## Some discrete inequalities for central-difference type operators

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- by Hiroki Kojima, Takayasu Matsuo and Daisuke Furihata PDF
- Math. Comp.
**86**(2017), 1719-1739 Request permission

## Abstract:

Discrete versions of basic inequalities in functional analysis such as the Sobolev inequality play a key role in theoretical analysis of finite difference schemes. They have been shown for some simple difference operators, but are still left open for general operators, even including the standard central difference operators. In this paper, we propose a systematic approach for deriving such inequalities for a certain class of central-difference type operators. We illustrate the results by giving a generic a priori estimate for certain conservative schemes for the nonlinear Schrödinger and Cahn–Hilliard equations.## References

- Georgios D. Akrivis, Vassilios A. Dougalis, and Ohannes A. Karakashian,
*On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation*, Numer. Math.**59**(1991), no. 1, 31–53. MR**1103752**, DOI 10.1007/BF01385769 - Abraham Berman and Robert J. Plemmons,
*Nonnegative matrices in the mathematical sciences*, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR**544666** - Dietrich Braess,
*Finite elements*, 2nd ed., Cambridge University Press, Cambridge, 2001. Theory, fast solvers, and applications in solid mechanics; Translated from the 1992 German edition by Larry L. Schumaker. MR**1827293** - Susanne C. Brenner and L. Ridgway Scott,
*The mathematical theory of finite element methods*, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR**1894376**, DOI 10.1007/978-1-4757-3658-8 - Haim Brezis,
*Functional analysis, Sobolev spaces and partial differential equations*, Universitext, Springer, New York, 2011. MR**2759829** - M. Delfour, M. Fortin, and G. Payre,
*Finite-difference solutions of a nonlinear Schrödinger equation*, J. Comput. Phys.**44**(1981), no. 2, 277–288. MR**645840**, DOI 10.1016/0021-9991(81)90052-8 - Erwan Faou,
*Geometric numerical integration and Schrödinger equations*, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2012. MR**2895408**, DOI 10.4171/100 - Bengt Fornberg,
*A practical guide to pseudospectral methods*, Cambridge Monographs on Applied and Computational Mathematics, vol. 1, Cambridge University Press, Cambridge, 1996. MR**1386891**, DOI 10.1017/CBO9780511626357 - Bengt Fornberg,
*Generation of finite difference formulas on arbitrarily spaced grids*, Math. Comp.**51**(1988), no. 184, 699–706. MR**935077**, DOI 10.1090/S0025-5718-1988-0935077-0 - Daisuke Furihata,
*Finite difference schemes for $\partial u/\partial t=(\partial /\partial x)^\alpha \delta G/\delta u$ that inherit energy conservation or dissipation property*, J. Comput. Phys.**156**(1999), no. 1, 181–205. MR**1727636**, DOI 10.1006/jcph.1999.6377 - Daisuke Furihata and Takayasu Matsuo,
*Discrete variational derivative method*, Chapman & Hall/CRC Numerical Analysis and Scientific Computing, CRC Press, Boca Raton, FL, 2011. A structure-preserving numerical method for partial differential equations. MR**2744841** - Zhen Guan, John S. Lowengrub, Cheng Wang, and Steven M. Wise,
*Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations*, J. Comput. Phys.**277**(2014), 48–71. MR**3254224**, DOI 10.1016/j.jcp.2014.08.001 - Randall J. LeVeque,
*Finite difference methods for ordinary and partial differential equations*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. Steady-state and time-dependent problems. MR**2378550**, DOI 10.1137/1.9780898717839 - Gene H. Golub and Charles F. Van Loan,
*Matrix computations*, 3rd ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996. MR**1417720** - Nicholas J. Higham,
*Accuracy and stability of numerical algorithms*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. MR**1368629** - Helge Holden and Xavier Raynaud,
*Convergence of a finite difference scheme for the Camassa-Holm equation*, SIAM J. Numer. Anal.**44**(2006), no. 4, 1655–1680. MR**2257121**, DOI 10.1137/040611975 - Xiuling Hu and Luming Zhang,
