.725
e
-
008
(0.5, 0.7)
1.031102
1.031102
4.129
e
-
008
(0.5, 0.8)
1.091755
1.091755
4.509
e
-
008
(0.5, 0.9)
1.152408
1.152408
4.870
e
-
008
(0.5, 1.0)
1.213061
1.213061
5.218
e
-
008Table 2
Maximum norm errors of (54) with different step-sizes.
h
τ
E
∞
(
h
,
τ
)
E
∞
(
h
,
τ
)
/
E
∞
(
h
/
2
,
τ
/
2
)
1/10
1/100
5.220
e
-
008
*
1/20
1/400
3.295
e
-
009
15.842
1/40
1/1600
2.059
e
-
010
15.999
1/80
1/6400
1.283
e
-
011
16.046
1/160
1/25600
6.566
e
-
013
19.546Figure1 provides us the error curves of numerical solutions for (54) at t=1 by using scheme (20)–(22). Figures 2 and 3 give the error surface of the numerical solutions with step-sizes h=1/10, τ=1/100, and h=1/20, τ=1/400, respectively.Figure 1
Error curves of difference scheme (20)–(22) solving for problem (54) with different step-sizes when t=1.Figure 2
Error surface maps of difference scheme (20)–(22) solving for problem (54) with step-size h=1/10; τ=1/100.Figure 3
Error surface maps of difference scheme (20)–(22) solving for problem (54) with step-size h=1/20; τ=1/400.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.



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