Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Data Availibility

Enquiries about data availability should be directed to the authors.

References

  1. Karniadakis, G., Israeli, M., Orszag, S.: High-order splitting methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 97, 414–443 (1991)

    Article  MathSciNet  Google Scholar 

  2. Golub, G., Huang, L., Simon, H., Tang, W.: A fast Poisson solver for the finite difference solution of the incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 19, 1606–1624 (1998)

    Article  MathSciNet  Google Scholar 

  3. Gupta, M.M., Kouatchou, J., Zhang, J.: Comparison of second- and fourth-order discretizations for multigrid Poisson solvers. J. Comput. Phys. 132, 226–232 (1997)

    Article  MathSciNet  Google Scholar 

  4. Zhang, J.: Fast and high accuracy multigrid solution of the three dimensional Poisson equation. J. Comput. Phys. 143, 449–461 (1998)

    Article  MathSciNet  Google Scholar 

  5. Zhang, J.: Multigrid method and fourth-order compact scheme for 2D Poisson equation with unequal mesh-size discretization. J. Comput. Phys. 179, 170–179 (2002)

    Article  Google Scholar 

  6. Kwon, Y., Stephenson, J.W.: Single cell finite difference approximations for Poisson’s equation in three variables. Appl. Math. Notes. 2, 13–20 (1982)

    MathSciNet  MATH  Google Scholar 

  7. Wang, J., Zhong, W., Zhang, J.: A general meshsize fourth-order compact difference discretization scheme for 3D Poisson equation. Appl. Math. Comput. 183, 804–812 (2006)

    MathSciNet  MATH  Google Scholar 

  8. Ge, Y.: Multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D Poisson equation. J. Comput. Phys. 229, 6381–6391 (2010)

    Article  MathSciNet  Google Scholar 

  9. Wang, H., Zhang, Y., Ma, X., Qiua, J., Liang, Y.: An efficient implementation of fourth-order compact finite difference scheme for Poisson equation with Dirichlet boundary conditions. Comput. Math. Appl. 71, 1843–1860 (2016)

    Article  MathSciNet  Google Scholar 

  10. Spotz, W.F., Carey, G.F.: A high-order compact formulation for the 3D Poisson equation. Numer. Meth. Part Differ. Equ. 12, 235–243 (1996)

    Article  Google Scholar 

  11. Zhai, S., Feng, X., He, Y.: A family of fourth-order and sixth-order compact difference schemes for the three-dimensional Poisson equation. J. Sci. Comput. 54, 97–120 (2013)

    Article  MathSciNet  Google Scholar 

  12. Zhai, S., Feng, X., He, Y.: A new method to deduce high-order compact difference schemes for two-dimensional Poisson equation. Appl. Math. Comput. 230, 9–26 (2014)

    MathSciNet  MATH  Google Scholar 

  13. Sutmann, G., Steffen, B.: High-order compact solvers for the three-dimensional Poisson equation. J. Comput. Appl. Math. 187, 142–170 (2006)

    Article  MathSciNet  Google Scholar 

  14. Saldanha, G.: Single cell high order difference schemes for Poisson’s equation in three variables. Appl. Math. Comput. 186, 548–557 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Dai, R., Lin, P., Zhang, J.: An efficient sixth-order solution for anisotropic Poisson equation with completed Richardson extrapolation and multiscale multigrid method. Comput. Math. Appl. 73, 1865–1877 (2017)

    Article  MathSciNet  Google Scholar 

  16. Dai, R., Lin, P., Zhang, J.: An EXCMG accelerated multiscale multigrid computation for 3D Poisson equation. Comput. Math. Appl. 77, 2051–2060 (2019)

    Article  MathSciNet  Google Scholar 

  17. Uh Zapata, M., Itzá Balam, R.: High-order implicit finite difference schemes for the two-dimensional Poisson equation. Appl. Math. Comput. 309, 222–244 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Feng, H., Zhao, S.: FFT-based high order central difference schemes for three-dimensional Poisson’s equation with various types of boundary conditions. J. Comput. Phys. 410, 109391 (2020)

    Article  MathSciNet  Google Scholar 

  19. Feng, H., Long, G., Zhao, S.: FFT-based high order central difference schemes for Poisson’s equation with staggered boundaries. J. Sci. Comput. 86, 7 (2021)

    Article  MathSciNet  Google Scholar 

  20. Ren, Y., Feng, H., Zhao, S.: A FFT accelerated high order finite difference method for elliptic boundary value problems over irregular domains. J. Comput. Phys. 448, 110762 (2022)

    Article  MathSciNet  Google Scholar 

  21. Spotz, W.F.: High-order compact scheme for computational mechanics. Presented to the faculty of the graduate school of the university of Texas at Austin in Partial Fulfillment of the requirments for the degree of Doctor of philosophy (1995)

  22. Nabavi, M., Siddiqui, M.K., Dargahi, J.: A new 9-point sixth-order accurate compact finite-difference method for the Helmholtz equation. J. Sound Vibr. 307, 972–982 (2007)

    Article  Google Scholar 

  23. Sutmann, G.: Compact finite difference schemes of sixth order for the Helmholtz equation. J. Comput. Appl. Math. 203, 15–31 (2007)

    Article  MathSciNet  Google Scholar 

  24. Liao, H.L., Sun, Z.Z., Shi, H.S.: Error estimate of fourth-order compact scheme for linear Schrödinger equations. SIAM J. Numer. Anal. 47, 4381–4401 (2010)

    Article  MathSciNet  Google Scholar 

  25. Püschel, M., Moura, J.M.: The algebraic approach to the discrete cosine and sine transforms and their fast algorithms. SIAM J. Comput. 32, 1280–1316 (2003)

    Article  MathSciNet  Google Scholar 

Download references

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).

Funding

The authors have not disclosed any funding.

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

Authors
  1. Tao Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hai-Wei Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Chengjian Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hai-Wei Sun.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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}$$

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-022-02003-6

Keywords

Mathematics Subject Classification

Search

Navigation