Skip to main content
SpringerLink
Log in

The Legacy of ADI and LOD Methods and an Operator Splitting Algorithm for Solving Highly Oscillatory Wave Problems

  • Conference paper
  • First Online:
Modern Mathematical Methods and High Performance Computing in Science and Technology

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 171))

Abstract

Different splitting methods have been playing an important role in computations of numerical solutions of partial differential equations. Modern numerical strategies including mesh adaptations, linear and nonlinear transformations are also utilized together with splitting algorithms in applications. This survey concerns two cornerstones of the splitting methods, that is, the Alternating Direction Implicit (ADI) and Local One-Dimensional (LOD) methods, as well as their applications together with an eikonal mapping for solving highly oscillatory paraxial Helmholtz equations in slowly varying envelope approximations of active laser beams. The resulted finite difference scheme is not only oscillation-free, but also asymptotically stable. This ensures the high efficiency and applicability in optical wave applications.

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

Access this chapter

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 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
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

Institutional subscriptions

References

  1. Chin, S.A.: A fundamental theorem on the structure of symplectic integrators. Phys. Lett. A 354, 373–376 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Descartes, R.: Discourse on the Méthod (trans: Lafleur, L.J., 1637). The Liberal Arts Press, New York (1960)

    Google Scholar 

  3. Descombes, S., Thalhammer, M.: An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime. BIT 50, 729–749 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. D’Yakonov, E.G.: Difference schemes with splitting operator for multi-dimensional nonstationary problems. Zh. Vychisl. Mat. i Mat. Fiz. 2, 549–568 (1962)

    Google Scholar 

  5. Douglas Jr., J., Rachford Jr., H.H.: On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  6. Frey, P., George, P.-L.: Mesh Generation, 2nd edn. Wiley-ISTE, New York (2008)

    Book  MATH  Google Scholar 

  7. Hausdorff, F.: Die symbolische Exponentialformel in der Gruppentheorie. Ber Verh Saechs Akad Wiss Leipzig 58, 19–48 (1906)

    MATH  Google Scholar 

  8. Hundsdorfer, W.H., Verwer, J.G.: Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems. Math. Comput. 53, 81–101 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Iserles, A.: A First Course in the Numerical Analysis of Differential Equations, 2nd edn. Cambridge University Press, London (2011)

    MATH  Google Scholar 

  10. Jahnke, T., Lubich, C.: Error bounds for exponential operator splitting. BIT 40, 735–744 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Marchuk, G.I.: Some applicatons of splitting-up methods to the solution of problems in mathematical physics. Aplikace Matematiky 1, 103–132 (1968)

    MATH  Google Scholar 

  12. McLachlan, R.I., Quispel, G.R.W.: Splitting methods. Acta Numerica 11, 341–434 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45, 3–46 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Sheng, Q.: Solving linear partial differential equations by exponential splitting. IMA J. Numer. Anal. 9, 199–212 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sheng, Q.: Global error estimate for exponential splitting. IMA J. Numer. Anal. 14, 27–56 (1993)

    Article  MathSciNet  Google Scholar 

  17. Sheng, Q.: The ADI Methods. Encyclopedia of Applied and Computational Mathematics. Springer Verlag GmbH, Heidelberg (2015)

    Google Scholar 

  18. Sheng, Q.: ADI, LOD and modern decomposition methods for certain multiphysics applications. J. Algorithms Comput. Technol. 9, 105–120 (2015)

    Article  MathSciNet  Google Scholar 

  19. Sheng, Q., Sun, H.: On the stability of an oscillation-free ADI method for highly oscillatory wave equations. Commun. Comput. Phys. 12, 1275–1292 (2012)

    Article  Google Scholar 

  20. Sheng, Q., Sun, H.: Exponential splitting for \(n\)-dimensional paraxial Helmholtz equation with high wavenumbers. Comput. Math. Appl. 68, 1341–1354 (2014)

    Article  MathSciNet  Google Scholar 

  21. Suzuki, M.: General theory of fractal path integrals with applications to manybody theories and statistical physics. J. Math. Phys. 32, 400–407 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Trotter, H.F.: On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545–551 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yanenko, N.N.: The Method of Fractional Steps; the Solution of Problems of Mathematical Physics in Several Variables. Springer, Berlin (1971)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Center for Astrophysics, Space Physics and Engineering Research, Baylor University, Waco, TX, 76798-7328, USA

    Qin Sheng

Authors
  1. Qin Sheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qin Sheng .

Editor information

Editors and Affiliations

  1. Department of Applied Mathematics, Raj Kumar Goel Institute of Technology, Ghaziabad, Uttar Pradesh, India

    Vinai K. Singh

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada

    H.M. Srivastava

  3. Dipartimento di Matematica, University of Turin, Torino, Italy

    Ezio Venturino

  4. High Performance Computing Center (HLRS), University of Stuttgart, Stuttgart, Baden-Württemberg, Germany

    Michael Resch

  5. Department of Mathematics, Netaji Subhas Institute of Technology, New Delhi, India

    Vijay Gupta

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Science+Business Media Singapore

About this paper

Cite this paper

Sheng, Q. (2016). The Legacy of ADI and LOD Methods and an Operator Splitting Algorithm for Solving Highly Oscillatory Wave Problems. In: Singh, V., Srivastava, H., Venturino, E., Resch, M., Gupta, V. (eds) Modern Mathematical Methods and High Performance Computing in Science and Technology. Springer Proceedings in Mathematics & Statistics, vol 171. Springer, Singapore. https://doi.org/10.1007/978-981-10-1454-3_18

Download citation

Publish with us

Policies and ethics

Search

Navigation