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

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: 𝑟(𝑥,𝑡)𝑢_𝑡−𝑑𝑢_𝑥𝑥(𝑢()()),()∈()×(],=𝑓𝑥,𝑡,𝑢𝑥,𝑡−𝑠,𝑥,𝑡𝑥,𝑡0,10,𝑇𝑢(𝑥,𝑡)=𝜙(𝑥,𝑡),𝑥∈[0,1],𝑡∈[−𝑠,0],𝑢(0,𝑡)=𝛼(𝑡),𝑢(1,𝑡)=𝛽(𝑡),𝑡∈(0,𝑇], (1)(2)(3) 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 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 𝑚 be an integer satisfying 𝑚𝑠≤𝑇<(𝑚+1)𝑠, denote 𝐼_𝑙=(𝑙𝑠,𝑙𝑠+𝑠), 𝑙=−1,0,…,𝑚−1, 𝐼_𝑚=(𝑚𝑠,𝑇), and 𝐼=∪^𝑚_𝑝=−1𝐼_𝑝, assume that (1)–(3) has a unique solution 𝑢∈𝐶^6,4(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 𝑐₃=max_{𝜀1≤𝜀0,𝜀2≤𝜀0}𝑓_𝜇𝑢(𝑥,𝑡)+𝜀₁,𝑢(𝑥,𝑡−𝑠)+𝜀₂,𝑐,𝑥,𝑡₄=max_{𝜀1≤𝜀0,𝜀2≤𝜀0}𝑓_𝜈𝑢(𝑥,𝑡)+𝜀₁,𝑢(𝑥,𝑡−𝑠)+𝜀₂,𝑥,𝑡, (4) where 𝜀₀>0, 𝑐₃, and 𝑐₄ are constants, (𝑥,𝑡)∈(0,1)×(0,𝑇].

First let 𝑀 and 𝑗 be two positive integers; then, we take ℎ=1/𝑀, 𝜏=𝑠/𝑗, 𝑥_𝑖=𝑖ℎ, 𝑡_𝑘=𝑘𝜏, 𝑡_𝑘+1/2=(𝑡_𝑘+𝑡_𝑘+1)/2. Define Ω_ℎ𝜏=Ω_ℎ×Ω_𝜏, where Ω_ℎ={𝑥_𝑖∣0≤𝑖≤𝑀}, Ω_𝜏={𝑡_𝑘∣−𝑗≤𝑘≤𝑁}, 𝑁=[𝑇/𝜏]. Denote 𝑈^𝑘_𝑖=𝑢(𝑥_𝑖,𝑡_𝑘), 0≤𝑖≤𝑀, −𝑗≤𝑘≤𝑁, throughout this paper. Let 𝒲=𝑣^𝑘_𝑖∣0≤𝑖≤𝑀,−𝑗≤𝑘≤𝑁 (5) be the grid function space defined on Ω_ℎ𝜏. The following notations are made: 𝑣_𝑖^𝑘+(1/2)=𝑣^𝑘_𝑖+𝑣_𝑖^𝑘+12,𝛿_𝑡𝑣_𝑖^𝑘+(1/2)=𝑣_𝑖^𝑘+1−𝑣^𝑘_𝑖𝜏,𝛿_𝑥𝑣^𝑘_𝑖+(1/2)=𝑣^𝑘_𝑖+1−𝑣^𝑘_𝑖ℎ,𝛿²_𝑥𝑣^𝑘_𝑖=𝑣^𝑘_𝑖+1−2𝑣^𝑘_𝑖+𝑣^𝑘_𝑖−1ℎ²,𝒜𝑣^𝑘_𝑖=1𝑣12^𝑘𝑖−1+10𝑣^𝑘_𝑘+𝑣^𝑘_𝑖+1. (6) Considering (1) at the point (𝑥_𝑖,𝑡_𝑘+(1/2)), we have 𝑟𝑥_𝑖,𝑡_𝑘+(1/2)𝜕𝑢𝑥𝜕𝑡^𝑖,𝑡_𝑘+(1/2)𝜕−𝑑²𝑢𝜕𝑥²𝑥_𝑖,𝑡_𝑘+(1/2)=𝑓𝑢𝑥_𝑖,𝑡_𝑘+(1/2),𝑢𝑥_𝑖,𝑡_{𝑘+(1/2)−𝑗},𝑥_𝑖,𝑡_𝑘+(1/2),0≤𝑖≤𝑀,0≤𝑘≤𝑁−1. (7) From Taylor expansion, we have 𝜕𝑢𝑥𝜕𝑡^𝑖,𝑡_𝑘+(1/2)=𝛿_𝑡𝑈_𝑖^𝑘+(1/2)−𝜏²𝜕24³𝑢𝜕𝑡³𝑥_𝑖,𝜂^𝑘_𝑖,𝜂^𝑘_𝑖∈𝑡_𝑘,𝑡_𝑘+1,𝜕²𝑢𝜕𝑥²𝑥_𝑖,𝑡_𝑘+(1/2)=12𝜕²𝑢𝜕𝑥²𝑥_𝑖,𝑡_𝑘+𝜕²𝑢𝜕𝑥²𝑥_𝑖,𝑡_𝑘+1−𝜏²8𝜕⁴𝑢𝜕𝑥²𝜕𝑡²𝑥_𝑖,𝛾^𝑘_𝑖=12𝛿²_𝑥𝑈^𝑘_𝑖+𝛿²_𝑥𝑈_𝑖^𝑘+1−ℎ²𝜕24⁴𝑢𝜕𝑥⁴𝜉^𝑘_𝑖,𝑡_𝑘+𝜕⁴𝑢𝜕𝑥⁴𝜉_𝑖^𝑘+1,𝑡_𝑘+1−𝜏²8𝜕⁴𝑢𝜕𝑥²𝜕𝑡²𝑥_𝑖,𝛾^𝑘_𝑖,𝜉^𝑘_𝑖,𝜉_𝑖^𝑘+1∈𝑥_𝑖−1,𝑥_𝑖+1,𝛾^𝑘_𝑖∈𝑡_𝑘,𝑡_𝑘+1,𝑓𝑢𝑥_𝑖,𝑡_𝑘+(1/2),𝑢𝑥_𝑖,𝑡_{𝑘+(1/2)−𝑗},𝑥_𝑖,𝑡_𝑘+(1/2)3=𝑓2𝑈^𝑘_𝑖−12𝑈_𝑖^𝑘−1,12𝑈_𝑖^{𝑘+1−𝑗}+12𝑈_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2)+3𝜏²8𝜕²𝑢𝑥_𝑖,𝜌^𝑘𝜕𝑡²𝑓_𝜇𝜁^𝑘_𝑖,𝜍^𝑘_𝑖,𝑥_𝑖,𝑡_𝑘+(1/2)−𝜏²8𝜕²𝑢𝑥_𝑖,𝜚^𝑘𝜕𝑡²𝑓_𝜈𝜁^𝑘_𝑖,𝜍^𝑘_𝑖,𝑥_𝑖,𝑡_𝑘+(1/2), (8) where 𝜌^𝑘∈(𝑡_𝑘−1,𝑡_𝑘+1/2), 𝜚^𝑘∈(𝑡_𝑘−𝑗,𝑡_{𝑘+1−𝑗}), 𝜁^𝑘_𝑖 is between 𝑢(𝑥_𝑖,𝑡_𝑘+(1/2)) and (3/2)𝑈^𝑘_𝑖−(1/2)𝑈_𝑖^𝑘−1, and 𝜍^𝑘_𝑖 is between 𝑢(𝑥_𝑖,𝑡_{𝑘+(1/2)−𝑗}) and (1/2)𝑈_𝑖^{𝑘+1−𝑗}+(1/2)𝑈_𝑖^𝑘−𝑗. Substituting (8) into (7), denote 𝑟_𝑖^𝑘+(1/2)=𝑟(𝑥_𝑖,𝑡_𝑘+(1/2)); we obtain 𝑟_𝑖^𝑘+(1/2)𝛿_𝑡𝑈_𝑖^𝑘+(1/2)−𝑑2𝜕²𝑢𝜕𝑥²𝑥_𝑖,𝑡_𝑘+𝜕²𝑢𝜕𝑥²𝑥_𝑖,𝑡_𝑘+13=𝑓2𝑈^𝑘_𝑖−12𝑈_𝑖^𝑘−1,12𝑈_𝑖^{𝑘+1−𝑗}+12𝑈_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2)+𝜏²𝑅^𝑘_𝑖,1≤𝑖≤𝑀,0≤𝑘≤𝑁−1, (9) where 𝑅^𝑘_𝑖=𝑟_𝑖^𝑘+(1/2)𝜕24³𝑢𝜕𝑡³𝑥_𝑖,𝜂^𝑘_𝑖−𝑑8𝜕⁴𝑢𝜕𝑥²𝜕𝑡²𝑥_𝑖,𝛾^𝑘_𝑖+38𝜕²𝑢𝑥_𝑖,𝜌^𝑘𝜕𝑡²𝑓_𝜇𝜁^𝑘_𝑖,𝜍^𝑘_𝑖,𝑥_𝑖,𝑡_𝑘+(1/2)−18𝜕²𝑢𝑥_𝑖,𝜚^𝑘𝜕𝑡²𝑓_𝜈𝜁^𝑘_𝑖,𝜍^𝑘_𝑖,𝑥_𝑖,𝑡_𝑘+(1/2). (10) Acting operator 𝒜 on both sides of (9), we have 𝒜𝑟_𝑖^𝑘+(1/2)𝛿_𝑡𝑈_𝑖^𝑘+(1/2)−𝑑2𝒜𝜕²𝑢𝜕𝑥²𝑥_𝑖,𝑡_𝑘𝜕+𝒜²𝑢𝜕𝑥²𝑥_𝑖,𝑡_𝑘+13=𝒜𝑓2𝑈^𝑘_𝑖−12𝑈_𝑖^𝑘−1,12𝑈_𝑖^{𝑘+1−𝑗}+12𝑈_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2)+𝜏²𝒜𝑅^𝑘_𝑖,1≤𝑖≤𝑀−1,0≤𝑘≤𝑁−1. (11) Resorting to the following Lemma, we can obtain the estimation of the operator 𝒜.

Lemma 1 (see [19, 20]). Suppose that 𝑞(𝑥)∈𝐶⁶[𝑥_𝑖−1,𝑥_𝑖+1]; then, we have 1𝑞12^′′𝑥_𝑖−1+10𝑞^′′𝑥_𝑖+𝑞^′′𝑥_𝑖+1−1ℎ²𝑞𝑥_𝑖−1−2𝑞𝑥_𝑖+𝑞𝑥_𝑖+1=ℎ⁴𝑞240⁽⁶⁾𝜔_𝑖, (12) where 𝜔_𝑖∈(𝑥_𝑖−1,𝑥_𝑖+1).

From Lemma 1 and Taylor expansion, we obtain 𝒜𝜕²𝑢𝜕𝑥²𝑥_𝑖,𝑡_𝑘=𝛿²_𝑥𝑈^𝑘_𝑖+ℎ⁴𝜕240⁶𝑢𝜕𝑥⁶𝜃^𝑘_𝑖,𝑡_𝑘,𝜃^𝑘_𝑖∈𝑥_𝑖−1,𝑥_𝑖+1,𝒜𝜕²𝑢𝜕𝑥²𝑥_𝑖,𝑡_𝑘+1=𝛿²_𝑥𝑈_𝑖^𝑘+1+ℎ⁴𝜕240⁶𝑢𝜕𝑥⁶𝜃_𝑖^𝑘+1,𝑡_𝑘+1,𝜃_𝑖^𝑘+1∈𝑥_𝑖−1,𝑥_𝑖+1. (13) Inserting (13) into (11), we have 𝒜𝑟_𝑖^𝑘+(1/2)𝛿_𝑡𝑈_𝑖^𝑘+(1/2)−𝑑𝛿²_𝑥𝑈_𝑖^𝑘+1/23=𝒜𝑓2𝑈^𝑘_𝑖−12𝑈_𝑖^𝑘−1,12𝑈_𝑖^{𝑘+1−𝑗}+12𝑈_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2)+𝑅^𝑘_𝑖,1≤𝑖≤𝑀−1,0≤𝑘≤𝑁−1, (14) where 𝑅^𝑘_𝑖=𝜏²𝒜𝑅^𝑘_𝑖+𝑑ℎ⁴𝜕480⁶𝑢𝜕𝑥⁶𝜃^𝑘_𝑖,𝑡_𝑘+𝜕⁶𝑢𝜕𝑥⁶𝜃_𝑖^𝑘+1,𝑡_𝑘+1. (15)

From 0<𝑐₀≤𝑟_𝑖^𝑘+1/2≤𝑐₁ and assumptions (H1) and (H2), we have 𝑅^𝑘_𝑖≤𝑐₅,1≤𝑖≤𝑀−1,0≤𝑘≤𝑁−1, (16) such that 𝑅^𝑘_𝑖≤𝑐₆𝜏²+ℎ⁴,1≤𝑖≤𝑀−1,0≤𝑘≤𝑁−1. (17)

