Skip to main content
SpringerLink
Log in
Book cover

Numerical Treatment of Multiphase Flows in Porous Media pp 311–323Cite as

A Comparison of ELLAM with ENO/WENO Schemes for Linear Transport Equations

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Physics ((LNP,volume 552))

Abstract

We present an Eulerian-Lagrangian localized adjoint method (EL-LAM) for linear advection-reaction partial differential equations in multiple space dimensions. We carry out numerical experiments to compare the performance of the ELLAM scheme with the essentially non-oscillatory (ENO) schemes and weighted essentially non-oscillatory (WENO) schemes, which shows that the ELLAM scheme outperforms ENO and WENO schemes in the context of linear transport PDEs.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Al-Lawatia, M., Sharpley, R. C., and Wang, H., Second-order characteristic methods for advection-diffusion equations and comparison to other schemes, Advances in Water Resources 22 (1999), 741–768.

    Article  Google Scholar 

  2. Barrett, J. W. and Morton, K. W., Approximate symmetrization and Petrov-Galerkin methods for diffusion-convection problems, Comp. Meth. Appl. Mech. Engrg. 45 (1984), 97–122.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Bear, J., Hydraulics of Groundwater, McGraw-Hill, New York, 1979

    Google Scholar 

  4. Bouloutas, E. T. and Celia, M. A., An improved cubic Petrov-Galerkin method for simulation of transient advection-diffusion processes in rectangularly decomposable domains, Comp. Meth. Appl. Mech. Engrg. 91 (1991), 289–308.

    Article  Google Scholar 

  5. Celia, M. A., Russell, T. F., Herrera, I., Ewing, R. E., An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation, Adv. Wat. Res. 13 (1990), 187–206

    Article  Google Scholar 

  6. Christie, I., Griffiths, D. F., Mitchell, A. R., and Zienkiewicz, O. C., Finite element methods for second order differential equations with significant first derivatives, Int. J. Num. Engrg. 10 (1976), 1389–1396.

    Article  MATH  MathSciNet  Google Scholar 

  7. Colella, P., A direct Eulerian MUSCL scheme for gas dynamics, SIAM J. Sci. Stat. Comp. 6 (1985), 104–117.

    Article  MATH  MathSciNet  Google Scholar 

  8. Ewing, R. E. (ed.), The Mathematics of Reservoir Simulation, Research Frontiers in Applied Mathematics. 1, SIAM, Philadelphia, 1984

    Google Scholar 

  9. Ewing, R. E. and Wang, H., Eulerian-Lagrangian localized adjoint methods for linear advection equations, Computational Mechanics, Springer International, 1991, pp. 245–250.

    Google Scholar 

  10. Ewing, R. E. and Wang, H., An optimal-order error estimate to Eulerian-Lagrangian localized adjoint method for variable-coefficient advection-reaction problems, SIAM Num. Anal. 33 (1996), 318–348.

    Article  MATH  MathSciNet  Google Scholar 

  11. Harten, A., Engquist, B., Osher, S., and Chakravarthy, S., Uniformly high order accurate essentially nonoscillatory schemes, III, J. Comp. Phys. 71 (1987), 231–241.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Harten, A. and Osher, S., Uniformly high-order accurate non-oscillatory schemes, I, SIAM J. Num. Anal. 24 (1987), 279–309.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Hughes, T. J. R. and Brooks, A. N., A multidimensional upwinding scheme with no crosswind diffusion, Hughes (ed.), Finite Element Methods for Convection Dominated Flows 34, ASME, New York, 1979.

    Google Scholar 

  14. Johnson, C. and Pitkäranta, J., An analysis of discontinuous Galerkin methods for a scalar hyperbolic equation, Math. Comp. 46 (1986), 1–26.

    Article  MATH  MathSciNet  Google Scholar 

  15. Liu, X.-D., Osher, S., and Chan, T., Weighted essentially nonoscillatory schemes, J. Comput. Phys. 115 (1994), 200–212.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  16. Richter, G. R., An optimal-order error estimate for the discontinuous Galerkin method, Math. Comp. 50 (1988), 75–88.

    Article  MATH  MathSciNet  Google Scholar 

  17. Russell, T. F. and Trujillo, R. V., Eulerian-Lagrangian localized adjoint methods with variable coefficients in multiple dimensions, Gambolati, et al. (ed.), Computational Methods in Surface Hydrology, Springer-Verlag, Berlin, 1990, 357–363.

    Google Scholar 

  18. van Leer, B., On the relation between the upwind-differencing schemes of Godunov, Engquist-Osher, and Roe, SIAM J. Sci. Stat. Comp. 5 (1984), 1–20.

    Article  MATH  ADS  Google Scholar 

  19. Wang, H., Liang, D., Ewing, R. E., Lyons, S. L., and Qin, G., An accurate approximation to compressible flow in porous media with wells, this volume.

    Google Scholar 

  20. Wang, H., Ewing, R. E., Qin, G., Lyons, S. L., Al-Lawatia, M, and Man, S, A family of Eulerian-Lagrangian localized adjoint methods for multi-dimensional advection-reaction equations, J. Comp. Phys. 152 (1999), 120–163.

    Article  MATH  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of South Carolina, Columbia, South Carolina, 29208, USA

    Hong Wang

  2. Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Al-Khod Postal Code 123, Muscat, Sultanate of Oman

    Mohamed Al-Lawatia

Authors
  1. Hong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohamed Al-Lawatia
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Southern Methodist University, Box 750156, 75275-0156, Dallas, TX, USA

    Zhangxin Chen

  2. Institute for Scientific Computation, Texas A&M University, 77843-3404, College Station, TX, USA

    Richard E. Ewing

  3. Institute of ComputaTional Mathematics, Chinese Academy of Sciences, P.O.Box 2719, 100080, Beijing, P.R.China

    Zhong-Ci Shi

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Wang, H., Al-Lawatia, M. (2000). A Comparison of ELLAM with ENO/WENO Schemes for Linear Transport Equations. In: Chen, Z., Ewing, R.E., Shi, ZC. (eds) Numerical Treatment of Multiphase Flows in Porous Media. Lecture Notes in Physics, vol 552. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45467-5_26

Download citation

  • DOI: https://doi.org/10.1007/3-540-45467-5_26

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67566-2

  • Online ISBN: 978-3-540-45467-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation