# The Compact Finite Difference Method of Two-Dimensional Cattaneo Model

Authors: Yating Huang; Zhe Yin
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we propose and analyze the compact finite difference scheme of the two-dimensional Cattaneo model. The stability and convergence of the scheme are proved by the energy method, the convergence orders are2 in time and 4 in space. We also use the variables separation method to find the true solution of the problem. On this basis, the validity and accuracy of the scheme are verified by numerical experiments.


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## Body

## 1. Introduction

In the diffusion phenomenon described by traditional Fick law and Fourier law, the moment a point passes by, it is disturbed at a great distance, the propagation speed of the disturbance seems to be infinite. However, this property is unphysical. To solve the problem, Cattaneo modified the constitutive equation by introducing a relaxation parameter which plays the role of relaxation time, it should be noted that the relaxation parameter is small and depends on the thermodynamic properties of the material. Thus, he proposed the Cattaneo model [1].From a mathematical point of view, Cattaneo model transforms the traditional diffusion equation into a hyperbolic equation, the speed of propagation is finite, and improves the property of infinite propagation speed. On the other hand, because of the hyperbolic nature of the Cattaneo model, it may have oscillatory solutions and negative values.From a physical point of view, the Cattaneo model describes a physical phenomenon which is called heat waves. Although this phenomenon can only be observed under special circumstances, materials, or states [2], it is still gradually accepted by the public. It can be used to describe not only ultrasonic wave propagation in certain diluted gases but also heat pulse propagation in some pure nonmetallic crystals [3]. Straughan et al. [4] studied thermal convection with the Cattaneo-Christov model in horizontal layers of an incompressible Newtonian fluid. Haddad [5] utilized the theory of Cattaneo-Christov to investigate Brinkman's porous media. Cattaneo model is widely used in extended irreversible thermodynamics, cosmological models, and crystalline solids [6–8].Because the fractional models have the advantage of previous memory, scholars have studied various fractional models [9–13]. With the wide application of the Cattaneo model in physics and theoretical analysis, more and more people pay attention to the generalization and solution of the Cattaneo model. Compte and Metzler [14] generalized the Cattaneo model from three perspectives which called continuous-time random walks, nonlocal transport theory, and delayed flux-force relation. Ferrillo et al. [15] compared the Cattaneo model and fractional Cattaneo model and studied the asymptotic behavior of solutions to the Cattaneo equations. Su et al. [16] studied solutions to the classical Cattaneo equation and compared with the solutions of a phase-lagging heat transport equation. C ̆iegis [17] studied the numerical solutions of a class of hyperbolic heat conduction equations, proposed explicit and implicit Euler schemes for such equations, and verified that the convergence order of this scheme was Oτ+h2. Huang and Yin [18] proposed a fourth-order compact difference scheme for the 1D Cattaneo model and verified that the convergence order of the scheme was Oτ2+h4 through numerical experiments. Zhao and Sun [19] presented compact Crank-Nicolson schemes for a class fractional Cattaneo equation in an inhomogeneous medium and used the energy method to verify that the convergence order of the scheme is Oτ3−α+h4. Li and Cao [20] proposed an unconditionally stable scheme with convergence order of Oτ3−α+h2. Vong et al. [21] proposed a higher-order difference scheme for a class of generalized Cattaneo equations and proved the stability and convergence of the scheme by energy method.There are many methods to solve partial differential equations, such as the finite element method [22, 23], finite volume element method [24, 25], and finite difference method [26]. Compared with other methods, the finite difference method has the advantages of a low requirement for grid nodes and simple calculation. In 1992, Lele [27] first proposed a general form of compact difference scheme. Compact difference scheme is a kind of finite difference scheme with high precision, which is widely used in solving parabolic equation and hyperbolic equation [28–33].Previous scholars used different methods to study the integer-order Cattaneo model and the fractional Cattaneo model. The one-dimensional Cattaneo model describes the heat conduction phenomenon of the uniform thin column with side insulation, while the two-dimensional Cattaneo model needs to be studied for the heat conduction phenomenon of the thin plate with upper and lower bottom insulation. In this paper, we construct a compact difference scheme for a two-dimensional Cattaneo model. The time derivative is discretized with a central difference, and the space derivative is discretized with compact operators, so that the convergence order of the scheme reachesOτ2+hx4+hy4. We use the energy method to verify the stability and convergence of the scheme. Finally, we verify the validity and accuracy of the scheme through numerical experiments. It should be noted that the above articles are all about constructing the exact solution of the problem, and then calculating the right term, so as to conduct numerical experiments. In this paper, our example is the definite problem corresponding to the homogeneous equation. The exact solution is obtained through the separation of variables method, and then the numerical experiment is conducted.The arrangement of the article is as follows. In the second part, we give some lemmas and symbols to derive a compact difference scheme. In the third part, we give a strict proof of the stability and convergence of the scheme by energy method. In the fourth part, we obtain the true solution of the specific problem through the separation of variable method and use numerical experiments to prove the validity of the scheme. In the fifth part, we make a final summary of the article.


## 2. Construction of the Compact Finite Difference Scheme

In this chapter, we consider the following two-dimensional Cattaneo model:(1)ε2utt+ut=Duxx+uyy,x,y∈Ω,t∈0,T,ux,y,0=u0x,y,x,y∈Ω,utx,y,0=u1x,y,x,y∈Ω,ux,y,t=0,x,y∈Γ,t∈0,T,where Ω=0,Lx×0,Ly is a rectangular region in R2, Γ=∂Ω is the boundary of the rectangular region, ε2 is a relaxation parameter, D is the diffusion coefficient, and u0x,y and u1x,y are known functions. The above model is considered under homogeneous Dirichlet boundary conditions. The regularity assumption that problem (1) has a solution u∈Cx,y,t6,6,4Ω¯×0,T will be used in the following analysis.In order to propose the compact finite difference scheme, we first introduce some basic symbols and preliminary lemmas. Take three positive integersM, N, and K, then let hx=Lx/M, hy=Ly/N, τ=T/K. In this case, the spatial node can be expressed as xi,yj, where xi=ihx,i=0,1,⋯,M; yj=jhy,j=0,1,⋯,N; tn=nτ,n=0,1,⋯,K, let Ω¯h=xi,yj∣0≤i≤M,0≤j≤N, Ωτ=un∣0≤n≤K, Ωh=Ω¯h∩Ω, Γh=Ω¯h∩Γ, ω=i,j∣xi,yj∈Ωh, ∂ω=i,j∣xi,yj∈Γh, ω¯=ω∪∂ω. Let Uh=u∣u=uij∣i,j∈ω¯ be a grid function space defined on Ω¯h. For any grid function u∈Uh, introduce the following notations:
(2)δx2uij=ui−1,j−2ui,j+ui+1,jhx2,δy2uij=ui,j−1−2ui,j+ui,j+1hy2.The compact operators are defined as follows(3)Axui,j=I+hx212δx2uij,1≤i≤M−1,0≤j≤N,ui,j,i=0orM,Ayui,j=I+hy212δy2uij,1≤j≤N−1,0≤i≤M,ui,j,j=0orN.LetUτ=u∣u=u0,u1,⋯,uK be grid function space defined on Ωτ. For any grid function u∈Uτ, introduce the following notations:
(4)δtun=un+1−un−12τ,δt¯un+1/2=un+1−unτ,δt¯un−1/2=un−un−1τ,δt2un=un−1−2un+un+1τ2.The following lemmas also play an important role in constructing a scheme.Lemma 1 [34].
Supposegx∈C2c,c+h, it holds that
(5)g′c=1hgc+h−gc−h2g′′ξ7,c<ξ7<c+h,where c is a constant.Lemma 2 [34].
Supposegx∈C3tn,tn+1, it holds that
(6)g′′=2τgtn+1−gtnh−g′−τ3g′′′tξ,tn<tξ<tn+1.Lemma 3 [34].
Supposegx∈C4c−h,c+h, it holds that
(7)g′′c=gc+h−2gc+gc−hh2−h212g4ξ4,c−h<ξ4<c+h,where c is a constant.Lemma 4 [34].
Supposegx∈C6c−h,c+h, it holds that
(8)112g′′c−h+10g′′c+g′′c+h=1h2gc+h−2gc+gc−h+h4240g6ξ6,where ξ6∈c−h,c+h, c is a constant.
Define the grid functions asuijn=uxi,yj,tn, 0≤i≤M,0≤j≤N,0≤n≤K. We discrete at xi,yj,tn and refer to the formula [34].