Discretizing the initial and boundary conditions of (2) and (3), we obtain 𝑈^𝑘_𝑖=𝜙𝑥_𝑖,𝑡_𝑘,0≤𝑖≤𝑀,−𝑗≤𝑘≤0,𝑈^𝑘₀=𝛼𝑡_𝑘,𝑈^𝑘_𝑀=𝛽𝑡_𝑘,1≤𝑘≤𝑁. (18)(19) Replacing 𝑈^𝑘_𝑖 by 𝑢^𝑘_𝑖 in (14) and omitting 𝑅^𝑘_𝑖, we obtain the following compact difference scheme: 𝒜𝑟_𝑖^𝑘+(1/2)𝛿_𝑡𝑢_𝑖^𝑘+(1/2)−𝑑𝛿²_𝑥𝑢_𝑖^𝑘+1/23=𝒜𝑓2𝑢^𝑘_𝑖−12𝑢_𝑖^𝑘−1,12𝑢_𝑖^{𝑘+1−𝑗}+12𝑢_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2),1≤𝑖≤𝑀−1,0≤𝑘≤𝑁−1,𝑢^𝑘_𝑖=𝜙𝑥_𝑖,𝑡_𝑘,0≤𝑖≤𝑀,−𝑗≤𝑘≤0,𝑢^𝑘₀=𝛼𝑡_𝑘,𝑢^𝑘_𝑀=𝛽𝑡_𝑘,1≤𝑘≤𝑁. (20)(21)(22)

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

Define the following grid function space on Ω_ℎ: 𝑉=𝑣∣𝑣=𝑣₀,𝑣₁,…,𝑣_𝑀,𝑣₀=𝑣_𝑀=0. (23) If 𝑣∈𝑉, we introduce the following notations: ‖𝑣‖=ℎ^𝑀−1_𝑖=1𝑣_𝑖²,|𝑣|₁=ℎ^𝑀_𝑖=1𝑣_𝑖−𝑣_𝑖−1ℎ²,‖𝑣‖_∞=max_{0≤𝑖≤𝑀}𝑣_𝑖. (24) By [16, 17, 19], we have the following two inequalities: ‖𝑣‖_∞≤12|𝑣|₁,‖𝑣‖≤1√6|𝑣|₁. (25)(26) For the analysis of the difference scheme, the following Lemma is introduced.

Lemma 2 (see [16, 17, 19]). Assume that {𝐹^𝑘∣𝑘≥0} to be nonnegative sequence and satisfies 𝐹^𝑘+1≤𝐴+𝐵𝜏^𝑘_𝑖=1𝐹^𝑙,𝑘=0,1,…; (27) then 𝐹^𝑘+1≤𝐴exp(𝐵𝑘𝜏),𝑘=0,1,2,…, (28) where 𝐴 and 𝐵 are nonnegative constants.

Theorem 3. Under the condition that 5𝑐₀−𝑐₁>0, the compact difference scheme (20)–(22) has a unique solution.

Proof. Denote that 𝜆=𝑑𝜏/ℎ²; then, difference scheme (20)–(22) can be reformed as 1𝑟12^{𝑘+1/2𝑖−1}−𝜆2𝑢^𝑘+1_𝑖−1+10𝑟12^𝑖𝑘+1/2+𝜆𝑢_𝑖^𝑘+1+1𝑟12^{𝑘+1/2𝑖+1}−𝜆2𝑢^𝑘+1_𝑖+1=1𝑟12^{𝑘+1/2𝑖−1}+𝜆2𝑢^𝑘_𝑖−1+10𝑟12^𝑖𝑘+1/2−𝜆𝑢^𝑘_𝑖+1𝑟12^{𝑘+1/2𝑖+1}+𝜆2𝑢^𝑘_𝑖+1+𝜏𝑓3122𝑢^𝑘_𝑖−1−12𝑢^𝑘−1_𝑖−1,12𝑢^{𝑘+1−𝑗}_𝑖−1+12𝑢^𝑘−𝑗_𝑖−1,𝑥_𝑖−1,𝑡_𝑘+(1/2)+10𝜏𝑓3122𝑢^𝑘_𝑖−12𝑢_𝑖^𝑘−1,12𝑢_𝑖^{𝑘+1−𝑗}+12𝑢_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2)+𝜏𝑓3122𝑢^𝑘_𝑖+1−12𝑢^𝑘−1_𝑖+1,12𝑢^{𝑘+1−𝑗}_𝑖+1+12𝑢^𝑘−𝑗_𝑖+1,𝑥_𝑖+1,𝑡_𝑘+(1/2). (29)
The mathematical induction method will be used in the proof of this theorem. Denote 𝑢^𝑘=𝑢^𝑘₀,𝑢^𝑘₁,…,𝑢^𝑘_𝑀. (30) Notice that 𝑢^𝑘(−𝑗≤𝑘≤0) is determined by the initial condition (21). Suppose that 𝑢^𝑙 has been determined.
Let 𝑘=𝑙 in (20); the linear algebraic equations with respect to 𝑢^𝑙+1 can be obtained. Under the condition that 5𝑐₀−𝑐₁>0, we have 10𝑟12^𝑖𝑙+1/2−1+𝜆𝑟12^{𝑙+1/2𝑖−1}−𝜆2−1𝑟12^{𝑙+1/2𝑖+1}−𝜆2≥10𝑟12^𝑖𝑙+1/2−1𝑟12^{𝑙+1/2𝑖−1}−1𝑟12^{𝑙+1/2𝑖+1}≥10𝑐₀−2𝑐₁12>0. (31) Thus, the coefficient matrix of the linear algebraic system is strictly diagonally dominant and then there exists a unique solution 𝑢^𝑙+1. By the inductive principle, the proof ends.

Denote 𝑒^𝑘_𝑖=𝑈^𝑘_𝑖−𝑢^𝑘_𝑖, 0≤𝑖≤𝑀, −𝑗≤𝑘≤𝑁; subtracting (20)–(22) from (14), (18), and (19), respectively, the following error equations can be obtained: 𝒜𝑟_𝑖^𝑘+(1/2)𝛿_𝑡𝑒_𝑖^𝑘+(1/2)−𝑑𝛿²_𝑥𝑒_𝑖^𝑘+1/2𝑓3=𝒜2𝑈^𝑘_𝑖−12𝑈_𝑖^𝑘−1,12𝑈_𝑖^{𝑘+1−𝑗}+12𝑈_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2)3−𝑓2𝑢^𝑘_𝑖−12𝑢_𝑖^𝑘−1,12𝑢_𝑖^{𝑘+1−𝑗}+12𝑢_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2)+𝑅^𝑘_𝑖,1≤𝑖≤𝑀−1,0≤𝑘≤𝑁−1,𝑒^𝑘_𝑖=0,0≤𝑖≤𝑀,−𝑗≤𝑘≤0,𝑒^𝑘₀=0,𝑒^𝑘_𝑀=0,1≤𝑘≤𝑁. (32)(33)(34)

Theorem 4. Denote 𝐶=𝑐₆3𝑇𝑑9𝑐₀−𝑐₁2exp5𝑐²₃+𝑐²₄𝑑9𝑐₀−𝑐₁𝑇. (35) If the following conditions are satisfied: 𝜖𝜏≤⁰4𝐶^1/2𝜖,ℎ≤⁰4𝐶^1/4, (36) then we have 𝑒^𝑘_∞≤𝐶𝜏²+ℎ⁴,0≤𝑘≤𝑁, (37) where 𝜖₀>0 is a constant.

Proof. Acting ℎ𝛿_𝑡𝑒_𝑖^𝑘+(1/2) on (32) and summing up for 𝑖 from 1 to 𝑀−1, we obtain ℎ^𝑀−1_𝑖=1𝒜𝑟_𝑖^𝑘+(1/2)𝛿_𝑡𝑒_𝑖^𝑘+(1/2)𝛿_𝑡𝑒_𝑖^𝑘+(1/2)−𝑑ℎ^𝑀−1_𝑖=1𝛿²_𝑥𝑒_𝑖^𝑘+1/2𝛿_𝑡𝑒_𝑖^𝑘+(1/2)=ℎ^𝑀−1_𝑖=1𝒜𝑓32𝑈^𝑘_𝑖−12𝑈_𝑖^𝑘−1,12𝑈_𝑖^{𝑘+1−𝑗}+12𝑈_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2)3−𝑓2𝑢^𝑘_𝑖−12𝑢_𝑖^𝑘−1,12𝑢_𝑖^{𝑘+1−𝑗}+12𝑢_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2)𝛿_𝑡𝑒_𝑖^𝑘+(1/2)+ℎ^𝑀−1_𝑖=1𝑅^𝑘_𝑖𝛿_𝑡𝑒_𝑖^𝑘+(1/2),1≤𝑖≤𝑀−1,0≤𝑘≤𝑁−1. (38)
Mathematical induction will be used to prove this theorem. Notice that ‖𝑒^𝑘‖_∞=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: ℎ^𝑀−1_𝑖=1𝒜𝑟_𝑖^𝑘+(1/2)𝛿_𝑡𝑒_𝑖^𝑘+(1/2)𝛿_𝑡𝑒_𝑖^𝑘+(1/2)≥ℎ12^𝑀−1_𝑖=19𝑟_𝑖^𝑘+(1/2)−𝑟^𝑘+(1/2)_𝑖−1+𝑟^𝑘+(1/2)_𝑖+12𝛿_𝑡𝑒_𝑖^𝑘+(1/2)2≥9𝑐₀−𝑐₁𝛿12^𝑡𝑒^𝑘+(1/2)2,−𝑑ℎ^𝑀−1_𝑖=1𝛿²_𝑥𝑒_𝑖^𝑘+(1/2)𝛿_𝑡𝑒_𝑖^𝑘+(1/2)=𝑑𝑒2𝜏^𝑘+121−𝑒^𝑘2₁,ℎ^𝑀−1_𝑖=1𝑅^𝑘_𝑖𝛿_𝑡𝑒_𝑖^𝑘+(1/2)≤69𝑐₀−𝑐₁ℎ^𝑀−1_𝑖=1𝑅^𝑘_𝑖²+9𝑐₀−𝑐₁ℎ24^𝑀−1_𝑖=1𝛿_𝑡𝑒_𝑖^𝑘+(1/2)2≤69𝑐₀−𝑐₁𝑐²₆𝜏²+ℎ⁴²+9𝑐₀−𝑐₁𝛿24^𝑡𝑒^𝑘+(1/2)2. (39) From the inductive assumption and (36), we have 𝑒^𝑘_∞≤𝐶𝜏²+ℎ⁴≤𝜖₀2,0≤𝑘≤𝑙. (40) From (H2), we have 𝑓32𝑈^𝑘_𝑖−12𝑈_𝑖^𝑘−1,12𝑈_𝑖^{𝑘+1−𝑗}+12𝑈_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2)3−𝑓2𝑢^𝑘_𝑖−12𝑢_𝑖^𝑘−1,12𝑢_𝑖^{𝑘+1−𝑗}+12𝑢_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2)≤𝑐₃32𝑒^𝑘_𝑖−12𝑒_𝑖^𝑘−1+𝑐₄12𝑒_𝑖^{𝑘+1−𝑗}+12𝑒_𝑖^𝑘−𝑗,1≤𝑖≤𝑀,0≤𝑘≤𝑙. (41) It then follows that 𝒜𝑓32𝑈^𝑘_𝑖−12𝑈_𝑖^𝑘−1,12𝑈_𝑖^{𝑘+1−𝑗}+12𝑈_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2)3−𝑓2𝑢^𝑘_𝑖−12𝑢_𝑖^𝑘−1,12𝑢_𝑖^{𝑘+1−𝑗}+12𝑢_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2)≤𝒜𝑐₃32𝑒^𝑘_𝑖−12𝑒_𝑖^𝑘−1+𝑐₄12𝑒_𝑖^{𝑘+1−𝑗}+12𝑒_𝑖^𝑘−𝑗,1≤𝑖≤𝑀−1,0≤𝑘≤𝑙. (42) From the inequality above, we obtain ℎ^𝑀−1_𝑖=1𝒜𝑓32𝑈^𝑘_𝑖−12𝑈_𝑖^𝑘−1,12𝑈_𝑖^{𝑘+1−𝑗}+12𝑈_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2)3−𝑓2𝑢^𝑘_𝑖−12𝑢_𝑖^𝑘−1,12𝑢_𝑖^{𝑘+1−𝑗}+12𝑢_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2)×𝛿_𝑡𝑒_𝑖^𝑘+(1/2)≤ℎ^𝑀−1_𝑖=1𝒜𝑐₃32𝑒^𝑘_𝑖−12𝑒_𝑖^𝑘−1+𝑐₄12𝑒_𝑖^{𝑘+1−𝑗}+12𝑒_𝑖^𝑘−𝑗×𝛿_𝑡𝑒_𝑖^𝑘+(1/2)≤69𝑐₀−𝑐₁ℎ×^𝑀−1_𝑖=1𝒜𝑐₃32𝑒^𝑘_𝑖−12𝑒_𝑖^𝑘−1+𝑐₄12𝑒_𝑖^{𝑘+1−𝑗}+12𝑒_𝑖^𝑘−𝑗2+9𝑐₀−𝑐₁𝛿24^𝑡𝑒_𝑖^𝑘+(1/2)2≤69𝑐₀−𝑐₁ℎ×^𝑀−1_𝑖=1𝑐₃32𝑒^𝑘_𝑖−12𝑒_𝑖^𝑘−1+𝑐₄12𝑒_𝑖^{𝑘+1−𝑗}+12𝑒_𝑖^𝑘−𝑗2+9𝑐₀−𝑐₁𝛿24^𝑡𝑒^𝑘+(1/2)2≤129𝑐₀−𝑐₁×𝑐²₃ℎ^𝑀−1_𝑖=132𝑒^𝑘_𝑖−12𝑒_𝑖^𝑘−12+𝑐²₄ℎ^𝑀−1_𝑖=112𝑒_𝑖^{𝑘+1−𝑗}+12𝑒_𝑖^𝑘−𝑗2+9𝑐₀−𝑐₁𝛿24^𝑡𝑒^𝑘+(1/2)2≤129𝑐₀−𝑐₁×52𝑐²₃ℎ^𝑀−1_𝑖=1𝑒^𝑘_𝑖²+𝑒_𝑖^𝑘−12+12𝑐²₄ℎ^𝑀−1_𝑖=1𝑒_𝑖^{𝑘+1−𝑗2}+𝑒_𝑖^𝑘−𝑗2+9𝑐₀−𝑐₁𝛿24^𝑡𝑒^𝑘+(1/2)2=309𝑐₀−𝑐₁𝑐²₃𝑒^𝑘2+𝑒^𝑘−12+69𝑐₀−𝑐₁𝑐²₄𝑒^{𝑘+1−𝑗2}+𝑒^𝑘−𝑗2+9𝑐₀−𝑐₁𝛿24^𝑡𝑒^𝑘+(1/2)2,0≤𝑘≤𝑙. (43)
Inserting (39)–(43) into (38), we obtain 𝑑𝑒2𝜏^𝑘+121−𝑒^𝑘2₁≤309𝑐₀−𝑐₁𝑐²₃𝑒^𝑘2+𝑒^𝑘−12+69𝑐₀−𝑐₁𝑐²₄𝑒^{𝑘+1−𝑗2}+𝑒^𝑘−𝑗2+69𝑐₀−𝑐₁𝑐²₆𝜏²+ℎ⁴²,0≤𝑘≤𝑙. (44) The above inequality has the following form: 𝑒^𝑘+12₁≤𝑒^𝑘2₁+2𝜏𝑑309𝑐₀−𝑐₁𝑐²₃𝑒^𝑘2+𝑒^𝑘−12+2𝜏𝑑69𝑐₀−𝑐₁𝑐²₄𝑒^{𝑘+1−𝑗2}+𝑒^𝑘−𝑗2+2𝜏𝑑69𝑐₀−𝑐₁𝑐²₆𝜏²+ℎ⁴²,0≤𝑘≤𝑙. (45) Summing up (45) for 𝑘, noticing (33), and exploiting (26), we have 𝑒^𝑘+12₁≤𝑒⁰²₁+2𝜏𝑑309𝑐₀−𝑐₁𝑐²₃×^𝑘_𝑚=0𝑒^𝑚2+𝑒^𝑚−12+2𝜏𝑑69𝑐₀−𝑐₁𝑐²₄×^𝑘_𝑚=0𝑒^{𝑚+1−𝑗2}+𝑒^𝑚−𝑗2+2𝜏𝑑69𝑐₀−𝑐₁𝑐²₆^𝑙_𝑚=0𝜏²+ℎ⁴²≤245𝑐²₃+𝑐²₄𝑑9𝑐₀−𝑐₁𝜏^𝑘_𝑚=1𝑒^𝑚2+12𝑐²₆(𝑙+1)𝜏𝑑9𝑐₀−𝑐₁𝜏²+ℎ⁴²≤45𝑐²₃+𝑐²₄𝑑9𝑐₀−𝑐₁𝜏×^𝑘_𝑚=1𝑒^𝑚2+12𝑐²₆𝑇𝑑9𝑐₀−𝑐₁𝜏²+ℎ⁴²,0≤𝑘≤𝑙. (46) By Lemma 2, we have 𝑒^𝑙+12₁≤12𝑐²₆𝑇𝑑9𝑐₀−𝑐₁4exp5𝑐²₃+𝑐²₄𝑑9𝑐₀−𝑐₁𝑇𝜏²+ℎ⁴². (47) From (25), we obtain 𝑒^𝑙+1_∞≤𝑐₆3𝑇𝑑9𝑐₀−𝑐₁2exp5𝑐²₃+𝑐²₄𝑑9𝑐₀−𝑐₁𝑇𝜏²+ℎ⁴. (48) By the inductive principle, this completes the proof.

