skip to main content

Algorithm 986: A Suite of Compact Finite Difference Schemes

Authors Info & Claims
Published:05 October 2017Publication History
Skip Abstract Section

Abstract

A collection of Matlab routines that compute derivative approximations of arbitrary functions using high-order compact finite difference schemes is presented. Tenth-order accurate compact finite difference schemes for first and second derivative approximations and sixth-order accurate compact finite difference schemes for third and fourth derivative approximations are discussed for the functions with periodic boundary conditions. Fourier analysis of compact finite difference schemes is explained, and it is observed that compact finite difference schemes have better resolution characteristics when compared to classical finite difference schemes. Compact finite difference schemes for the functions with Dirichlet and Neumann boundary conditions are also discussed. Moreover, compact finite difference schemes for partial derivative approximations of functions in two variables are also given. For each case a Matlab routine is provided to compute the differentiation matrix and results are validated using the test functions.

Skip Supplemental Material Section

Supplemental Material

References

  1. Y. Adam. 1975. A hermitian finite difference method for the solution of parabolic equations. Comp. Maths. Appls. 1, 393--406 Google ScholarGoogle ScholarCross RefCross Ref
  2. Y. Adam. 1977. Highly accurate compact implicit methods and boundary conditions. J. Comput. Phys. 24, 10--22. Google ScholarGoogle ScholarCross RefCross Ref
  3. C. Cheong and S. Lee. 2001. Grid-optimized dispersion-relation-preserving schemes on general geometries for computational aeroacoustics. J. Comput. Phys. 174, 248--276. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. P. C. Chu and C. Fan. 1998. A three-point combined compact difference scheme. J. Comput. Phys. 140, 370--399. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. P. C. Chu and C. Fan. 1999. A three-point sixth-order nonuniform combined compact difference scheme. J. Comput. Phys. 148, 663--674. Google ScholarGoogle ScholarDigital LibraryDigital Library
  6. B. During and M. Fournie. 2012. High-order compact finite difference scheme for option pricing in stochastic volatility models. J. Comput. Appl. Math. 236, 4462--4473. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. B. During, M. Fournie, and A. Jungel. 2004. Convergence of high-order compact finite difference scheme for a nonlinear Black-Scholes equation. Math. Model. Numer. Anal. 38, 359--369. Google ScholarGoogle ScholarCross RefCross Ref
  8. L. Gamet, F. Ducros, F. Nicoud, and T. Poinsot. 1999. Compact finite difference schemes on non-uniform meshes. Application to direct numerical simulation of compressible flows. Int. J. Numer. Meth. Fluids 29, 159--191. Google ScholarGoogle ScholarCross RefCross Ref
  9. L. Ge and J. Zhang. 2001. High accuracy solution of convection diffusion equation with boundary layers on nonuniform grids. J. Comput. Phys. 171, 560--571. Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. W. J. Geodheer and J. H. H. M. Potters. 1985. A compact finite difference scheme on a non-equidistant mesh. J. Comput. Phys. 61, 269--279. Google ScholarGoogle ScholarCross RefCross Ref
  11. R. S. Hirsh. 1975. Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique. J. Comput. Phys. 19, 90--109. Google ScholarGoogle ScholarCross RefCross Ref
  12. V. Kumar. 2009. High-order compact finite-difference scheme for singularly-perturbed reaction-diffusion problems on a new mesh of Shishkin type. J. Optim. Theory. Appl. 143, 123--147. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. S. K. Lele. 1992. Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16--42. Google ScholarGoogle ScholarCross RefCross Ref
  14. K. Mahesh. 1998. A family of high order finite difference schemes with good spectral resolution. J. Comput. Phys. 145, 332--358. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. H. L. Meitz and H. F. Fasel. 2000. A compact-difference scheme for the Navier-Stokes equations in vorticity-velocity formulation. J. Comput. Phys. 157, 371--403. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Shuvam Sen. 2013. A new family of (5,5)CC-4OC schemes applicable for unsteady Navier-Stokes equations. J. Comput. Phys. 251, 251--271. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Shuvam Sen. 2016. Fourth order compact schemes for variable coefficient parabolic problems with mixed derivatives. Comput. Fluids 134--135, 81--89.Google ScholarGoogle Scholar
  18. T. K. Sengupta, G. Ganeriwal, and S. De. 2003. Analysis of central and upwind compact schemes. J. Comput. Phys. 192, 677--694. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. J. S. Shang. 1999. High-order compact-difference schemes for time-dependent maxwell equations. J. Comput. Phys. 153, 312--333. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. R. K. Shukla, M. Tatineni, and X. Zhong. 2007. Very high-order compact finite difference schemes for non-uniform grid for incompressible Navier-Stokes equations. J. Comput. Phys. 224, 1064--1094. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. R. K. Shukla and X. Zhong. 2005. Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation. J. Comput. Phys. 204, 404--429. Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. E. Weinan and J. G. Liu. 1996. Essentially compact schemes for unsteady viscous incompressible flows. J. Comput. Phys. 126, 122--138. Google ScholarGoogle ScholarDigital LibraryDigital Library
  23. J. Zhao, W. Dai, and T. Niu. 2007. Fourth order compact scheme of a heat conduction problem with Neumann boundary conditions. Numer. Methods Partial Differential Eq. 12, 949--959. Google ScholarGoogle ScholarCross RefCross Ref

Index Terms

  1. Algorithm 986: A Suite of Compact Finite Difference Schemes

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in

      Full Access

      • Published in

        cover image ACM Transactions on Mathematical Software
        ACM Transactions on Mathematical Software  Volume 44, Issue 2
        June 2018
        242 pages
        ISSN:0098-3500
        EISSN:1557-7295
        DOI:10.1145/3132683
        Issue’s Table of Contents

        Copyright © 2017 ACM

        Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 5 October 2017
        • Accepted: 1 June 2017
        • Revised: 1 January 2017
        • Received: 1 December 2016
        Published in toms Volume 44, Issue 2

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • research-article
        • Research
        • Refereed

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader