# Higher-Order Compact Finite Difference for Certain PDEs in Arbitrary Dimensions

Authors: Yan Gao; Songlin Liu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we first present the expression of a model of a fourth-order compact finite difference (CFD) scheme for the convection diffusion equation with variable convection coefficient. Then, we also obtain the fourth-order CFD schemes of the diffusion equation with variable diffusion coefficients. In addition, a fine description of the sixth-order CFD schemes is also developed for equations with constant coefficients, which is used to discuss certain partial differential equations (PDEs) with arbitrary dimensions. In this paper, various ways of numerical test calculations are prepared to evaluate performance of the fourth-order CFD and sixth-order CFD schemes, respectively, and the empirical results are proved to verify the effectiveness of the schemes in this paper.


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## Body

## 1. Introduction

The standard strategy associated with generating higher order finite difference schemes to expand the stencil is proposed by Leonard [1]. The major disadvantage of these approaches is to widen the computational stencil with the increasing order of the approximation, leading to larger matrix bandwidths, which shall complicate the numerical analysis near the boundaries, while communication to implement on parallel computer architecture is raising demand. In consideration of the problems caused by noncompact finite difference methods, it is desirable to develop a class of schemes involving compact with high order.In recent years, a great deal of efforts has been devoted to developing the designing schemes of CFD for solving various partial differential equations (see, e.g., [2–7]). A derivation of fourth-order and sixth-order compact difference schemes for the three dimensional Poisson equation is developed by Zhai and Feng [8] finite volume (FV) method, based on two different types of dual partitions. A new design of sixth-order compact finite-difference method with a nine-point stencil is developed by Nabavi et al. [9] to solve the Helmholtz equation in two-dimensional domain under the circumstance of Dirichlet and Neumann boundary. With the idea of the immersed interface method, third- and fourth-order compact finite difference schemes were proposed for solving the Helmholtz equations with discontinuous coefficient [10, 11]. It is worth noting that the advert of [12, 13] a new high-order finite difference discretization strategy, based on the Richardson extrapolation technique and an operator interpolation scheme, is explored to solve convection diffusion equations [14] which exploits an innovative adaptive scheme in terms of Adaptive Mesh Refinement (AMR) and Multigrid Algorithms to achieve a settlement of the fourth-order two-dimensional Poisson equation. In addition, a great number of studies reported by Bilbao and Hamilton [15] provides a two-step schemes (which operate over three time levels) of higher order accurate finite difference schemes applied for the wave equation in any number of spatial dimensions.The main aim of the present work is to provide a general formulation and simple approach of higher order CFD schemes for steady elliptic diffusion and convection-diffusion problems in any dimension. The primary concern of this paper focuses on the following convection-diffusion equation with Dirichlet boundary conditions(1)−Δu+α·∇u=f,inΩ=0,1d,u∂Ω=g,where α=α1,⋯,αd is the velocity vector with dimension d, the source term f is known analytically throughout the domain Ω, and g is given on the boundary ∂Ω, Δu=∑m=1duxmxm is the Laplacian operator, and ∇u=ux1,⋯,uxd is the divergence operator.Obviously, it is not difficult to derive the fourth-order compact finite difference scheme, especially for the case whenα is a constant vector. The scheme in response to simple procedure is introduced in Theorems 2 and 4 in the next section. The compact scheme of the sixth-order finite difference scheme for (1) with constant parameters will be completely described by Theorem 6 in Section 3.The paper is organized as follows. In Section2, the forth-order CFD schemes are derived for the convection-diffusion problems with variable convective coefficients in any dimension. Since then, two different fourth-order CFD schemes are linked to the consequent of steady elliptic diffusion equations with variable diffusive coefficients. The sixth-order CFD schemes are considered to be the solution of the convection-diffusion problems with constant convective coefficients in Section 3. The analysis consisted of two numerical examples is presented to verify the feasibility and high-order accuracy of the proposed methods in Section 4. This paper concludes with a discussion in Section 5.


## 2. Fourth-Order CFD Schemes for Convection-Diffusion Equations

### 2.1. Fourth-Order CFD Schemes for Convection-Diffusion Equations with Variable Convective Coefficients

Letπh=xi1,⋯,xid:xik=ikh,ik=0,⋯,n+1,k=1,⋯,d be a uniform partition of Ω with the step size h=1/n+1. Moreover, we introduce the central difference approximations δxmu and δxmxmu as the first and second derivatives, respectively, by using the immediate neighbours. Denote ui1,⋯,id by the value of ux1,⋯,xd at the nodes xi1,⋯,xid, and
(2)δxmuxi1,⋯,xid=ui1,⋯,im+1,⋯,id−ui1,⋯,im−1,⋯,id2h,δxmxmuxi1,⋯,xid=ui1,⋯,im+1,⋯,id−2ui1,⋯,im,⋯,id+ui1,⋯,im−1,⋯,idh2,by the first order and second order central difference schemes, respectively, in the xmm=1,⋯,d direction.By using the Taylor expansion, the problem (1) can be approximated by
(3)∑m=1d−δxmxm+αmδxmu+Ru=f,where
(4)Ru=h26R1u+Oh4,with
(5)R1u=∑m=1d12uxmxmxmxm−αmuxmxmxm.Remark 1.
By neglectingRu in (4), we obtain the simplest CFD scheme, with a second order accuracy, for (1), given by
(6)∑m=1d−δxmxmu+αmδxmu=f,which is a 5-point scheme in 2D and a 7-point one in 3D.To obtain higher order CFD schemes, we will approximate the high order partial differential terms in (4) in a compact stencil.Theorem 2.
ForR1u defined in (5), we have
(7)R1u=12F1−12∑m=1d∑n=1,n≠mdδxmxmδxnxnu+∑m=1dαm∑n=1,n≠mdδxmδxnxnu−∑m=1dammδxmxmu−12∑m=1d∑n=1,n≠mdamnδxmδxnu−∑m=1damδxmu+Oh2,where
(8)F1=∑m=1d−fxmxm+αmfxm,(9)am=12−Δαm+∑n=1dαnαmxn,amm=12αm2−αmxm,amn=αmαn−αmxn−αnxm,m≠n,and αmxn denote the derivation of αm along the xn−direction.Proof.
Differentiating both sides of (1) with respect to the m-th direction gives
(10)−∑n=1duxmxnxn+∑n=1dαnuxmxn+∑n=1dαnxmuxn=fxm.Differentiating both sides of (10) respect to the m-th direction again and then using (10) gives
(11)−∑n=1duxmxmxnxn+∑n=1dαnuxmxmxn+2∑n=1dαnxmuxmxn+∑n=1dαnxmxmuxn=fxmxm.Therefore, adding (11) with each direction, we get
(12)R1u=12F1−12∑m=1d∑n=1,n≠mduxmxmxnxn−∑m=1d∑n=1,n≠mdαmuxmxnxn−∑m=1dammuxmxm−12∑m=1d∑n=1,n≠mdamnuxmxn−∑m=1damuxm,where F1 appeared in (8). By using the central difference schemes
(13)uxm=δxmu+Oh2,uxmxm=δxmxmu+Oh2,uxmxn=δxmδxnu+Oh2,uxmxnxn=δxmδxnxnu+Oh2,uxmxmxnxn=δxmxmδxnxnu+Oh2,with n≠m, we can get (7), which completes the proof of Theorem 2.From equations (3), (4), (5), and Theorem 2, a fourth-order compact difference scheme for approximating (1) is obtained, which can be represented as follows
(14)∑m=1d∑n=1dn≠mcmnδxmδxn+cmnnδxmδxnxn+cmmnnδxmxmδxnxnu+∑m=1dcmδxm+cmmδxmxmu=−f+h212F1,where
(15)cm=−αm+h26am,cmm=1+h26amm,cmn=h26amn,cmnn=−h26αm,cmmnn=h26,and am,amm,amnm≠n are defined in (9).Remark 3.
(i)
Ifd=2, the scheme (14) is no other than one suggested by Gupta et al. [16], and if d=3, the scheme (14) is no other than the one suggested by Zhang [17].(ii)
Ifα is a constant vector, it is easy to simplify the fourth-order CFD scheme (14) by reducing the terms (15) as
(16)cm=−αm,cmm=1+h212αm2,cmn=h26αmαn,cmnn=−h26αm,cmmnn=h26m≠n.