To discuss the stability of the difference scheme (20)–(22), we consider the following problem: 𝑟(𝑥,𝑡)𝑣_𝑡−𝑑𝑣_𝑥𝑥(𝑣()()),()∈()×(],𝑣()()()[][],𝑣()(𝑡)()(𝑡)(],=𝑓𝑥,𝑡,𝑣𝑥,𝑡−𝑠,𝑥,𝑡𝑥,𝑡0,10,𝑇𝑥,𝑡=𝜙𝑥,𝑡+𝜓𝑥,𝑡,𝑥∈0,1,𝑡∈−𝑠,00,𝑡=𝛼,𝑣1,𝑡=𝛽,𝑡∈0,𝑇 (49) where 𝜓(𝑥,𝑡) is the perturbation caused by 𝜙(𝑥,𝑡). The following difference scheme solving for (49) can be obtained: 𝒜𝑟_𝑖^𝑘+(1/2)𝛿_𝑡𝑣_𝑖^𝑘+(1/2)−𝑑𝛿²_𝑥𝑣_𝑖^𝑘+1/23=𝒜𝑓2𝑣^𝑘_𝑖−12𝑣_𝑖^𝑘−1,12𝑣_𝑖^{𝑘+1−𝑗}+12𝑣_𝑖^𝑘−𝑗,𝑥_𝑖,𝑡_𝑘+(1/2),𝑣1≤𝑖≤𝑀−1,0≤𝑘≤𝑁−1,_𝑘𝑖=𝜙𝑥_𝑖,𝑡_𝑘+𝜓^𝑘_𝑖𝑣,0≤𝑖≤𝑀,−𝑗≤𝑘≤0,_𝑘0=𝛼𝑡_𝑘,𝑣^𝑘_𝑀=𝛽𝑡_𝑘,1≤𝑘≤𝑁. (50)