(9)∂2uxi,yj,tn∂x2=12∂2uxi,yj,tn−1∂x2+∂2uxi,yj,tn+1∂x2−τ22∂4uxi,yj,ξin∂x2∂t2,tn−1<ξin<tn+1.
We can obtain(10)ε2∂2u∂t2xi,yj,tn+∂u∂txi,yj,tn=D2∂2u∂x2xi,yj,tn−1+∂2u∂x2xi,yj,tn+1+∂2u∂y2xi,yj,tn−1+∂2u∂y2xi,yj,tn+1+Rijn1,i,j∈ω,1≤n≤K−1,where
(11)Rijn1=−Dτ22∂4uxi,yj,ξijn∂x2∂t2+∂4uxi,yj,ηijn∂y2∂t2,tn−1<ξijn<tn+1,tn−1<ηijn<tn+1.
Utilizing Taylor formula results in(12)∂uxi,yj,tn∂t=δtuijn−τ26∂3uxi,yj,θijn∂t3,tn−1<θijn<tn+1.
Using the above equation and Lemma3 result in
(13)ε2δt2uijn+δtuijn=D2∂2u∂x2xi,yj,tn−1+∂2u∂x2xi,yj,tn+1+∂2u∂y2xi,yj,tn−1+∂2u∂y2xi,yj,tn+1+Rijn2,i,j∈ω,1≤n≤K−1,where Rijn2=Rijn1+τ21/6∂3uxi,yj,θijn/∂t3+ε2/12∂4uxi,yj,ρijn/∂t4.
We apply compact operatorsAx and Ay to both sides of Eq. (13) and apply Lemma 4, we get
(14)ε2AxAyδt2uijn+AxAyδtuijn=D2Ayδx2uijn−1+Ayδx2uijn+1+Axδy2uijn−1+Axδy2uijn+1+Rijn3,i,j∈ω,1≤n≤K−1,where
(15)Rijn3=AxAyRijn2+hx4240Ay∂6u∂x6ϑijn,yj,tn−1+Ay∂6u∂x6μijn,yj,tn+1+hy4240Ax∂6u∂y6xi,ζijn,tn−1+Ax∂6u∂y6xi,ςijn,tn+1.
We think about the differential equation (1) at point xi,yj,t0,
(16)ε2∂2uxi,yj,t0∂t2+∂uxi,yj,t0∂t=D2∂2uxi,yj,t1∂x2+∂2uxi,yj,t0∂x2+∂2uxi,yj,t1∂y2+∂2uxi,yj,t0∂y2−D2∂2uxi,yj,t1∂x2−∂2uxi,yj,t0∂x2+∂2uxi,yj,t1∂y2−∂2uxi,yj,t0∂y2.
By applying Lemma1 and Lemma 2, we have
(17)∂2uxi,yj,t0∂t2=2τδt¯u12−u1xi,yj−τ3∂3uxi,yj,ς1∂t3,ς1∈t0,t1.∂uxi,yj,t0∂t=δt¯u12−τ2∂2uxi,yj,ς2∂t2,ς2∈t0,t1.
Substituting the above equation into (16) and applying AxAy to both sides, we have
(18)2ε2τAxAyδt-uij12−u1xi,yj+AxAyδt-uij12=D2Ayδx2uij0+Ayδx2uij1+Axδy2uij0+Axδy2uij1+rijn,where
(19)rijn=Oτ+hx4+hy4,i,j∈ω.
Omitting small termsRijn3 and rijn. The numerical approximate solution is represented by the grid function Uijn. Replacing uijn with it, the fourth-order compact difference scheme can be obtained
(20)ε2AxAyδt2Uijn+AxAyδtUijn=D2Ayδx2Uijn−1+Ayδx2Uijn+1+Axδy2Uijn−1+Axδy2Uijn+1,i,j∈ω,1≤n≤K−1,2ε2τAxAyδt¯Uij12−u1xi,yj+AxAyδt¯Uij12=D2Ayδx2Uij0+Ayδx2Uij1+Axδy2Uij0+Axδy2Uij1,i,j∈ω,Uij0=u0xi,yj,i,j∈ω,Uijn=0,i,j∈∂ω,0≤n≤K.
From the above analysis, the following theorems can be obtained.Theorem 5.
The truncation error of compact difference scheme (20) is
(21)∣Rijn∣=Oτ2+hx4+hy4,i,j∈ω,0≤n≤K.


## 3. Analysis of the Compact Finite Difference Scheme

Letu,v∈Ůh=u∣u∈Uh;i,j∈∂ω,uij=0, we define the following inner products and the corresponding norms
(22)u,v=hxhy∑i=1M−1∑j=1N−1uijvij,u=u,u,u,v1,x=hxhy∑i=1M∑j=1N−1δxui−1/2,jδxvi−1/2,j,δxu=u,u1,x,u,v1,y=hxhy∑i=1M−1∑j=1Nδyui,j−1/2δyvi,j−1/2,δyu=u,u1,y,u,vx,y=hxhy∑i=1M∑j=1Nδxδyui−1/2,j−1/2δxδyvi−1/2,j−1/2,δxδyu=u,ux,y,u,vA=AxAyu,v=u,v−hx212u,v1,x−hy212u,v1,y+hx2hy2144u,vx,y,∥∇hu∥A2=∥δxu∥2−hy212δyδxu2+δyu2−hx212δxδyu2,uA=u,uA,u∞=max1≤i≤M−1,1≤j≤N−1uij.Next, we introduce the Lemma that will be applied to the stability analysis of the scheme.Lemma 6 [35].
For any grid functionu∈Ůh, it holds that
(23)13u2≤uA2≤u2.
Applying the Lemma, we have the following result.Theorem 7.
Supposevijn∣0≤i≤M,0≤j≤N,0≤n≤K is the solution of the following difference scheme
(24)ε2AxAyδt2vijn+AxAyδtvijn=D2Ayδx2vijn−1+Ayδx2vijn+1+Axδy2vijn−1+Axδy2vijn+1+gijn,i,j∈ω,1≤n≤K−1,(25)2ε2τAxAyδt¯vij12−u1xi,yj+AxAyδt¯vij12=D2Ayδx2vij0+Ayδx2vij1+Axδy2vij0+Axδy2vij1+gij0,i,j∈ω,(26)vij0=u0xi,yj,i,j∈ω,(27)vijn=0,i,j∈∂ω,0≤n≤K.
It holds that(28)ε2∥δt¯vn+1/2∥A2+D2∥∇hvn+1∥A2+∥∇hvn∥A2≤D∥∇hv0∥A2+3ε4τ∥φ∥2+3τ24ε2∥g0∥2+3τ2∑k=1n∥gk∥2,0≤n≤K−1,where φij=AxAyu1xi,yj, gijn is a source item.Proof.
Multiplying Eq. (24) by 2δtvn, and summing up for i from 1 to M−1, we have
(29)2ε2AxAyδt2vn,δtvn+2AxAyδtvn,δtvn=DAyδx2vn−1+Ayδx2vn+1+Axδy2vn−1+Axδy2vn+1,δtvn+2gn,δtvn,1≤n≤K−1.
Next, we deal with the two terms on the left-hand side of Eq. (29), we have
(30)2ε2AxAyδt2vn,δtvn=2ε2δt2vn,δtvn−hx212δt2vn,δtvn1,x−hy212δt2vn,δtvn1,y+hx2hy2144δt2vn,δtvnx,y=ε2τ∥δt¯vn+1/2∥2−∥δt¯vn−1/2∥2−hx212∥δxδt¯vn+1/2∥2−∥δxδt¯vn−12∥2−hy212∥δyδt¯vn+1/2∥2−∥δyδt¯vn−1/2∥2+hx2hy2144∥δxδyδt¯vn+1/2∥2−∥δxδyδt¯vn−1/2∥2=ε2τ∥δt¯vn+1/2∥A2−∥δt¯vn−1/2∥A2,1≤n≤K−1.(31)2AxAyδtvn,δtvn=2∥δtvn∥A2,1≤n≤K−1.
Then we deal with the first term on the right-hand side of Eq. (29).
(32)Ayδx2vn−1+Ayδx2vn+1,δtvn=I+hy212δy2δx2vn−1+I+hy212δy2δx2vn+1,δtvn=δx2vn−1+δx2vn+1,δtvn−hy212δx2vn−1+δx2vn+1,δtvn1,y=−12τ∥δxvn+1∥2−∥δxvn−1∥2+hy224τ∥δxδyvn+1∥2−∥δxδyvn−1∥2,1≤n≤K−1.
Analogously(33)Axδy2vn−1+Axδy2vn+1,δtvn=−12τ∥δyvn+1∥2−∥δyvn−1∥2+hx224τ∥δxδyvn+1∥2−∥δxδyvn−1∥2,1≤n≤K−1.
Combining (32) and (33), we have
(34)DAyδx2vn−1+Ayδx2vn+1+Axδy2vn−1+Axδy2vn+1,δtvn=−D2τ∥∇hvn+1∥A2−∥∇hvn−1∥A2,1≤n≤K−1.
Substituting (30), (31), (34) into (29) and applying Lemma 6, Cauchy-Schwarz inequality, we have
(35)ε2τ∥δt¯vn+1/2∥A2−∥δt¯vn−1/2∥A2+2∥δtvn∥A2=−D2τ∥∇hvn+1∥A2−∥∇hvn−1∥A2+2gn,δtvn≤−D2τ∥∇hvn+1∥A2−∥∇hvn−1∥A2+23∥δtvn∥2+32∥gn∥2≤−D2τ∥∇hvn+1∥A2−∥∇hvn−1∥A2+2∥δtvn∥A2+32∥gn∥2,1≤n≤K−1.
Sorting Eq. (35) and adding D/2∥∇hvn∥A2 to both sides, we have
(36)ε2∥δt¯vn+1/2∥A2+D2∥∇hvn+1∥A2+∥∇hvn∥A2≤ε2∥δt¯vn−1/2∥A2+D2∥∇hvn∥A2+∥∇hvn−1∥A2+3τ2∥gn∥2,1≤n≤K−1.
LetEn=ε2∥δt¯vn+1/2∥A2+D/2∥∇hvn+1∥A2+∥∇hvn∥A2, 1≤n≤K−1.
Eq. (36) is written as
(37)En≤En−1+3τ2∥gn∥2,1≤n≤K−1.
Summing overn results in
(38)En≤E0+3τ2∑k=1n∥gk∥2,1≤n≤K−1.
Multiplying Eq. (25) by δt¯v1/2, summing up for i from 1 to M−1, applying Lemma 4 and Cauchy-Schwarz inequality, we have
(39)2ε2τAxAyδt¯v12,δt¯v12−2ε2τφ,δt¯v12+AxAyδt¯v12,δt¯v12+D2τ∥∇hv1∥A2−∥∇hv0∥A2=g0,δt¯v12,where φij=AxAyu1xi,yj.
Sorting Eq. (39), we have