### 2.2. Fourth-Order CFD Schemes for Diffusion Equations with Variable Diffusive Coefficients

Consider thed-dimensional diffusion equation
(17)−∇·κ∇u=f~,inΩ⊂ℝd,u∂Ω=φ,where the diffusive coefficient κ is a scalar function which is assumed to be positive. It can be rewritten as
(18)−∑n=1dκuxnxn+κxnuxn=f~.Scheme I.
By using the Taylor series expansion, the problem (17) can be approximated by
(19)−∑m=1dκδxmxm+κxmδxmu+R~u=f~.
R~u is the same as (4) with different force terms, where
(20)R~u=h26R~1u+Oh4,(21)R~1u=∑m=1dκ2uxmxmxmxm+κxmuxmxmxm.Theorem 4.
ForR~1u defined in (21), we have
(22)κR~1u=∑m=1d∑n=1dn≠m−κ22δxmxmδxnxn−κδxmδxnxn−12amnδxmδxnu−∑m=1dammδxmxmu−∑m=1damδxmu+12F~1+Oh2,where
(23)F~1=∑m=1d−κf~xmxm+κxmf~xm,(24)am=12∑n=1dκκxmxnxn−κxnκxmxn,amm=κκxmxm−κxm2+12κ∑n=1dκxnxn−12∑n=1dn≠mκxn2,amn=2κκxmxn−κxmκxn,m≠n.Proof.
Differentiating both sides of (17) once and twice with respect to xm, respectively, gives
(25)2κxmuxmxm+∑n=1dκuxnxnxm+κxmxnuxn+∑n=1dn≠mκxmuxnxm+κxnuxmxn=−f~xm,(26)3κxmuxmxmxm+∑n=1dκuxmxmxnxn+κxmxmxnuxn+∑n=1dn≠m2κxmuxmxnxn+κxmuxmxmxn+κxmxmuxnxn+2κxmxnuxmxn=−f~xmxm.From (25), we can obtain the expression of κκxmuxmxmxm, and then, substituting (25) into (26) to eliminate the term concluding uxmxmxm gives the expression of κ2uxmxmxmxm. By the definition of R~1u, we have
(27)κR~1u=−κ22∑m=1d∑n=1dn≠muxmxmxnxn−κ∑m=1d∑n=1dn≠mκxmuxmxnxn−∑m=1dammuxmxm−12∑m=1d∑n=1dn≠mamnuxmxn−∑m=1damuxm+12F~1.By using the central difference schemes defined in (13), we can get (22),which completes the proof of Theorem 4.From (19), (4), (21), and Theorem 4, a fourth-order compact difference scheme for approximating (17) is obtained by dropping off the higher order terms Oh4(28)∑m=1dcmδxmu+∑m=1dcmmδxmxmu+∑m=1d∑n=1dn≠mcmnδxmδxnu+∑m=1d∑n=1dn≠mcmnnδxmδxnxnu+∑m=1d∑n=1dn≠mcmmnnδxmxmδxnxnu=F~,where
(29)F~=−κf~+h212F~1,(30)cm=κκxm+h26am,cmm=κ2+h26amm,cmn=h26amn,cmnn=h26κκxm,cmmnn=h212κ2,m≠n.F~1 and am,amm,amnm≠n are defined by (23) and (24).Scheme II.
Now, we derive another fourth-order CFD scheme for approximating (17) based on the fourth-order CFD scheme for convection-diffusion equations discussed in Section 2.1. Since κ is positive, we can also rewrite (17) as the form
(31)−∑n=1duxnxn+κxnκuxn=f~κ.Letα=α1,⋯,αd with αm=−κxm/κm=1,⋯,d, then (31) has the same form of (1). Substituting this α into (14) and (15) and then using some direct calculations, a new fourth-order compact finite difference scheme is obtained as
(32)∑m=1d∑n=1n≠mdcmnδxmδxn+cmnnδxmδxnxn+cmmnnδxmxmδxnxnu+∑m=1dcmδxm+cmmδxmxmu=−κf~+h212F~1=F~,where
(33)F~=∑m=1dκ2f~xmxm−κκxmf~xm+κxm2−κκxmxmf~,(34)cm=κ2κxm+h212∑n=1dκ2κxmxnxn+κxmκxn2−2κκxmκxmxm−κ∑n=1,n≠mdκxmκxnxn+κxnκxmxn,cmm=κ3+h212κ2κκxmxm−κxm2,cmn=h26κ2κκxmxn−κxmκxn,cmnn=h26κ2κxm,cmmnn=h26κ3m≠n.Remark 5.
(i)
In practice, we should approximate the factors related toκ and f~ with the accuracy Oh2 in (24), like κxm,κxmxm,κxmxnm≠n,κxmxmxnm≠n, and f~xm,f~xmxm. All of these can be approximated by the corresponding central difference scheme to meet the requirement.(ii)
To guarantee the accuracy of the schemes (28) and (32), we also need to approximate κxm and κxmxmxm with the fourth-order accuracy. For the purpose of this, the following approximation
(35)κ′x=−16hκx+h−8κx+h2+8κx−h2−κx−h+Oh4,κ′″x=4h3κx+h−2κx+h2+2κx−h2−κx−h+Oh2,will be introduced.


