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A Sixth-Order Quasi-Compact Difference Scheme for Multidimensional Poisson Equations Without Derivatives of Source Term

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Abstract

Sixth-order compact difference schemes for Poisson equations have been widely investigated in the literature. Nevertheless, those methods are all constructed based on knowing the exact values of the derivatives of the source term. Therefore, this drawback mostly prevents their actual applications as the analytic form of the source term is rarely available. In this paper, we propose a sixth-order quasi-compact difference method, without having to know the derivatives of the source term, for solving the 2D and 3D Poisson equations. Our strategy is to discretize the equation by the fourth-order compact scheme at the improper interior grid points that adjoin the boundary, while the sixth-order scheme, where it is compact only for the unknowns, is exploited to the proper interior grid points that are not adjoining the boundary. Theoretically, we rigorously prove that the proposed method can achieve the global sixth-order accuracy. Since there are no derivatives of the source term involved in the proposed scheme, our global sixth-order quasi-compact difference method can be developed to solve the time-dependent problems using a time advancing scheme. Numerical experiments are carried out to demonstrate the convergence order and the efficiency of the proposed methods.

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Acknowledgements

The authors are grateful to the anonymous referees for their constructive comments which benefit this paper a lot. The third author (corresponding author) Hai-Wei Sun is supported by Science and Technology Development Fund of Macao SAR (Grant No. 0122/2020/A3) and MYRG2020-00224-FST from University of Macau,  and the fourth author Chengjian Zhang is supported by NSFC (Grant No. 11971010).

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Macau, Macao, China

    Tao Sun & Hai-Wei Sun

  2. Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan, China

    Zhi Wang

  3. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, China

    Chengjian Zhang

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  1. Tao Sun
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  2. Zhi Wang
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  3. Hai-Wei Sun
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  4. Chengjian Zhang
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Appendices

Appendix 1: Details for Poisson Equations

1.1 2D Poisson Equation

For simplify, for \(\varphi \in \{U,f\}\), we introduce notations as follows:

$$\begin{aligned} \begin{array}{l} {\varphi _{CI}} = {\varphi _{i + 1,j}} + {\varphi _{i - 1,j}}, {\varphi _{CJ}} = {\varphi _{i,j + 1}} + {\varphi _{i,j - 1}},\\ {\varphi _{DI}} = {\varphi _{i + 2,j}} + {\varphi _{i - 2,j}}, {\varphi _{DJ}} = {\varphi _{i,j + 2}} + {\varphi _{i,j - 2}},\\ {\varphi _T} = {\varphi _{i + 1,j + 1}} + {\varphi _{i + 1,j - 1}} + {\varphi _{i - 1,j + 1}} + {\varphi _{i - 1,j - 1}}. \end{array} \end{aligned}$$
(6.1)

Then the QCD scheme (2.25) for 2D Poisson problem (2.1) can be stated as following pointwise format:

$$\begin{aligned} \left\{ \begin{array}{ll} -{\mathscr {L}}_{2,h}U_{i,j}={\mathscr {H}}_{2,h}f_{i,j},&{}\quad (x_i,y_j)\in \Omega _{2,h},\\ U_{i,j}=g_{i,j},&{}\quad (x_i,y_j)\in \partial \Omega _{2,h},\\ \end{array}\right. \end{aligned}$$
(6.2)

in which \(U_{i,j}\) is the approximation of the exact solution for 2D Poisson problem (2.1) in \((x_i,y_j)\), \(f_{i,j}:=f(x_i,y_j)\) and \(g_{i,j}:=g(x_i,y_j)\); the compact difference operator \({\mathscr {L}}_{2,h}\) for \(U_{i,j}\) is given by

$$\begin{aligned} {\mathscr {L}}_{2,h}U_{i,j}:=- \frac{{10}}{{3{h^2}}}{U_{i,j}} + \frac{2}{{3{h^2}}}({U_{CI}} + {U_{CJ}}) + \frac{1}{{6{h^2}}}{U_T}, \end{aligned}$$
(6.3)

and difference operator \({\mathscr {H}}_{2,h}\) for source term \(f_{i,j}\) is expressed as

$$\begin{aligned}{\mathscr {H}}_{2,h}f_{i,j}:= \left\{ \begin{array}{ll} \frac{{119}}{{180}}{f_{i,j}} + \frac{7}{{90}}\left( {f_{CI}} + {f_{CJ}}\right) - \frac{1}{{240}}\left( {f_{DI}} + {f_{DJ}}\right) + \frac{1}{{90}}{f_T},&{} (x_i,y_j)\in \mathring{\Omega }_{2,h},\\ \frac{{247}}{{360}}{f_{i,j}} + \frac{{11}}{{180}}{f_{CI}} + \frac{7}{{90}}{f_{CJ}} - \frac{1}{{240}}{f_{DJ}} + \frac{1}{{90}}{f_T},&{} (x_i,y_j)\in \Omega ^{*,x}_{2,h},\\ \frac{{247}}{{360}}{f_{i,j}} + \frac{7}{{90}}{f_{CI}} + \frac{{11}}{{180}}{f_{CJ}} - \frac{1}{{240}}{f_{DI}} + \frac{1}{{90}}{f_T},&{}(x_i,y_j)\in \Omega ^{*,y}_{2,h},\\ \frac{{32}}{{45}}{f_{i,j}} + \frac{{11}}{{180}}\left( {f_{CI}} + {f_{CJ}}\right) + \frac{1}{{90}}{f_T},&{} (x_i,y_j)\in \Omega ^{*,*}_{2,h}.\end{array} \right. \end{aligned}$$

1.2 3D Poisson Equation

For \(\varphi \in \{U,f\}\), introducing new notations as follows:

$$\begin{aligned} \begin{array}{l} {\varphi _{CI}} = {\varphi _{i + 1,j,l}} + {\varphi _{i - 1,j,l}},\quad {\varphi _{CJ}} = {\varphi _{i,j + 1,l}} + {\varphi _{i,j - 1,l}},\quad {\varphi _{CL}} = {\varphi _{i,j,l+1}} + {\varphi _{i,j,l- 1}},\\ {\varphi _{DI}} = {\varphi _{i + 2,j,l}} + {\varphi _{i - 2,j,l}},\quad {\varphi _{DJ}} = {\varphi _{i,j + 2,l}} + {\varphi _{i,j - 2,l}},\quad {\varphi _{DL}} = {\varphi _{i,j,l+2}} + {\varphi _{i,j,l- 2}},\\ {\varphi _T} = {\varphi _{i + 1,j + 1,l}}+ {\varphi _{i + 1,j - 1,l}}+ {\varphi _{i-1,j+1 ,l}}+ {\varphi _{i-1,j-1 ,l}} + {\varphi _{i+1,j ,l+1}}+ {\varphi _{i+1,j ,l-1}} \\ \quad \quad \quad + {\varphi _{i-1,j ,l+1}}+ {\varphi _{i-1,j ,l-1}}+{\varphi _{i ,j + 1,l+1}}+ {\varphi _{i,j + 1,l-1}} + {\varphi _{i,j- 1,l+1}}+ {\varphi _{i,j- 1,l-1}},\\ {\varphi _V} = {\varphi _{i + 1,j + 1,l+1}}+ {\varphi _{i -1,j + 1,l+1}} + {\varphi _{i + 1,j - 1,l+1}} + {\varphi _{i + 1,j + 1,l-1}} + {\varphi _{i - 1,j - 1,l+1}} \\ \quad \quad \quad + {\varphi _{i - 1,j + 1,l-1}} + {\varphi _{i + 1,j - 1,l-1}}+ {\varphi _{i -1,j - 1,l-1}}. \end{array} \end{aligned}$$
(6.4)

Then the QCD scheme (2.37) for 3D Poisson problem (2.26) can be stated as following pointwise format:

$$\begin{aligned} \left\{ \begin{array}{ll} -{\mathscr {L}}_{3,h}U_{i,j,l}={\mathscr {H}}_{3,h}f_{i,j,l},&{}\quad (x_i,y_j,z_l)\in \Omega _{3,h},\\ U_{i,j,l}=g_{i,j,l},&{}\quad (x_i,y_j,z_l)\in \partial \Omega _{3,h},\\ \end{array}\right. \end{aligned}$$
(6.5)

in which \(U_{i,j,l}\) is the approximation of the exact solution for 3D Poisson problem (2.26) in \((x_i,y_j,z_l)\), \(f_{i,j,l}:=f(x_i,y_j,z_l)\) and \(g_{i,j,l}:=g(x_i,y_j,z_l)\); the compact difference operator \({\mathscr {L}}_{3,h}\) for \(U_{i,j,l}\) is given by

$$\begin{aligned} {\mathscr {L}}_{3,h}U_{i,j,l}:= - \frac{{64}}{{15{h^2}}}{U_{i,j,l}} + \frac{7}{{15{h^2}}}({U_{CI}} + {U_{CJ}} + {U_{CL}}) + \frac{1}{{10{h^2}}}{U_T} + \frac{1}{{30{h^2}}}{U_V},\nonumber \\ \end{aligned}$$
(6.6)

and the difference operator \({\mathscr {H}}_{3,h}\) for source term \(f_{i,j,l}\) is expressed as

$$\begin{aligned} \begin{aligned}&{\mathscr {H}}_{3,h}f_{i,j,l}\!:=\\&\left\{ \!\! \begin{array}{ll} \frac{{67}}{{120}}{f_{i,j,l}} \!+ \!\frac{{1}}{{18}}\left( {f_{CI}}\! +\! {f_{CJ}}\! +\! {f_{CK}}\right) \! - \!\frac{1}{{240}}\left( {f_{DI}}\! + \!{f_{DJ}} \!+\! {f_{DL}}\right) \! +\! \frac{1}{{90}}{f_T}, &{} (x_i,y_j,z_l)\in \mathring{\Omega }_{3,h},\\ \frac{{7}}{{12}}{f_{i,j,l}} + \frac{{7}}{{180}}{f_{CI}}\! + \!\frac{{1}}{{18}}\left( {f_{CJ}} \!+ \!{f_{CL}}\right) \! -\! \frac{1}{{240}}\left( {f_{DJ}} \!+\! {f_{DL}}\right) \! + \!\frac{1}{{90}}{f_T}, &{}(x_i,y_j,z_l)\in \Omega ^{*,x}_{3,h},\\ \frac{{7}}{{12}}{f_{i,j,l}}\! +\! \frac{{7}}{{180}}{f_{CJ}}\! +\! \frac{{1}}{{18}}\left( {f_{CI}} \!+\! {f_{CL}}\right) \! -\! \frac{1}{{240}}\left( {f_{DI}} \!+ \!{f_{DL}}\right) \! +\! \frac{1}{{90}}{f_T},&{}(x_i,y_j,z_l)\in \Omega ^{*,y}_{3,h},\\ \frac{{7}}{{12}}{f_{i,j,l}}\! +\! \frac{{7}}{{180}}{f_{CL}}\! +\! \frac{{1}}{{18}}\left( {f_{CI}}\! +\! {f_{CJ}}\right) \!-\! \frac{1}{{240}}\left( {f_{DI}} \!+ \!{f_{DJ}}\right) \! + \frac{1}{{90}}{f_T},&{}(x_i,y_j,z_l)\in \Omega ^{*,z}_{3,h},\\ \frac{{73}}{{120}}{f_{i,j,l}}\! +\! \frac{{7}}{{180}}\left( {f_{CI}}\! + \!{f_{CJ}}\right) \!+ \!\frac{{1}}{{18}}{f_{CL}} \!- \!\frac{1}{{240}}{f_{DL}}+ \frac{1}{{90}}{f_T},&{}(x_i,y_j,z_l)\in \Omega ^{*,x,y}_{3,h},\\ \frac{{73}}{{120}}{f_{i,j,l}}\! +\! \frac{{7}}{{180}}\left( {f_{CI}}\! + \!{f_{CL}}\right) \!+ \!\frac{{1}}{{18}}{f_{CJ}}\! -\! \frac{1}{{240}}{f_{DJ}} + \frac{1}{{90}}{f_T},&{} (x_i,y_j,z_l)\in \Omega ^{*,x,z}_{3,h},\\ \frac{{73}}{{120}}{f_{i,j,l}}\! +\! \frac{{7}}{{180}}\left( {f_{CJ}}\! +\! {f_{CL}}\right) \! + \!\frac{{1}}{{18}}{f_{CI}}\! - \!\frac{1}{{240}}{f_{DI}} +\frac{1}{{90}}{f_T} ,&{}(x_i,y_j,z_l)\in \Omega ^{*,y,z}_{3,h},\\ \frac{{19}}{{30}}{f_{i,j,l}}\! + \!\frac{{7}}{{180}}\left( {f_{CI}}\! +\! {f_{CJ}} + {f_{CL}}\right) + \frac{1}{{90}}{f_T} , &{} (x_i,y_j,z_l)\in \Omega ^{*,*}_{3,h}. \end{array} \right. \end{aligned} \end{aligned}$$

Appendix 2: Details for Helmholtz Equations

1.1 2D Helmholtz Equation

Combining the notations (6.1), the QCD scheme (4.7) for 2D Helmholtz equation can be presented as the following pointwise format:

$$\begin{aligned} \left\{ \begin{array}{ll} {\mathscr {L}}_{2,h,\lambda }U_{i,j}={\mathscr {H}}_{2,h,\lambda }f_{i,j},&{}\quad (x_i,y_j)\in \Omega _{2,h},\\ U_{i,j}=g_{i,j},&{}\quad (x_i,y_j)\in \partial \Omega _{2,h},\\ \end{array}\right. \end{aligned}$$
(6.7)

in which \(U_{i,j}\) is the approximation of the exact solution for 2D Helmholtz problem (4.14) in \((x_i,y_j)\), \(f_{i,j}:=f(x_i,y_j)\) and \(g_{i,j}:=g(x_i,y_j)\); the compact difference operator \({\mathscr {L}}_{2,h,\lambda }\) for \(U_{i,j}\) is given by

$$\begin{aligned}{\mathscr {L}}_{2,h,\lambda }U_{i,j}:= & {} \left( - \frac{{10}}{{3h^2}}+\frac{{46\lambda }}{{45}}-\frac{{\lambda ^2 h^2}}{{12}}+\frac{{\lambda ^3h^4}}{{360}}\right) {U_{i,j}}+ \left( \frac{2}{{3h^2}}-\frac{{\lambda }}{{90}}\right) \left( {U_{CI}} + {U_{CJ}}\right) \\&+ \left( \frac{1}{{6h^2}}+\frac{{\lambda }}{{180}}\right) {U_T}, \end{aligned}$$

and the difference operator \({\mathscr {H}}_{2,h,\lambda }\) for source term \(f_{i,j}\) is expressed as

$$\begin{aligned} \begin{aligned}&{\mathscr {H}}_{2,h,\lambda }f_{i,j}\!:=\\&\left\{ \!\!\! \begin{array}{ll} \left( \frac{{119}}{{180}} - \frac{{5\lambda {h^2}}}{{72}} + \frac{{{\lambda ^2}{h^4}}}{{360}}\right) {f_{i,j}} + \left( \frac{7}{{90}} - \frac{{\lambda {h^2}}}{{270}}\right) \left( {f_{CI}} + {f_{CJ}}\right) - \left( \frac{1}{{240}} - \frac{{\lambda {h^2}}}{{4320}}\right) \left( {f_{DI}} + {f_{DJ}}\right) + \frac{1}{{90}}{f_T},&{} (x_i,y_j)\!\in \!\mathring{\Omega }_{2,h},\\ \left( \frac{{247}}{{360}} - \frac{{17\lambda {h^2}}}{{240}} + \frac{{{\lambda ^2}{h^4}}}{{360}}\right) {f_{i,j}} + \left( \frac{{11}}{{180}} - \frac{{\lambda {h^2}}}{{360}}\right) {f_{CI}} + \left( \frac{7}{{90}} - \frac{{\lambda {h^2}}}{{270}}\right) {f_{CJ}} - \left( \frac{1}{{240}} - \frac{{\lambda {h^2}}}{{4320}}\right) {f_{DJ}} + \frac{1}{{90}}{f_T},&{} (x_i,y_j)\!\in \!\Omega ^{*,x}_{2,h},\\ \left( \frac{{247}}{{360}} - \frac{{17\lambda {h^2}}}{{240}} + \frac{{{\lambda ^2}{h^4}}}{{360}}\right) {f_{i,j}} + \left( \frac{7}{{90}} - \frac{{\lambda {h^2}}}{{270}}\right) {f_{CI}} + \left( \frac{{11}}{{180}} - \frac{{\lambda {h^2}}}{{360}}\right) {f_{CJ}} - \left( \frac{1}{{240}} - \frac{{\lambda {h^2}}}{{4320}}\right) {f_{DI}} + \frac{1}{{90}}{f_T},&{} (x_i,y_j)\!\in \!\Omega ^{*,y}_{2,h}\\ \left( \frac{{32}}{{45}} - \frac{{13\lambda {h^2}}}{{180}} + \frac{{{\lambda ^2}{h^4}}}{{360}}\right) {f_{i,j}} + \left( \frac{{11}}{{180}} - \frac{{\lambda {h^2}}}{{360}}\right) \left( {f_{CI}} + {f_{CJ}}\right) + \frac{1}{{90}}{f_T},&{} (x_i,y_j)\!\in \!\Omega ^{*,*}_{2,h}. \end{array}\right. \end{aligned} \end{aligned}$$

1.2 3D Helmholtz Equation

Using the notations (6.4), the QCD scheme (4.13) for 3D Helmholtz problem (4.1) can be rewritten as the following pointwise format:

$$\begin{aligned} \left\{ \begin{array}{ll} {\mathscr {L}}_{3,h,\lambda }U_{i,j,l}={\mathscr {H}}_{3,h,\lambda }f_{i,j,l},&{}\quad (x_i,y_j,z_l)\in \Omega _{3,h},\\ U_{i,j,l}=g_{i,j,l},&{}\quad (x_i,y_j,z_l)\in \partial \Omega _{3,h},\\ \end{array}\right. \end{aligned}$$
(6.8)

in which \(U_{i,j,l}\) is the approximation of the exact solution of 3D Helmholtz equation in \((x_i,y_j,z_l)\), \(f_{i,j,l}:=f(x_i,y_j,z_l)\) and \(g_{i,j,l}:=g(x_i,y_j,z_l)\); the compact difference operator \({\mathscr {L}}_{3,h,\lambda }\) for \(U_{i,j,l}\) is given by

$$\begin{aligned}{\mathscr {L}}_{3,h,\lambda }U_{i,j,l}:= & {} \left( - \frac{{64}}{{15h^2}}+\frac{{16\lambda }}{{15}}-\frac{{\lambda ^2 h^2}}{{12}}+\frac{{\lambda ^3h^4}}{{360}}\right) {U_{i,j,l}}\\&+ \left( \frac{7}{{15h^2}}-\frac{{\lambda }}{{45}}\right) \left( {U_{CI}} + {U_{CJ}}+ {U_{CL}}\right) + \left( \frac{1}{{10h^2}}+\frac{{\lambda }}{{180}}\right) {U_T}+\frac{1}{{30h^2}}{U_V},\end{aligned}$$