*Conservative compact difference schemes for the coupled nonlinear Schrödinger system*, Numer. Methods Partial Differential Equations**30**(2014), no. 3, 749–772. MR**3190336**, DOI 10.1002/num.21826 - Arieh Iserles,
*On skew-symmetric differentiation matrices*, IMA J. Numer. Anal.**34**(2014), no. 2, 435–451. MR**3194794**, DOI 10.1093/imanum/drt013 - Fritz John,
*Lectures on advanced numerical analysis*, Gordon and Breach Science Publishers, New York-London-Paris, 1967. MR**0221721** - H. Kanazawa, T. Matsuo, and T. Yaguchi,
*Discrete variational derivative method based on the compact finite differences*(in Japanese), Trans. Japan Soc. Indust. Appl. Math.**23**(2013), 203–232. - H. Kojima,
*Construction and theoretical analyses of structure preserving schemes with high accuracy*, master’s thesis at The University of Tokyo, 2015. - Sanjiva K. Lele,
*Compact finite difference schemes with spectral-like resolution*, J. Comput. Phys.**103**(1992), no. 1, 16–42. MR**1188088**, DOI 10.1016/0021-9991(92)90324-R - Takayasu Matsuo, Masaaki Sugihara, Daisuke Furihata, and Masatake Mori,
*Spatially accurate dissipative or conservative finite difference schemes derived by the discrete variational method*, Japan J. Indust. Appl. Math.**19**(2002), no. 3, 311–330. MR**1933890**, DOI 10.1007/BF03167482 - T. Matsuo, M. Sugihara, and M. Mori,
*A derivation of a finite difference scheme for the nonlinear Schrödinger equation by the discrete variational method*(in Japanese), Trans. Japan Soc. Indust. Appl. Math.**8**(1998), 405–426. - Robert I. McLachlan and G. Reinout W. Quispel,
*Splitting methods*, Acta Numer.**11**(2002), 341–434. MR**2009376**, DOI 10.1017/S0962492902000053 - Thiab R. Taha and Mark J. Ablowitz,
*Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation*, J. Comput. Phys.**55**(1984), no. 2, 203–230. MR**762363**, DOI 10.1016/0021-9991(84)90003-2 - Vidar Thomée,
*Galerkin finite element methods for parabolic problems*, 2nd ed., Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 2006. MR**2249024** - A. I. Tolstykh,
*High Accuracy Non-centered Compact Difference Schemes for Fluid Dynamics Applications*, World Scientific, Singapore, 1994. - Andrei I. Tolstykh and Michael V. Lipavskii,
*On performance of methods with third- and fifth-order compact upwind differencing*, J. Comput. Phys.**140**(1998), no. 2, 205–232. MR**1616134**, DOI 10.1006/jcph.1998.5887 - Tingchun Wang,
*Convergence of an eighth-order compact difference scheme for the nonlinear Schrödinger equation*, Adv. Numer. Anal. , posted on (2012), Art. ID 913429, 24. MR**2984521**, DOI 10.1155/2012/913429 - Tingchun Wang, Boling Guo, and Qiubin Xu,
*Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions*, J. Comput. Phys.**243**(2013), 382–399. MR**3064174**, DOI 10.1016/j.jcp.2013.03.007 - C. Wang and S. M. Wise,
*An energy stable and convergent finite-difference scheme for the modified phase field crystal equation*, SIAM J. Numer. Anal.**49**(2011), no. 3, 945–969. MR**2802554**, DOI 10.1137/090752675 - S. M. Wise, C. Wang, and J. S. Lowengrub,
*An energy-stable and convergent finite-difference scheme for the phase field crystal equation*, SIAM J. Numer. Anal.**47**(2009), no. 3, 2269–2288. MR**2519603**, DOI 10.1137/080738143

## Additional Information

**Hiroki Kojima**- Affiliation: Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, 113-8656, Japan
- Email: hiroki_kojima@mist.i.u-tokyo.ac.jp, h.k.psi.mot@gmail.com
**Takayasu Matsuo**- Affiliation: Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, 113-8656, Japan
- MR Author ID: 664782
- Email: matsuo@mist.i.u-tokyo.ac.jp
**Daisuke Furihata**- Affiliation: Cybermedia Center, Osaka University, 1-32 Machikaneyama-cho, Toyonaka City, Osaka 560-0043, Japan
- MR Author ID: 601502
- Email: furihata@cmc.osaka-u.ac.jp
- Received by editor(s): February 16, 2015
- Received by editor(s) in revised form: November 28, 2015
- Published electronically: September 15, 2016
- Additional Notes: This work was supported by JSPS KAKENHI Grant Number 23560063 and 25287030. This work was also supported by CREST, JST
- © Copyright 2016 American Mathematical Society
- Journal: Math. Comp.
**86**(2017), 1719-1739 - MSC (2010): Primary 65M06, 65M12
- DOI: https://doi.org/10.1090/mcom/3154
- MathSciNet review: 3626534