## 2.1. Fourth-Order CFD Schemes for Convection-Diffusion Equations with Variable Convective Coefficients

Letπh=xi1,⋯,xid:xik=ikh,ik=0,⋯,n+1,k=1,⋯,d be a uniform partition of Ω with the step size h=1/n+1. Moreover, we introduce the central difference approximations δxmu and δxmxmu as the first and second derivatives, respectively, by using the immediate neighbours. Denote ui1,⋯,id by the value of ux1,⋯,xd at the nodes xi1,⋯,xid, and
(2)δxmuxi1,⋯,xid=ui1,⋯,im+1,⋯,id−ui1,⋯,im−1,⋯,id2h,δxmxmuxi1,⋯,xid=ui1,⋯,im+1,⋯,id−2ui1,⋯,im,⋯,id+ui1,⋯,im−1,⋯,idh2,by the first order and second order central difference schemes, respectively, in the xmm=1,⋯,d direction.By using the Taylor expansion, the problem (1) can be approximated by
(3)∑m=1d−δxmxm+αmδxmu+Ru=f,where
(4)Ru=h26R1u+Oh4,with
(5)R1u=∑m=1d12uxmxmxmxm−αmuxmxmxm.Remark 1.
By neglectingRu in (4), we obtain the simplest CFD scheme, with a second order accuracy, for (1), given by
(6)∑m=1d−δxmxmu+αmδxmu=f,which is a 5-point scheme in 2D and a 7-point one in 3D.To obtain higher order CFD schemes, we will approximate the high order partial differential terms in (4) in a compact stencil.Theorem 2.
ForR1u defined in (5), we have
(7)R1u=12F1−12∑m=1d∑n=1,n≠mdδxmxmδxnxnu+∑m=1dαm∑n=1,n≠mdδxmδxnxnu−∑m=1dammδxmxmu−12∑m=1d∑n=1,n≠mdamnδxmδxnu−∑m=1damδxmu+Oh2,where
(8)F1=∑m=1d−fxmxm+αmfxm,(9)am=12−Δαm+∑n=1dαnαmxn,amm=12αm2−αmxm,amn=αmαn−αmxn−αnxm,m≠n,and αmxn denote the derivation of αm along the xn−direction.Proof.
Differentiating both sides of (1) with respect to the m-th direction gives
(10)−∑n=1duxmxnxn+∑n=1dαnuxmxn+∑n=1dαnxmuxn=fxm.Differentiating both sides of (10) respect to the m-th direction again and then using (10) gives
(11)−∑n=1duxmxmxnxn+∑n=1dαnuxmxmxn+2∑n=1dαnxmuxmxn+∑n=1dαnxmxmuxn=fxmxm.Therefore, adding (11) with each direction, we get
(12)R1u=12F1−12∑m=1d∑n=1,n≠mduxmxmxnxn−∑m=1d∑n=1,n≠mdαmuxmxnxn−∑m=1dammuxmxm−12∑m=1d∑n=1,n≠mdamnuxmxn−∑m=1damuxm,where F1 appeared in (8). By using the central difference schemes
(13)uxm=δxmu+Oh2,uxmxm=δxmxmu+Oh2,uxmxn=δxmδxnu+Oh2,uxmxnxn=δxmδxnxnu+Oh2,uxmxmxnxn=δxmxmδxnxnu+Oh2,with n≠m, we can get (7), which completes the proof of Theorem 2.From equations (3), (4), (5), and Theorem 2, a fourth-order compact difference scheme for approximating (1) is obtained, which can be represented as follows
(14)∑m=1d∑n=1dn≠mcmnδxmδxn+cmnnδxmδxnxn+cmmnnδxmxmδxnxnu+∑m=1dcmδxm+cmmδxmxmu=−f+h212F1,where
(15)cm=−αm+h26am,cmm=1+h26amm,cmn=h26amn,cmnn=−h26αm,cmmnn=h26,and am,amm,amnm≠n are defined in (9).Remark 3.
(i)
Ifd=2, the scheme (14) is no other than one suggested by Gupta et al. [16], and if d=3, the scheme (14) is no other than the one suggested by Zhang [17].(ii)
Ifα is a constant vector, it is easy to simplify the fourth-order CFD scheme (14) by reducing the terms (15) as
(16)cm=−αm,cmm=1+h212αm2,cmn=h26αmαn,cmnn=−h26αm,cmmnn=h26m≠n.