(40)2ε2τ∥δt¯v12∥A2+∥δt¯v12∥A2+D2τ∥∇hv1∥A2−∥∇hv0∥A2=2ε2τφ,δt¯v12+g0,δt¯v12≤13∥δt¯v12∥2+3ε4τ2∥φ∥2+ε23τ∥δt¯v12∥2+3τ4ε2∥g0∥2≤∥δt¯v12∥A2+ε2τ∥δt¯v12∥A2+3ε4τ2∥φ∥2+3τ4ε2∥g0∥2.
Sorting Eq. (40), we have
(41)ε2∥δt¯v12∥A2+D2∥∇hv1∥A2+∥∇hv0∥A2≤D∥∇hv0∥A2+3ε4τ∥φ∥2+3τ24ε2∥g0∥2.
Substituting (41) into (38), we get
(42)En≤D∥∇hv0∥A2+3ε4τ∥φ∥2+3τ24ε2∥g0∥2+3τ2∑k=1n∥gk∥2,1≤n≤K−1.
End of the proof.Remark 8.
According to the results, it can be seen that the derivative term of initial value affects stability and convergence, so this term in the Cattaneo model will be set as zero in general. For details, please refer to the literature [15].Theorem 9.
Supposeuijn∣i,j∈ω¯,0≤n≤K is the solution of Eq. (1), Uijn∣i,j∈ω¯,0≤n≤K is the solution of the difference scheme (20). Let
(43)eijn=uijn−Uijn,i,j∈ω¯,0≤n≤K,it holds that
(44)ε2∥δt¯en+12∥A2+D2∥∇hen+1∥A2+∥∇hen∥A2≤34ε2+3T2Cτ2+hx4+hy42,0≤n≤K−1.Proof.
Subtracting (14), (18), and (20), we can get the error system
(45)ε2AxAyδt2eijn+AxAyδteijn=D2Ayδx2eijn−1+Ayδx2eijn+1+Axδy2eijn−1+Axδy2eijn+1+Rijn3,i,j∈ω,1≤n≤K−1,(46)2ε2τAxAyδt¯eij12+AxAyδt¯eij12=D2Ayδx2eij0+Ayδx2eij1+Axδy2eij0+Axδy2eij1+rijn,i,j∈ω,(47)eij0=0,i,j∈ω,(48)eijn=0,i,j∈∂ω,0≤n≤K.
Using the Theorem7 and noticing (47) and (48), we have
(49)ε2∥δt¯en+1/2∥A2+D2∥∇hen+1∥A2+∥∇hen∥A2≤D∥∇he0∥A2+3τ24ε2rijn2+3τ2∑k=1nRijn3k2≤3τ24ε2C12τ+hx4+hy42+3τ2nC22τ2+hx4+hy42≤34ε2+3T2Cτ2+hx4+hy42.
End of the proof.Remark 10.
Through Theorem7 and Theorem 9, we prove the stability and convergence of the scheme. Due to the basic relationship between H1 semi-norm and infinity norm, we can also obtain that the proposed scheme is stable and convergent in L∞ norm.