and the difference operator \({\mathscr {H}}_{3,h,\lambda }\) for source term \(f_{i,j,l}\) is expressed as

$$\begin{aligned} \begin{array}{ll} {\mathscr {H}}_{3,h,\lambda }f_{i,j,l}:=\\ \!\!\left\{ \!\!\!\begin{array}{ll} \left( \frac{{67}}{{120}}\! -\! \frac{{\lambda {h^2}}}{{16}}\! + \!\frac{{{\lambda ^2}{h^4}}}{{360}}\right) {f_{i,j,l}} \!+\! \left( \frac{{1}}{{18}} - \frac{{\lambda {h^2}}}{{270}}\right) \left( {f_{CI}} \!+ \!{f_{CJ}}\! + \!{f_{CK}}\right) \! -\! \left( \frac{1}{{240}}\! - \frac{{\lambda {h^2}}}{{4320}}\right) \left( {f_{DI}} \!+ \!{f_{DJ}}\! + \!{f_{DL}}\right) \!+ \!\frac{1}{{90}}{f_T},&{} (x_i,y_j,z_l)\!\in \!\mathring{\Omega }_{3,h},\\ \left( \frac{{7}}{{12}}\! - \!\frac{{23\lambda {h^2}}}{{360}}\! + \!\frac{{{\lambda ^2}{h^4}}}{{360}}\right) {f_{i,j,l}}\! + \!\left( \frac{{7}}{{180}}\! - \! \frac{{\lambda {h^2}}}{{360}}\right) {f_{CI}}\!+ \!\left( \frac{{1}}{{18}} \!- \frac{{\lambda {h^2}}}{{270}}\right) \left( {f_{CJ}}\! +\! {f_{CL}}\right) \!- \!\left( \frac{1}{{240}} \!-\! \frac{{\lambda {h^2}}}{{4320}}\right) \left( {f_{DJ}} \!+ \!{f_{DL}}\right) \! + \!\frac{1}{{90}}{f_T},&{}(x_i,y_j,z_l)\!\in \!\Omega ^{*,x}_{3,h},\\ \left( \frac{{7}}{{12}} \!-\! \frac{{23\lambda {h^2}}}{{360}}\! +\! \frac{{{\lambda ^2}{h^4}}}{{360}}\right) {f_{i,j,l}} \!+ \!\left( \frac{{7}}{{180}}\! -\! \frac{{\lambda {h^2}}}{{360}}\right) {f_{CJ}}\! +\! \left( \frac{{1}}{{18}}\! -\! \frac{{\lambda {h^2}}}{{270}}\right) \left( {f_{CI}} \!+ \!{f_{CL}}\right) \!-\! \left( \frac{1}{{240}}\! - \!\frac{{\lambda {h^2}}}{{4320}}\right) \left( {f_{DI}}\! +\! {f_{DL}}\right) \! + \! \frac{1}{{90}}{f_T},&{}(x_i,y_j,z_l)\!\in \!\Omega ^{*,y}_{3,h},\\ \left( \frac{{7}}{{12}} \!- \!\frac{{23\lambda {h^2}}}{{360}} + \frac{{{\lambda ^2}{h^4}}}{{360}}\right) {f_{i,j,l}}\! +\! \left( \frac{{7}}{{180}} \!- \! \frac{{\lambda {h^2}}}{{360}}\right) {f_{CL}}\! +\! \left( \frac{{1}}{{18}}\!- \!\frac{{\lambda {h^2}}}{{270}}\right) \left( {f_{CI}}\! +\! {f_{CJ}}\right) \! - \! \left( \frac{1}{{240}} \!-\! \frac{{\lambda {h^2}}}{{4320}}\right) \left( {f_{DI}} \!+ \!{f_{DJ}}\right) \! + \frac{1}{{90}}{f_T},&{}(x_i,y_j,z_l)\!