## 2.2. Fourth-Order CFD Schemes for Diffusion Equations with Variable Diffusive Coefficients

Consider thed-dimensional diffusion equation
(17)−∇·κ∇u=f~,inΩ⊂ℝd,u∂Ω=φ,where the diffusive coefficient κ is a scalar function which is assumed to be positive. It can be rewritten as
(18)−∑n=1dκuxnxn+κxnuxn=f~.Scheme I.
By using the Taylor series expansion, the problem (17) can be approximated by
(19)−∑m=1dκδxmxm+κxmδxmu+R~u=f~.
R~u is the same as (4) with different force terms, where
(20)R~u=h26R~1u+Oh4,(21)R~1u=∑m=1dκ2uxmxmxmxm+κxmuxmxmxm.Theorem 4.
ForR~1u defined in (21), we have
(22)κR~1u=∑m=1d∑n=1dn≠m−κ22δxmxmδxnxn−κδxmδxnxn−12amnδxmδxnu−∑m=1dammδxmxmu−∑m=1damδxmu+12F~1+Oh2,where
(23)F~1=∑m=1d−κf~xmxm+κxmf~xm,(24)am=12∑n=1dκκxmxnxn−κxnκxmxn,amm=κκxmxm−κxm2+12κ∑n=1dκxnxn−12∑n=1dn≠mκxn2,amn=2κκxmxn−κxmκxn,m≠n.Proof.
Differentiating both sides of (17) once and twice with respect to xm, respectively, gives
(25)2κxmuxmxm+∑n=1dκuxnxnxm+κxmxnuxn+∑n=1dn≠mκxmuxnxm+κxnuxmxn=−f~xm,(26)3κxmuxmxmxm+∑n=1dκuxmxmxnxn+κxmxmxnuxn+∑n=1dn≠m2κxmuxmxnxn+κxmuxmxmxn+κxmxmuxnxn+2κxmxnuxmxn=−f~xmxm.From (25), we can obtain the expression of κκxmuxmxmxm, and then, substituting (25) into (26) to eliminate the term concluding uxmxmxm gives the expression of κ2uxmxmxmxm. By the definition of R~1u, we have
(27)κR~1u=−κ22∑m=1d∑n=1dn≠muxmxmxnxn−κ∑m=1d∑n=1dn≠mκxmuxmxnxn−∑m=1dammuxmxm−12∑m=1d∑n=1dn≠mamnuxmxn−∑m=1damuxm+12F~1.By using the central difference schemes defined in (13), we can get (22),which completes the proof of Theorem 4.From (19), (4), (21), and Theorem 4, a fourth-order compact difference scheme for approximating (17) is obtained by dropping off the higher order terms Oh4(28)∑m=1dcmδxmu+∑m=1dcmmδxmxmu+∑m=1d∑n=1dn≠mcmnδxmδxnu+∑m=1d∑n=1dn≠mcmnnδxmδxnxnu+∑m=1d∑n=1dn≠mcmmnnδxmxmδxnxnu=F~,where
(29)F~=−κf~+h212F~1,(30)cm=κκxm+h26am,cmm=κ2+h26amm,cmn=h26amn,cmnn=h26κκxm,cmmnn=h212κ2,m≠n.F~1 and am,amm,amnm≠n are defined by (23) and (24).Scheme II.
Now, we derive another fourth-order CFD scheme for approximating (17) based on the fourth-order CFD scheme for convection-diffusion equations discussed in Section 2.1. Since κ is positive, we can also rewrite (17) as the form
(31)−∑n=1duxnxn+κxnκuxn=f~κ.Letα=α1,⋯,αd with αm=−κxm/κm=1,⋯,d, then (31) has the same form of (1). Substituting this α into (14) and (15) and then using some direct calculations, a new fourth-order compact finite difference scheme is obtained as
(32)∑m=1d∑n=1n≠mdcmnδxmδxn+cmnnδxmδxnxn+cmmnnδxmxmδxnxnu+∑m=1dcmδxm+cmmδxmxmu=−κf~+h212F~1=F~,where
(33)F~=∑m=1dκ2f~xmxm−κκxmf~xm+κxm2−κκxmxmf~,(34)cm=κ2κxm+h212∑n=1dκ2κxmxnxn+κxmκxn2−2κκxmκxmxm−κ∑n=1,n≠mdκxmκxnxn+κxnκxmxn,cmm=κ3+h212κ2κκxmxm−κxm2,cmn=h26κ2κκxmxn−κxmκxn,cmnn=h26κ2κxm,cmmnn=h26κ3m≠n.Remark 5.
(i)
In practice, we should approximate the factors related toκ and f~ with the accuracy Oh2 in (24), like κxm,κxmxm,κxmxnm≠n,κxmxmxnm≠n, and f~xm,f~xmxm. All of these can be approximated by the corresponding central difference scheme to meet the requirement.(ii)
To guarantee the accuracy of the schemes (28) and (32), we also need to approximate κxm and κxmxmxm with the fourth-order accuracy. For the purpose of this, the following approximation
(35)κ′x=−16hκx+h−8κx+h2+8κx−h2−κx−h+Oh4,κ′″x=4h3κx+h−2κx+h2+2κx−h2−κx−h+Oh2,will be introduced.


## 3. Sixth-Order CFD Schemes for Convection-Diffusion Equations

In the previous section, we have obtained the fourth-order CFD scheme for (1). Especially for the case that α is a constant vector, this scheme seems very simple. Actually, after some further analysis, we can make it achieve higher order accuracy.


### 3.1. Sixth-Order CFD Schemes for Diffusion Equations with Variable Convection Coefficients

This study will offer a fresh insight into the following Theorem6 to show the sixth-order CFD scheme for (1). By using the Taylor expansion, the problem (1) can be approximated by (3), where
(36)Ru=h26R1u+h4120R2u+Oh6,with
(37)R1u=∑n=1d12uxnxnxnxn−αnuxnxnxn,(38)R2u=∑n=1d13uxnxnxnxnxnxn−αnuxnxnxnxnxn.Theorem 6.
LetRu be defined by (36), then we have
(39)Ru=h26S1u+h4720S2+S3u+h4180S4+S5−S6+S7+S8u+F∗+Oh6,where
(40)F∗=h212F1+h47202F2−F3−4F4+4F5−4F6,with Siui=1,⋯,8 defined by (44), (59),(60), (65), (66), (71), (72), and (76) and Fii=1,⋯,6 defined by (45), (53), (61), (67), (73), and (77) below.This study will offer the compact sixth-order finite difference scheme for (1)
(41)∑m=1d−δxmxm+αmδxmu+Ru=F,where F=f−F∗ and RU are defined in (39) by replacing the exact solution u by the approximating solution U and dropping off the remainder term Oh6.To prove Theorem6, we define Qii=1,2,⋯,5 as follows:
(42)Q1=∑n=1dαn2uxnxnxnxn,Q2=2∑n=1d∑k=1dk≠nαnαkuxnxnxnxk,Q3=2∑n=1d∑k=1dk≠nαkuxnxnxnxnxk,Q4=2∑n=1d∑k=1dk≠nαkuxnxnxkxkxk,Q5=2∑n=1d∑k=1dk≠nuxnxnxnxnxkxk,and the following Lemmas 7-13 are needed.Lemma 7.
LetR1u be defined in (37) and Qkk=1,⋯,5 be defined in (42), then we have
(43)R1u=S1u+h224Q1+2Q2−Q3−2Q4+Q5+12F1+Oh4,where
(44)S1u=12∑n=1d∑k=1dk≠n−δxkxkδxnxn+αkδxkδxnxn+αnδxkxkδxn−αnαkδxkδxnu−12∑n=1dαn2δxnxnu,(45)F1=∑n=1d−fxnxn+αnfxn.Proof.
Differentiating both sides of (1) once and twice with respect to xnn=1,⋯,d, respectively, gives
(46)−∑k=1duxkxkxn+∑k=1dαkuxkxn=fxn,−∑k=1duxkxkxnxn+∑k=1dαkuxkxnxn=fxnxn,which leads to
(47)uxnxnxn=−∑k=1dk≠nuxkxkxn+∑k=1dαkuxkxn−fxn,(48)uxnxnxnxn=−∑k=1dk≠nuxkxkxnxn+∑k=1dαkuxkxnxn−fxnxn.Using (47) and (48) to eliminate the terms uxnxnxn and uxnxnxnxn in (37) yields
(49)R1u=−12∑n=1d∑k=1dk≠nuxkxkxnxn+12∑n=1d∑k=1dk≠nαkuxkxnxn+αnuxkxkxn−12∑n=1d∑k=1dαkαnuxkxn+12∑n=1d−fxnxn+αnfxn.Taking account of the centered difference approximations(50)uxnxn=δxnxnu−h212uxnxnxnxn+Oh4,uxkxn=δxkδxnu−h26uxkxkxkxn+uxkxnxnxn+Oh4,uxkxkxn=δxkxkδxnu−h212uxkxkxkxkxn+2uxkxkxnxnxn+Oh4,uxkxkxnxn=δxkxkδxnxnu−h212uxkxkxnxnxnxn+uxkxkxkxkxnxn+Oh4,with n≠k, and noting the definition of S1u given by (44), then we have
(51)R1u=h224∑n=1d∑k=1dk≠nuxkxkxnxnxnxn+uxkxkxkxkxnxn−αkuxnxnxnxnxk+2uxnxnxkxkxk−αnuxkxkxkxkxn+2uxkxkxnxnxn+2αnαkuxkxkxkxn+uxkxnxnxnu+h224∑n=1dαn2uxnxnxnxn+S1u+12F1=h224Q1+2Q2−Q3−2Q4+Q5+S1u+12F1+Oh4.Therefore, the proof is completed when conditions (42) and (50) are satisfied.Lemma 8.
LetR2u be defined by (38) and Qkk=1,⋯,5 be defined by (42), then
(52)R2u=−164Q1+2Q2−Q3−2Q4+Q5+13F2,where
(53)F2=∑n=1d−fxnxnxnxn+2αnfxnxnxn.Proof.
Differentiating both sides of (1) three times and four times with respect to xnn=1,⋯,d, respectively, gives rise to
(54)uxnxnxnxnxn=−fxnxnxn−∑k=1dk≠nuxkxkxnxnxn+∑k=1dαkuxkxnxnxn,(55)uxnxnxnxnxnxn=−fxnxnxnxn+∑k=1dk≠n−uxkxkxnxnxnxn+αkuxkxnxnxnxn+αnuxnxnxnxnxn.Using (54) and (55) to eliminate the terms uxnxnxn and uxnxnxnxn in (38) yields
(56)R2u=13∑n=1d∑k=1dk≠n−uxnxnxnxnxkxk+αkuxnxnxnxnxk+2αnuxnxnxnxkxk−2αnαkuxnxnxnxk−23∑n=1dαn2uxnxnxnxn+13F2=−164Q1+2Q2−Q3−2Q4+Q5+13F2,which finishes the proof.Lemma 9.
LetRu be defined by (36) and Qii=1,⋯,5 be defined by (42), S1u,F1, and F2 be defined by (44), (45), and (53) then
(57)Ru=h4720Q1+h41802Q2−Q3−2Q4+Q5+h26S1u+h26F1+h4360F2.Proof.
The proof can be easily done by putting Lemma7, Lemma 8., and the definition of Ru together.Now, we need to represent the termsQii=1,⋯,5 by using compact finite difference approximations.Lemma 10.
LetQ1 be defined by (42), then
(58)Q1=S2u+S3u−F3+Oh2,where
(59)S2u=∑n=1d∑k=1dk≠nαn2αkδxkδxnxn−αnδxkxkδxn−δxkxkδxnxnu,(60)S3u=∑n=1d∑k=1dαn3αkδxkxnu,(61)F3=∑n=1dαn2fxnxn+αnfxn.Proof.
Substituting (47) into (48) yields
(62)uxnxnxnxn=∑k=1dk≠n−uxkxkxnxn+αkuxkxnxn−αnuxkxkxn+αn∑k=1dαkuxkxn−fxnxn+αnfxn.Multiplying both sides of the above equation byαn and then summing them over n, we can rewrite Q1 defined in (42) as the form
(63)Q1=∑n=1dαn2∑k=1dk≠nαkuxkxnxn−αnuxkxkxn−uxkxkxnxn+∑n=1dαn2∑k=1dαkαnuxkxn−∑n=1dαn2fxnxn+αnfxn.Therefore, the expected result can be easily obtained by replacing the central difference approximations (13) in (63).Lemma 11.
LetQ2 be defined by (42),then we have
(64)Q2=S4u+S5u−F4+Oh2,where
(65)S4u=∑n=1d∑k=1dk≠nαn2αkδxnxnδxku,(66)S5u=12∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠k−αkαmδxnxnδxkδxm+αnαkαmδxnδxkδxmu,(67)F4=12∑n=1d∑k=1dk≠nαnαkfxnxk.Proof.
Differentiating both sides of (1) with respect to xmxk,k≠m,m,k=1,⋯,d yields
(68)−∑n=1d∑k=1dk≠nuxnxnxnxk−12∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kuxnxnxmxk+∑n=1d∑k=1dk≠nαnuxnxnxk+12∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαnuxnuxkuxm=12∑m=1d∑k=1dk≠mk≠mfxmxk.Multiplying both sides of the above equation byαmαk yields
(69)Q2=−12∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαmαkuxnxnxmxk+12∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαnαkαmuxnuxkuxm+∑n=1d∑k=1dk≠nαn2αkuxnxnxk−12∑m=1d∑k=1dk≠mαmαkfxmxk.By using the centered difference approximations (13), we can obtain the results.Lemma 12.
LetQ2,Q3, and Q4 be defined in (42), then we have
(70)Q2−Q3−Q4=−S6u+S7u+F5+Oh2,where
(71)S6u=∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαkδxkδxnxn+αnδxnδxkxkδxmxm−αmαkδxnxnδxk+αnδxnδxkxkδxmu,(72)S7u=2∑n=1d∑k=1dk≠nαn2δxnxnδxkxku,(73)F5=∑n=1d∑k=1dk≠nαkfxnxnxk+αnfxnxkxk.Proof.
By using the definitions ofQ2,Q3, and Q4 in (42), we have
(74)Q3+Q4=∑n=1d∑k=1dk≠nαkuxnxn+uxkxkxnxnxk+αnuxnxn+uxkxkxnxkxk=−F5+Q2+∑n=1d∑k=1dk≠nαk2+αn2uxnxnxkxk+∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαmαkuxmxnxnxk+αnuxmxnxkxk−∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαkuxmxmxnxnxk+αnuxmxmxnxkxk.
By using the centered difference approximations (13), we can obtain the results.Lemma 13.
LetQ4 and Q5 be defined in (42), then we have
(75)Q5−Q4=S8u−F6+Oh2,where
(76)S8u=∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαmδxmδxnxnδxkxk−δxmxmδxnxnδxkxku,(77)F6=∑n=1d∑k=1dk≠nfxnxnxkxk.Proof.
Differentiating both sides of (1) twice with respect to xn and then twice with respect to xk,k≠n, and then summing them over n and k gives
(78)∑n=1d∑k=1dk≠n∑m=1d−uxmxmxnxnxkxk+αmuxmxnxnxkxk=∑n=1d∑k=1dk≠nfxnxnxkxk,which can be rewritten as
(79)∑n=1d∑k=1dk≠n−uxnxnxnxnxkxk+αnuxnxnxnxkxk−uxkxkxnxnxkxk+αkuxkxnxnxkxk+∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠k−uxmxmxnxnxkxk+αmuxmxnxnxkxk=∑n=1d∑k=1dk≠nfxnxnxkxk.By using the definition ofQ4 and Q5 defined in (42), we have
(80)Q5−Q4=∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαmuxmxnxnxkxk−uxmxmxnxnxkxk−∑n=1d∑k=1dk≠nfxnxnxkxk.By using the centered difference approximations (13),we can obtain the results.Proof of Theorem 6.
The proof can be easily done by putting Lemmas9–13 and the definition of F∗ in (40) together.