## 4. Numerical Experiments

In this chapter, we verify the numerical accuracy and validity of the proposed scheme.In the following numerical experiment, for simplicity, the domain is set to the square domain. The domain isΩ=0,1×0,1, and hx is equal to hy. ε2=0.1, D=1, u0x,y=sinπxsinπy, u1x,y=0. In order to verify the numerical accuracy of the scheme, we need the following exact solution of the above problem. We can obtain the exact solution by using the separation of variables method,
(50)ux,y,t=e−12ε2tcos2Dπ2ε2−14ε2t+12ε22Dπ2/ε2−1/4ε2sin2Dπ2ε2−14ε2tsinπxsinπy.Before we give the experimental results, we first give the definitions ofl2-norm of the error and l∞-norm of the error.For all grid points,l2-norm of the error is defined by
(51)Errorl2=∑i=1M∑j=1Nhxhy∣uijK−UijK∣.For all grid points,l∞-norm of the error is defined by
(52)Errorl∞=max1≤i≤M,1≤j≤NuijK−UijK,where uij is the true solution, Uij is the numerical solution, and K is the last moment. Let N=M, order of spatial convergence is denoted by Ph and order of temporal convergence is denoted by Pτ, which are defined as follows
(53)Ph=logError1/Error2logh1/h2,Pτ=logError1/Error2logτ1/τ2In Table1, let h=hx=hy, and T=1, for h and τ, take the corresponding values, respectively, the compact difference scheme has the following corresponding error, order of temporal convergence and order of spatial convergence. The third column is the order of spatial convergence, and the fourth column is the order of time convergence. We can find that when the spatial step is reduced to one-half of the original and the temporal step is reduced to one-quarter of the original, the error is reduced to about one-sixteenth of the original. At the same time, the compact difference scheme keeps the fourth-order spatial rate of convergence and second-order temporal rate of convergency. So we also verify the stability and convergence of the scheme.Table 1
Error and convergence order of compact difference scheme.
hτorderordererrl∞errl21/41/20——7.1854e-033.5927e-031/81/803.5171.7595.8076e-042.9038e-041/161/3204.0052.0033.6204e-051.8102e-051/321/12803.9992.0002.2637e-061.3185e-061/641/51204.0002.0001.4151e-077.0753e-08In order to verify the applicability of the compact difference scheme more accurately, we give the image of convergence order defined the byl∞-norm. In Figure 1, we can see more clearly that the convergence order of the scheme (20) is Oτ2+hx4+hy4, which also verifies that the scheme (20) proposed in this paper is stable and effective. And the relaxation parameter does not affect the convergence rate.Figure 1
Order of spatial and temporal convergence.Figure2 show the true solution and the numerical solution when h=1/32 and τ=1/1280.Figure 2
The exact solution and the numerical at fixedD=1, ε2=0.1, T=1,h=1/32 and τ=1/1280.Figure3 shows the images when t is taken at different moments. It can be seen from the images that as time increases, the temperature also decreases gradually.Figure 3
Temperature images at different times whenD=1, ε2=0.1, h=1/16, and τ=1/80.Next, the value ofε2 in the previous example is 0.01, and the temperature image of t at different moments is also obtained. See Figure 4, it has the same property as Figure 3.Figure 4
Temperature images at different times whenD=1, ε2=0.01, h=1/16, and τ=1/80.Finally, we use two examples to test the influence of relaxation parameters, as shown in Figure5. From Figure 3–5, we can see that the larger ε2 is, the more obvious the fluctuation of temperature distribution is, and the slower the temperature change at the initial time.Figure 5
Temperature images with different relaxation parameters whenD=1, h=1/16 and 1/320.In Figure6, in order to observe the differences between the Cattaneo model and the traditional diffusion equation more conveniently, we give the temperature distribution images of the one-dimensional Cattaneo model and the heat conduction equation at different moments at the midpoint of x=0.5. It can be seen from the Figure 6 that the Cattaneo model has a slower temperature transformation at the initial moment than the heat conduction equation, but at the same time, the former has negative temperature and oscillation, which is also affected by the hyperbolic property of Cattaneo model.Figure 6
Temperature distribution images atx=0.5.


## 5. Conclusion

In this paper, we propose a compact finite difference scheme for a two-dimensional Cattaneo model and prove the stability and convergence of the scheme by energy method. A concrete example is given, and the exact solution is obtained by the method of separating variables, it can be seen from the figures that the fitting degree of the exact solution and numerical solution is higher, it is also verified by numerical experiments that the convergence order of the scheme isOτ2+hx4+hy4. In conclusion, the proposed scheme is simple, stable, and effective and has higher computational accuracy than some existing methods.



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