Similar to the proof of Theorem 4, the following stability result can be obtained

Theorem 5. Denote 𝜂^𝑘_𝑖=𝑣^𝑘_𝑖−𝑢^𝑘_𝑖,0≤𝑖≤𝑀,−𝑗≤𝑘≤𝑁. (51) Then, there exist constants 𝑐₇ and 𝑐₈ such that 𝜂^𝑘_∞≤𝑐₇𝜏ℎ⁰_𝑚=−𝑗^𝑀−1_𝑖=𝑙𝜓^𝑘_𝑖² (52) under the condition that ℎ and 𝜏 are small enough and max_{−𝑗≤𝑘≤0,0≤𝑖≤𝑀}|𝜓^𝑘_𝑖|≤𝑐₈.

4. Numerical Test

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

Define 𝐸_∞(ℎ,𝜏)=max_{0≤𝑖≤𝑀,0≤𝑘≤𝑁}𝑢𝑥_𝑖,𝑡_𝑘−𝑢^𝑘_𝑖. (53)

Example 1. Consider the following problem: 𝑟(𝑥,𝑡)𝑢_𝑡−2𝑢_𝑥𝑥=𝑢(𝑥,𝑡)+𝑢(𝑥,𝑡−0.1)+2𝑒^−𝑥,()(],𝑢()𝑥∈0,1,𝑡∈0,1𝑥,𝑡=𝑒^−𝑥()()[],𝑢()()1+𝑡,𝑥∈0,1,𝑡∈−0.1,00,𝑡=1+𝑡,𝑢1,𝑡=𝑒⁻¹(1+𝑡),𝑡∈(0,1], (54) where 𝑟(𝑥,𝑡)=4𝑡+5.9. The exact solution of (54) is 𝑢(𝑥,𝑡)=𝑒^−𝑥(1+𝑡).

Table 1 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.

Table 2 

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

Figure 1 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.

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).