### 3.2. Sixth-Order CFD Schemes for Diffusion Equations with Constant Convective Coefficients

It should be noted that the sixth-order CFD scheme given in (41) is valid for any dimension. For simplicity, we write down here the most commonly used schemes for (1), i.e., for 2 and 3 dimensions, in a concise way.
(i)
Ifd=2, the terms including the summation symbol ∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠k in Si,i=1,⋯,8, i.e., S5,S6,S7, and S8, vanish. Therefore, the sixth-order CFD scheme can be described as
(81)∑n=12∑k=12k≠ncnkδxnδxk+cnkkδxnδxkxk+cnnkkδxnxnδxkxku+∑n=12cnδxnu+∑n=12cnnδxnxnu=f−F∗,where
(82)cn=αn,cnn=1+αn2h2121−αn2h260,cnk=αnαkh2121−αn2+αk2h2120,cnkk=αnh261−αn2−9αk2h2120,cnnkk=h212+αn2+αk2h4160,and F∗ appeared in (40).(ii)
Ifd=3, all of the Si,i=1,⋯,8 exist, and the sixth-order CFD scheme can be described as(83)∑n=13∑k=13k≠ncnkδxnδxk+cnkkδxnδxkxk+cnnkkδxnδxkxku+∑n=13anδxn+annδxnxnu+c123δx1δx2δx3+c112233δx1x1δx2x2δx3x3u+c1123δx1x1δx2δx3+c1223δx1δx2x2δx3+c1233δx1δx2δx3x3u+c12233δx1δx2x2δx3x3+c11233δx1x1δx2δx3x3+c11223δx1x1δx2x2δx3u=f−F∗,where the forms of cn,cnn,cnk,cnkk,cnnkk,andk≠n are the same as that defined in (82), except in this case n=1,2,3 and
(84)c123=α1α2α3h430,c112233=h430,cmnkk=αmαnh490,cmnnkk=αmh490,where m,n,k=1,2,3, but they are different from each other.


