Skip to main content
SpringerLink
Log in

The second-order modified upwind PPM characteristic difference method and analysis for solving convection-diffusion equations

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In this paper, the conservation characteristic difference method based on the second-order modified upwind scheme is analyzed for solving the two-dimensional convection-diffusion equations by combining the splitting technique. Along each directional, the intermediate numerical solutions are first computed by the piecewise parabolic method (PPM), where the endpoint values over each domain are solved by the second-order modified upwind scheme. Then, the solutions are computed by the splitting implicit characteristic difference method. By some auxiliary lemmas, we can prove that our scheme is stable in \(L^{2}\)-norm. The optimal error estimate is given and the scheme is convergent of second order in space. Numerical experiments are given to verify the theoretical analysis.

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 Availability Statement

No additional data are available.

References

  1. Brunner, F., Radu, F., Knabner, P.: Analysis of an upwind-mixed hybrid finite element method for transport problems. SIAM J. Numer. Anal. 52, 83–102 (2014)

    Article  MathSciNet  Google Scholar 

  2. Celia, M., Russell, T., Herrera, I., Ewing, R.: An Euler-Lagrangian localized adjoint method for the advection-diffusion equations. Adv. Water Resources 13, 187–206 (1990)

    Article  Google Scholar 

  3. Chen, W., Li, X., Liang, D.: Energy-conserved splitting FDTD methods for Maxwell’s equations. Numer. Math. 108, 445–485 (2008)

    Article  MathSciNet  Google Scholar 

  4. Chen, W., Li, X., Liang, D.: Energy-conserved splitting finite-difference time-domain methods for Maxwell’s equations in three dimensions. SIAM J. Numer. Anal. 48, 1530–1554 (2010)

    Article  MathSciNet  Google Scholar 

  5. Chaudhuri, A., Hadjadj, A., Chinnayya, A., Palerm, S.: Numerical study of compressible mixing layers using high-order WENO schemes. J. Sci. Comput. 47, 170–197 (2011)

    Article  MathSciNet  Google Scholar 

  6. Cheng, J., Shu, C.: A high order accurate conservative remapping method on staggered meshes. Appl. Numer. Math. 58, 1042–1060 (2008)

    Article  MathSciNet  Google Scholar 

  7. Colella, P., Woodward, P.: The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys. 54, 174–201 (1984)

    Article  Google Scholar 

  8. Dahle, H., Ewing, R., Russell, I.: Eulerian-Lagrangian localized adjoint methods for a nonlinear advection-diffusion equation. Comput. Methods Appl. Mech. Engrg. 122, 223–250 (1995)

    Article  MathSciNet  Google Scholar 

  9. Dawson, C., Russell, T., Wheeler, M.: Some improved error estimates for the modified method of characteristics. SIAM J. Numer. Anal. 26, 1487–1512 (1989)

    Article  MathSciNet  Google Scholar 

  10. Douglas, J., Jr.: On the numerical integration of \(\frac{\partial ^2 u}{\partial x^2} +\frac{\partial ^2 u}{\partial y^2}=\frac{\partial u}{\partial t}\) by implicit methods. J. Soc. Ind. Appl. Math. 3, 42–65 (1955)

    Article  Google Scholar 

  11. Douglas, J., Jr., Russell, T.: Numerical solution for convection-dominated diffusion problem based on combining the method of characteristics with finite element or difference procedures. SIAM J. Numer Anal. 19, 871–885 (1982)

    Article  MathSciNet  Google Scholar 

  12. Du, C., Liang, D.: An efficient S-DDM iterative approach for compressible contamination fluid flows in porous media. J. Comput. Phys. 229, 4501–4521 (2010)

    Article  MathSciNet  Google Scholar 

  13. Fu, K., Liang, D.: The conservative characteristic FD methods for atmospheric aerosol transport problems. J. Comput. Phys. 305, 494–520 (2016)

    Article  MathSciNet  Google Scholar 

  14. Fu, K., Liang, D.: The time second order mass conservative characteristic FDM for advection-diffusion equations in high dimensions. J. Sci. Comput. 73, 26–49 (2017)

    Article  MathSciNet  Google Scholar 

  15. Fu, K., Liang, D.: A mass-conservative temporal second order and spatial fourth order characteristic finite volume method for atmosphertic pollution advection diffusion problems. SIAM J. Sci. Comput. 41, 1178–1210 (2019)

    Article  MathSciNet  Google Scholar 

  16. Gao, F., Liang, D.: A new weighted upwind finite volume element method based on non-standard covolume for time-dependent convection-diffusion problems. Int. J. Numer. Meth. Fluids 73, 953–975 (2013)

    Article  MathSciNet  Google Scholar 

  17. Hang, T., Zhou, Z., Pan, H., Wang, Y.: The conservative characteristic difference method and analysis for solving two-sided space-fractional advection-diffusion equations. Numerical Algorithms 92, 1723–1755 (2023)

    Article  MathSciNet  Google Scholar 

  18. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.: Uniformly high order accurate essentially non-oscillatory schemes. J. Comput. Phys 71, 231–303 (1986)

    Article  Google Scholar 

  19. Jiang, G., Shu, C.: Efficient implementation of weighted ENO schemes. J. Comput. Phys 126, 202–228 (1996)

    Article  MathSciNet  Google Scholar 

  20. Ju, L., Tian, L., Xiao, X., Zhao, W.: Covolume-upwind finite volume approximations for linear elliptic partial differential equations. J. Comput. Phys. 231, 6097–6120 (2012)

    Article  MathSciNet  Google Scholar 

  21. Laminer, S., Edwards, M.: High order multidimensional upwind convection schemes for flow in porous media on structured and unstructured quadrilateral grids. SIAM J. Sci. Comput. 32, 1119–1139 (2010)