\in \!\Omega ^{*,z}_{3,h},\\ \left( \frac{{73}}{{120}}\! - \!\frac{{47\lambda {h^2}}}{{720}}\! + \!\frac{{{\lambda ^2}{h^4}}}{{360}}\right) {f_{i,j,l}} \!+\! \left( \frac{{7}}{{180}}\! - \! \frac{{\lambda {h^2}}}{{360}}\right) \left( {f_{CI}}\! +\! {f_{CJ}}\right) \!+\! \left( \frac{{1}}{{18}}\! -\! \frac{{\lambda {h^2}}}{{270}}\right) {f_{CL}}\! - \! \left( \frac{1}{{240}} - \frac{{\lambda {h^2}}}{{4320}}\right) {f_{DL}} \! +\! \frac{1}{{90}}{f_T},&{}(x_i,y_j,z_l)\!\in \!\Omega ^{*,x,y}_{3,h},\\ \left( \frac{{73}}{{120}} \!- \!\frac{{47\lambda {h^2}}}{{720}}\! +\! \frac{{{\lambda ^2}{h^4}}}{{360}}\right) {f_{i,j,l}}\! +\! \left( \frac{{7}}{{180}}\! -\! \frac{{\lambda {h^2}}}{{360}}\right) \left( {f_{CI}}\! +\! {f_{CL}}\right) \! + \!\left( \frac{{1}}{{18}}\! -\! \frac{{\lambda {h^2}}}{{270}}\right) {f_{CJ}} - \left( \frac{1}{{240}}\! - \!\frac{{\lambda {h^2}}}{{4320}}\right) {f_{DJ}}\! +\! \frac{1}{{90}}{f_T},&{}(x_i,y_j,z_l)\!\in \!\Omega ^{*,x,z}_{3,h},\\ \left( \frac{{73}}{{120}}\! -\! \frac{{47\lambda {h^2}}}{{720}} + \frac{{{\lambda ^2}{h^4}}}{{360}}\right) {f_{i,j,l}}\! +\! \left( \frac{{7}}{{180}} - \frac{{\lambda {h^2}}}{{360}}\right) \left( {f_{CJ}}\! +\! {f_{CL}}\right) \! +\! \left( \frac{{1}}{{18}}\! -\! \frac{{\lambda {h^2}}}{{270}}\right) {f_{CI}}\! -\! \left( \frac{1}{{240}}\! -\! \frac{{\lambda {h^2}}}{{4320}}\right) {f_{DI}}\!+\! \frac{1}{{90}}{f_T},&{}(x_i,y_j,z_l)\!\in \!\Omega ^{*,y,z}_{3,h},\\ \left( \frac{{19}}{{30}} \!-\! \frac{{\lambda {h^2}}}{{15}}\! +\! \frac{{{\lambda ^2}{h^4}}}{{360}}\right) {f_{i,j,l}}\! +\! \left( \frac{{7}}{{180}}\! -\! \frac{{\lambda {h^2}}}{{360}}\right) \left( {f_{CI}}\! +\! {f_{CJ}}\! + \!{f_{CL}}\right) \!+ \!\frac{1}{{90}}{f_T},&{}(x_i,y_j,z_l)\!\in \!\Omega ^{*,*}_{3,h}. \end{array} \right. \end{array} \end{aligned}$$

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Sun, T., Wang, Z., Sun, HW. et al. A Sixth-Order Quasi-Compact Difference Scheme for Multidimensional Poisson Equations Without Derivatives of Source Term. J Sci Comput 93, 45 (2022). https://doi.org/10.1007/s10915-022-02003-6

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