## 3.1. Sixth-Order CFD Schemes for Diffusion Equations with Variable Convection Coefficients

This study will offer a fresh insight into the following Theorem6 to show the sixth-order CFD scheme for (1). By using the Taylor expansion, the problem (1) can be approximated by (3), where
(36)Ru=h26R1u+h4120R2u+Oh6,with
(37)R1u=∑n=1d12uxnxnxnxn−αnuxnxnxn,(38)R2u=∑n=1d13uxnxnxnxnxnxn−αnuxnxnxnxnxn.Theorem 6.
LetRu be defined by (36), then we have
(39)Ru=h26S1u+h4720S2+S3u+h4180S4+S5−S6+S7+S8u+F∗+Oh6,where
(40)F∗=h212F1+h47202F2−F3−4F4+4F5−4F6,with Siui=1,⋯,8 defined by (44), (59),(60), (65), (66), (71), (72), and (76) and Fii=1,⋯,6 defined by (45), (53), (61), (67), (73), and (77) below.This study will offer the compact sixth-order finite difference scheme for (1)
(41)∑m=1d−δxmxm+αmδxmu+Ru=F,where F=f−F∗ and RU are defined in (39) by replacing the exact solution u by the approximating solution U and dropping off the remainder term Oh6.To prove Theorem6, we define Qii=1,2,⋯,5 as follows:
(42)Q1=∑n=1dαn2uxnxnxnxn,Q2=2∑n=1d∑k=1dk≠nαnαkuxnxnxnxk,Q3=2∑n=1d∑k=1dk≠nαkuxnxnxnxnxk,Q4=2∑n=1d∑k=1dk≠nαkuxnxnxkxkxk,Q5=2∑n=1d∑k=1dk≠nuxnxnxnxnxkxk,and the following Lemmas 7-13 are needed.Lemma 7.
LetR1u be defined in (37) and Qkk=1,⋯,5 be defined in (42), then we have
(43)R1u=S1u+h224Q1+2Q2−Q3−2Q4+Q5+12F1+Oh4,where
(44)S1u=12∑n=1d∑k=1dk≠n−δxkxkδxnxn+αkδxkδxnxn+αnδxkxkδxn−αnαkδxkδxnu−12∑n=1dαn2δxnxnu,(45)F1=∑n=1d−fxnxn+αnfxn.Proof.
Differentiating both sides of (1) once and twice with respect to xnn=1,⋯,d, respectively, gives
(46)−∑k=1duxkxkxn+∑k=1dαkuxkxn=fxn,−∑k=1duxkxkxnxn+∑k=1dαkuxkxnxn=fxnxn,which leads to
(47)uxnxnxn=−∑k=1dk≠nuxkxkxn+∑k=1dαkuxkxn−fxn,(48)uxnxnxnxn=−∑k=1dk≠nuxkxkxnxn+∑k=1dαkuxkxnxn−fxnxn.Using (47) and (48) to eliminate the terms uxnxnxn and uxnxnxnxn in (37) yields
(49)R1u=−12∑n=1d∑k=1dk≠nuxkxkxnxn+12∑n=1d∑k=1dk≠nαkuxkxnxn+αnuxkxkxn−12∑n=1d∑k=1dαkαnuxkxn+12∑n=1d−fxnxn+αnfxn.Taking account of the centered difference approximations(50)uxnxn=δxnxnu−h212uxnxnxnxn+Oh4,uxkxn=δxkδxnu−h26uxkxkxkxn+uxkxnxnxn+Oh4,uxkxkxn=δxkxkδxnu−h212uxkxkxkxkxn+2uxkxkxnxnxn+Oh4,uxkxkxnxn=δxkxkδxnxnu−h212uxkxkxnxnxnxn+uxkxkxkxkxnxn+Oh4,with n≠k, and noting the definition of S1u given by (44), then we have
(51)R1u=h224∑n=1d∑k=1dk≠nuxkxkxnxnxnxn+uxkxkxkxkxnxn−αkuxnxnxnxnxk+2uxnxnxkxkxk−αnuxkxkxkxkxn+2uxkxkxnxnxn+2αnαkuxkxkxkxn+uxkxnxnxnu+h224∑n=1dαn2uxnxnxnxn+S1u+12F1=h224Q1+2Q2−Q3−2Q4+Q5+S1u+12F1+Oh4.Therefore, the proof is completed when conditions (42) and (50) are satisfied.Lemma 8.
LetR2u be defined by (38) and Qkk=1,⋯,5 be defined by (42), then
(52)R2u=−164Q1+2Q2−Q3−2Q4+Q5+13F2,where
(53)F2=∑n=1d−fxnxnxnxn+2αnfxnxnxn.Proof.
Differentiating both sides of (1) three times and four times with respect to xnn=1,⋯,d, respectively, gives rise to
(54)uxnxnxnxnxn=−fxnxnxn−∑k=1dk≠nuxkxkxnxnxn+∑k=1dαkuxkxnxnxn,(55)uxnxnxnxnxnxn=−fxnxnxnxn+∑k=1dk≠n−uxkxkxnxnxnxn+αkuxkxnxnxnxn+αnuxnxnxnxnxn.Using (54) and (55) to eliminate the terms uxnxnxn and uxnxnxnxn in (38) yields
(56)R2u=13∑n=1d∑k=1dk≠n−uxnxnxnxnxkxk+αkuxnxnxnxnxk+2αnuxnxnxnxkxk−2αnαkuxnxnxnxk−23∑n=1dαn2uxnxnxnxn+13F2=−164Q1+2Q2−Q3−2Q4+Q5+13F2,which finishes the proof.Lemma 9.
LetRu be defined by (36) and Qii=1,⋯,5 be defined by (42), S1u,F1, and F2 be defined by (44), (45), and (53) then
(57)Ru=h4720Q1+h41802Q2−Q3−2Q4+Q5+h26S1u+h26F1+h4360F2.Proof.
The proof can be easily done by putting Lemma7, Lemma 8., and the definition of Ru together.Now, we need to represent the termsQii=1,⋯,5 by using compact finite difference approximations.Lemma 10.
LetQ1 be defined by (42), then
(58)Q1=S2u+S3u−F3+Oh2,where
(59)S2u=∑n=1d∑k=1dk≠nαn2αkδxkδxnxn−αnδxkxkδxn−δxkxkδxnxnu,(60)S3u=∑n=1d∑k=1dαn3αkδxkxnu,(61)F3=∑n=1dαn2fxnxn+αnfxn.Proof.
Substituting (47) into (48) yields
(62)uxnxnxnxn=∑k=1dk≠n−uxkxkxnxn+αkuxkxnxn−αnuxkxkxn+αn∑k=1dαkuxkxn−fxnxn+αnfxn.Multiplying both sides of the above equation byαn and then summing them over n, we can rewrite Q1 defined in (42) as the form
(63)Q1=∑n=1dαn2∑k=1dk≠nαkuxkxnxn−αnuxkxkxn−uxkxkxnxn+∑n=1dαn2∑k=1dαkαnuxkxn−∑n=1dαn2fxnxn+αnfxn.Therefore, the expected result can be easily obtained by replacing the central difference approximations (13) in (63).Lemma 11.
LetQ2 be defined by (42),then we have
(64)Q2=S4u+S5u−F4+Oh2,where
(65)S4u=∑n=1d∑k=1dk≠nαn2αkδxnxnδxku,(66)S5u=12∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠k−αkαmδxnxnδxkδxm+αnαkαmδxnδxkδxmu,(67)F4=12∑n=1d∑k=1dk≠nαnαkfxnxk.Proof.
Differentiating both sides of (1) with respect to xmxk,k≠m,m,k=1,⋯,d yields
(68)−∑n=1d∑k=1dk≠nuxnxnxnxk−12∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kuxnxnxmxk+∑n=1d∑k=1dk≠nαnuxnxnxk+12∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαnuxnuxkuxm=12∑m=1d∑k=1dk≠mk≠mfxmxk.Multiplying both sides of the above equation byαmαk yields
(69)Q2=−12∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαmαkuxnxnxmxk+12∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαnαkαmuxnuxkuxm+∑n=1d∑k=1dk≠nαn2αkuxnxnxk−12∑m=1d∑k=1dk≠mαmαkfxmxk.By using the centered difference approximations (13), we can obtain the results.Lemma 12.
LetQ2,Q3, and Q4 be defined in (42), then we have
(70)Q2−Q3−Q4=−S6u+S7u+F5+Oh2,where
(71)S6u=∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαkδxkδxnxn+αnδxnδxkxkδxmxm−αmαkδxnxnδxk+αnδxnδxkxkδxmu,(72)S7u=2∑n=1d∑k=1dk≠nαn2δxnxnδxkxku,(73)F5=∑n=1d∑k=1dk≠nαkfxnxnxk+αnfxnxkxk.Proof.
By using the definitions ofQ2,Q3, and Q4 in (42), we have
(74)Q3+Q4=∑n=1d∑k=1dk≠nαkuxnxn+uxkxkxnxnxk+αnuxnxn+uxkxkxnxkxk=−F5+Q2+∑n=1d∑k=1dk≠nαk2+αn2uxnxnxkxk+∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαmαkuxmxnxnxk+αnuxmxnxkxk−∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαkuxmxmxnxnxk+αnuxmxmxnxkxk.
By using the centered difference approximations (13), we can obtain the results.Lemma 13.
LetQ4 and Q5 be defined in (42), then we have
(75)Q5−Q4=S8u−F6+Oh2,where
(76)S8u=∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαmδxmδxnxnδxkxk−δxmxmδxnxnδxkxku,(77)F6=∑n=1d∑k=1dk≠nfxnxnxkxk.Proof.
Differentiating both sides of (1) twice with respect to xn and then twice with respect to xk,k≠n, and then summing them over n and k gives
(78)∑n=1d∑k=1dk≠n∑m=1d−uxmxmxnxnxkxk+αmuxmxnxnxkxk=∑n=1d∑k=1dk≠nfxnxnxkxk,which can be rewritten as
(79)∑n=1d∑k=1dk≠n−uxnxnxnxnxkxk+αnuxnxnxnxkxk−uxkxkxnxnxkxk+αkuxkxnxnxkxk+∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠k−uxmxmxnxnxkxk+αmuxmxnxnxkxk=∑n=1d∑k=1dk≠nfxnxnxkxk.By using the definition ofQ4 and Q5 defined in (42), we have
(80)Q5−Q4=∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠kαmuxmxnxnxkxk−uxmxmxnxnxkxk−∑n=1d∑k=1dk≠nfxnxnxkxk.By using the centered difference approximations (13),we can obtain the results.Proof of Theorem 6.
The proof can be easily done by putting Lemmas9–13 and the definition of F∗ in (40) together.


## 3.2. Sixth-Order CFD Schemes for Diffusion Equations with Constant Convective Coefficients

It should be noted that the sixth-order CFD scheme given in (41) is valid for any dimension. For simplicity, we write down here the most commonly used schemes for (1), i.e., for 2 and 3 dimensions, in a concise way.
(i)
Ifd=2, the terms including the summation symbol ∑n=1d∑k=1dk≠n∑m=1dm≠n,m≠k in Si,i=1,⋯,8, i.e., S5,S6,S7, and S8, vanish. Therefore, the sixth-order CFD scheme can be described as
(81)∑n=12∑k=12k≠ncnkδxnδxk+cnkkδxnδxkxk+cnnkkδxnxnδxkxku+∑n=12cnδxnu+∑n=12cnnδxnxnu=f−F∗,where
(82)cn=αn,cnn=1+αn2h2121−αn2h260,cnk=αnαkh2121−αn2+αk2h2120,cnkk=αnh261−αn2−9αk2h2120,cnnkk=h212+αn2+αk2h4160,and F∗ appeared in (40).(ii)
Ifd=3, all of the Si,i=1,⋯,8 exist, and the sixth-order CFD scheme can be described as(83)∑n=13∑k=13k≠ncnkδxnδxk+cnkkδxnδxkxk+cnnkkδxnδxkxku+∑n=13anδxn+annδxnxnu+c123δx1δx2δx3+c112233δx1x1δx2x2δx3x3u+c1123δx1x1δx2δx3+c1223δx1δx2x2δx3+c1233δx1δx2δx3x3u+c12233δx1δx2x2δx3x3+c11233δx1x1δx2δx3x3+c11223δx1x1δx2x2δx3u=f−F∗,where the forms of cn,cnn,cnk,cnkk,cnnkk,andk≠n are the same as that defined in (82), except in this case n=1,2,3 and
(84)c123=α1α2α3h430,c112233=h430,cmnkk=αmαnh490,cmnnkk=αmh490,where m,n,k=1,2,3, but they are different from each other.


## 4. Numerical Results

The CFD schemes derived in Section2.1 for 2D and 3D convection-diffusion equations are discussed in literature [16, 17], respectively, and the corresponding numerical results have been completed to established. Hence, two examples in this filed are shown to validate the accuracy of the CFD schemes proposed in Section 2.2 and Section 3.2.Example 14.
In this example, the diffusive coefficient is regarded asκx1,⋯,xd=e∑i=1dxi, and then, the forcing function f and the Dirichlet boundary conditions are prescribed to satisfy the exact solution ux1,⋯,xd=∏i=1dsinπxi. To exhibit the numerical results, we denote two schemes as
(i)
SchemeI: (28) with coefficients (30)(ii)
SchemeII: (32) with coefficients (34)
As the computation of the coefficients, we take the following two cases:(i)
Case 1: coefficients (30) and (34) computed analytically(ii)
Case 2: coefficients (30) and (34) computed by the approximation introduced in Remark 3.The information relevant to maximum errors and convergence rates are listed in Tables1 and 2 for 2D and 3D problems, severally.Table 1
Numerical results for the fourth-order CFD method in 2D.
hSchemeISchemeIICase 1Case 2Case 1Case 2ErrorRatioErrorRatioErrorRatioErrorRatio1/46.613E-031.279E-036.613E-031.270E-031/83.995E-044.048.355E-053.933.994E-044.048.341E-053.921/162.557E-053.965.273E-063.982.556E-053.965.271E-063.981/321.594E-064.003.330E-073.981.594E-064.003.330E-073.981/649.958E-084.002.082E-083.999.958E-064.002.082E-083.99Table 2
Numerical results for the fourth-order CFD method in 3D.
hSchemeISchemeIICase 1Case 2Case 1Case 2ErrorRatioErrorRatioErrorRatioErrorRatio1/41.294E-024.226E-031.294E-024.211E-031/87.523E-044.102.560E-044.047.524E-044.102.558E-044.041/164.740E-053.981.587E-054.014.740E-053.981.587E-054.011/322.954E-064.009.898E-074.002.955E-064.009.898E-074.001/641.849E-073.996.192E-083.991.850E-073.996.192E-083.99From these tables, it offers a brief result that both two schemes are published in almost the same errors with a fourth-order convergence rate. Although the errors in essence were affected by the ways of computing the coefficients, the consequence of convergence rate still remainsOh4.Example 15.
An example to validate the accuracy of the CFD schemes is proposed in Section3.2, considering the convection-diffusion equation, constant convection vector coefficients α is regarded as Re·e, where e is d-dimensional vector with all unit elements. The forcing function fx1,⋯,xd and the Dirichlet boundary conditions are prescribed to satisfy the exact solution ux1,⋯,xd=∏i=1dsinπxi. Re is a representative of the constant conducted to reflect the ratio of the convection to diffusion and simulate the Reynolds number.For the differentRe, the refinement of the mesh size h is undertaken to observe the convergence rate. The maximum errors and the convergence rate of 2D and 3D problems are listed as 1≤Re≤103 in Tables 3 and 4 and in Figure 1, clearly.Table 3
Numerical results for the sixth-order CFD method in 2D withα=Re·1,1.
hRe=1Re=10Re=102Re=103ErrorRatioErrorRatioErrorRatioErrorRatio1/42.278E-045.292E-030.5348E-010.4837E-011/83.708E-065.948.547E-055.950.2204E-011.280.1380E-011.811/165.732E-086.021.339E-065.990.1437E-023.940.3918E-021.821/329.020E-105.992.093E-086.000.1925E-046.220.1078E-021.861/641.409E-116.003.271E-106.000.3043E-065.980.2463E-032.13Table 4
Numerical results for the sixth-order CFD method in 3D withα=Re·1,1,1.
hRe=1Re=10Re=102Re=103ErrorRatioErrorRatioErrorRatioErrorRatio1/41.195E-031.047E-024.286E-023.798E-021/81.814E-056.0421.589E-046.0432.303E-020.901.482E-021.361/162.924E-075.9552.542E-065.9661.795E-033.685.782E-031.361/324.550E-096.0063.985E-085.9952.865E-055.971.976E-031.551/647.106E-116.0016.251E-105.9944.547E-075.984.488E-042.14Figure 1
(a) Test of the sixth-order CFD method in 2D; (b) test of the sixth-order CFD method in 3D.
(a)(b)It has become apparent thatRe is not very large leading to the rapid decrease of errors with the refinement of the mesh size h and a sixth-order convergence rate. But with the increase of Re, the computed accuracy is reversely affected. For the large Re, only a second-order convergence rate can be achieved. It is expected that the deterioration of computed accuracy is attributed to a reason of the convection-dominated equation (1), and when the Re increases to be infinite, it approximates a first-order hyperbolic equation.


## 5. Conclusions

Compact finite-difference (CFD) schemes seem to be simple and powerful ways contributing to a deeper understanding of high accuracy and low computational cost. Compared with the traditional explicit finite-difference schemes of the same-order, the effects of compact schemes are significantly more accurate with the benefit of smaller stencil sizes. In the experimented present work, the fourth-order CFD schemes generate fresh insight into a general formulation of convection-diffusion equations with variable convective coefficients and elliptic diffusion equations with variable diffusive coefficients in any dimension. When the convective coefficients are constants, the sixth-order CFD schemes provide an important opportunity to solve convection-diffusion problems. Numerical results of preliminary work show the efficiency of the proposed CFD schemes. A significant analysis and discussion on similar approaches may be extended to elliptic diffusion problems with positive diffusive tensor in any dimension and are under investigation.



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