    Article  MathSciNet  Google Scholar 

  22. Lei, N., Cheng, J., Shu, C.: A high order positivity-preserving conservative WENO remapping method on 2D quadrilateral meshes. Comput. Methods Appl. Mech. Engrg. 373, 113497 (2021)

    Article  MathSciNet  Google Scholar 

  23. Li, L., Liu, J.: Large time behaviors of upwind schemes and B-schemes for Fokker-Planck equations on R by jump processes. Math. Comput. 89, 2283–2320 (2020)

    Article  MathSciNet  Google Scholar 

  24. Liang, D., Zhao, W.: A high-order upwind method for the convection-diffusion problem. Comput. Methods Appl. Mech. Eng. 147, 105–115 (1997)

    Article  MathSciNet  Google Scholar 

  25. Liang, D., Zhao, W.: A optimal weighted upwinding covolume method on non-standard grids for convection-diffusion problems in 2D. Int J. Numer. Meth. Eng. 67, 553–577 (2006)

    Article  MathSciNet  Google Scholar 

  26. Liang, D., Du, C., Wang, H.: A fractional step ELLAM approach to high-dimensional convection-diffusion problems with forward particle tracking. J. Comput. Phys. 221, 198–225 (2007)

    Article  MathSciNet  Google Scholar 

  27. Liu, C., Wang, C., Wang, Y.: A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance. J. Comput. Phys. 436, 110253 (2021)

    Article  MathSciNet  Google Scholar 

  28. Liu, X., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys 115, 200–212 (1994)

    Article  MathSciNet  Google Scholar 

  29. Meng, X., Shu, C., Wu, B.: Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations. Math. Comput. 85, 1225–1261 (2016)

    Article  MathSciNet  Google Scholar 

  30. Peaceman, D., Rachford, H., Jr.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3, 28–41 (1955)

    Article  MathSciNet  Google Scholar 

  31. Rui, H.: A conservative characteristic finite volume element method for solution of the advection-diffusion equation. Comput. Methods Appl. Mech. Eng. 197, 3862–3869 (2008)

    Article  MathSciNet  Google Scholar 

  32. Rui, H., Tabata, M.: A second order characteristic finite element scheme for convection diffusion problems. Numer. Math. 92, 161–177 (2002)

    Article  MathSciNet  Google Scholar 

  33. Rui, H., Tabata, M.: A mass-conservative characteristic finite element scheme for convection-diffusion problems. J. Sci. Comput. 43, 416–432 (2010)

    Article  MathSciNet  Google Scholar 

  34. Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517 (1968)

    Article  MathSciNet  Google Scholar 

  35. Wei, Z., Li, C., Zhao, S.: A spatially second order alternating direction implicit (ADI) method for solving three dimensional parabolic interface problems. Comput. Math. Appl. 75, 2173–2192 (2018)

    Article  MathSciNet  Google Scholar 

  36. Yuan, Y., Li, C., Song, H.: Mixed finite volume element-upwind mixed volume element of compressible two-phase displacement and its numerical analysis. J. Comput. Appl. Math. 370, 112637 (2020)

    Article  MathSciNet  Google Scholar 

  37. Zhao, S.: A matched alternating direction implicit (ADI) method for solving the heat equation with interfaces. J. Sci. Comput. 63, 118–137 (2015)

    Article  MathSciNet  Google Scholar 

  38. Zhou, H., Ying, W.: A dimension splitting method for time dependent PDEs on irregular domains. J. Sci. Comput. 94, 20 (2023)

  39. Zhou, Z., Hang, T., Pan, H., Wang, Y.: The upwind PPM scheme and analysis for solving two-sided space-fractional advection-diffusion equations in three dimension. Comput. Math. Appl. 150, 70–86 (2023)

  40. Zhou, Z., Liang, D.: The mass-preserving S-DDM scheme for two-dimensional parabolic equations. Commun. Comput. Phys. 19, 411–441 (2016)

    Article  MathSciNet  Google Scholar 

  41. Zhou, Z., Liang, D., Wong, Y.: The new mass-conserving S-DDM scheme for two-dimensional parabolic equations with variable coefficients. Appl. Math. Comput. 338, 882–902 (2018)

    MathSciNet  Google Scholar 

  42. Zhu, J., Shu, C.: A new type of multi-resolution WENO schemes with increasingly higher order of accuracy. J. Comput. Phys. 375, 659–683 (2018)

    Article  MathSciNet  Google Scholar 

  43. Zhu, J., Shu, C.: A new type of multi-resolution WENO schemes with increasingly higher order of accuracy on triangular meshes. J. Comput. Phys. 392, 19–33 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the reviewers for the valuable suggestions which have helped to improve the paper greatly.

Funding

This work was supported partially by the National Natural Science Foundation of China (Grant No. 61703250) and Natural Science Foundation of Shandong Government (Grant No. ZR2021MA002).

Author information

Authors and Affiliations

  1. School of Information Science and Engineering, Shandong Agricultural University, 271018, Taian, Shandong, China

    Huimin Ren, Qi Zhang & Zhongguo Zhou

Authors
  1. Huimin Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhongguo Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Huimin Ren: software, formal analysis, writing—original draft. Qi Zhang: software, validation, formal analysis. Zhongguo Zhou: conceptualization, methodology, funding, supervision, writing—review, editing.

Corresponding author

Correspondence to Zhongguo Zhou.

Ethics declarations

Conflict of interest

The authors declare no competing interests.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) 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

Ren, H., Zhang, Q. & Zhou, Z. The second-order modified upwind PPM characteristic difference method and analysis for solving convection-diffusion equations. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01803-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11075-024-01803-1

Keywords

Mathematics Subject Classification (2010)

Search

Navigation