×

Some discrete inequalities for central-difference type operators. (English) Zbl 1361.65057

Summary: 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.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q35 PDEs in connection with fluid mechanics
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
26D20 Other analytical inequalities

Software:

mctoolbox
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Akrivis, Georgios D.; Dougalis, Vassilios A.; Karakashian, Ohannes A., On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schr\"odinger equation, Numer. Math., 59, 1, 31-53 (1991) · Zbl 0739.65096 · doi:10.1007/BF01385769
[2] Berman, Abraham; Plemmons, Robert J., Nonnegative Matrices in the Mathematical Sciences, Computer Science and Applied Mathematics, xviii+316 pp. (1979), Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London · Zbl 0484.15016
[3] Braess, Dietrich, Finite Elements, xviii+352 pp. (2001), translated from the 1992 German edition by Larry L. Schumaker, Cambridge University Press, Cambridge · Zbl 0976.65099
[4] Brenner, Susanne C.; Scott, L. Ridgway, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics 15, xvi+361 pp. (2002), Springer-Verlag, New York · Zbl 1012.65115 · doi:10.1007/978-1-4757-3658-8
[5] Brezis, Haim, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, xiv+599 pp. (2011), Springer, New York · Zbl 1220.46002
[6] Delfour, M.; Fortin, M.; Payre, G., Finite-difference solutions of a nonlinear Schr\"odinger equation, J. Comput. Phys., 44, 2, 277-288 (1981) · Zbl 0477.65086 · doi:10.1016/0021-9991(81)90052-8
[7] Faou, Erwan, Geometric Numerical Integration and Schr\`“odinger Equations, Zurich Lectures in Advanced Mathematics, xiii+138 pp. (2012), European Mathematical Society (EMS), Z\'”urich · Zbl 1239.65078 · doi:10.4171/100
[8] Fornberg, Bengt, A Practical Guide to Pseudospectral Methods, Cambridge Monographs on Applied and Computational Mathematics 1, x+231 pp. (1996), Cambridge University Press, Cambridge · Zbl 0844.65084 · doi:10.1017/CBO9780511626357
[9] Fornberg, Bengt, Generation of finite difference formulas on arbitrarily spaced grids, Math. Comp., 51, 184, 699-706 (1988) · Zbl 0701.65014 · doi:10.2307/2008770
[10] Furihata, Daisuke, 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, 1, 181-205 (1999) · Zbl 0945.65103 · doi:10.1006/jcph.1999.6377
[11] Furihata, Daisuke; Matsuo, Takayasu, Discrete Variational Derivative Method, Chapman & Hall/CRC Numerical Analysis and Scientific Computing, xii+376 pp. (2011), CRC Press, Boca Raton, FL · Zbl 1227.65094
[12] Guan, Zhen; Lowengrub, John S.; Wang, Cheng; Wise, Steven M., Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations, J. Comput. Phys., 277, 48-71 (2014) · Zbl 1349.65298 · doi:10.1016/j.jcp.2014.08.001
[13] LeVeque, Randall J., Finite Difference Methods for Ordinary and Partial Differential Equations, xvi+341 pp. (2007), Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA · Zbl 1127.65080 · doi:10.1137/1.9780898717839
[14] Golub, Gene H.; Van Loan, Charles F., Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, xxx+698 pp. (1996), Johns Hopkins University Press, Baltimore, MD · Zbl 0865.65009
[15] Higham, Nicholas J., Accuracy and Stability of Numerical Algorithms, xxviii+688 pp. (1996), Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA · Zbl 0847.65010
[16] Holden, Helge; Raynaud, Xavier, Convergence of a finite difference scheme for the Camassa-Holm equation, SIAM J. Numer. Anal., 44, 4, 1655-1680 (electronic) (2006) · Zbl 1122.76065 · doi:10.1137/040611975
[17] Hu, Xiuling; Zhang, Luming, Conservative compact difference schemes for the coupled nonlinear Schr\"odinger system, Numer. Methods Partial Differential Equations, 30, 3, 749-772 (2014) · Zbl 1302.65192 · doi:10.1002/num.21826
[18] Iserles, Arieh, On skew-symmetric differentiation matrices, IMA J. Numer. Anal., 34, 2, 435-451 (2014) · Zbl 1288.65121 · doi:10.1093/imanum/drt013
[19] John, Fritz, Lectures on Advanced Numerical Analysis, xiv+179 pp. (1967), Gordon and Breach Science Publishers, New York-London-Paris
[20] Kanazawa 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.
[21] Kojima95 H. Kojima, Construction and theoretical analyses of structure preserving schemes with high accuracy, master’s thesis at The University of Tokyo, 2015.
[22] Lele, Sanjiva K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103, 1, 16-42 (1992) · Zbl 0759.65006 · doi:10.1016/0021-9991(92)90324-R
[23] Matsuo, Takayasu; Sugihara, Masaaki; Furihata, Daisuke; Mori, Masatake, Spatially accurate dissipative or conservative finite difference schemes derived by the discrete variational method, Japan J. Indust. Appl. Math., 19, 3, 311-330 (2002) · Zbl 1014.65083 · doi:10.1007/BF03167482
[24] Matsuo2 T. Matsuo, M. Sugihara, and M. Mori, A derivation of a finite difference scheme for the nonlinear Schr\"odinger equation by the discrete variational method (in Japanese), Trans. Japan Soc. Indust. Appl. Math. 8 (1998), 405-426.
[25] McLachlan, Robert I.; Quispel, G. Reinout W., Splitting methods, Acta Numer., 11, 341-434 (2002) · Zbl 1105.65341 · doi:10.1017/S0962492902000053
[26] Taha, Thiab R.; Ablowitz, Mark J., Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schr\"odinger equation, J. Comput. Phys., 55, 2, 203-230 (1984) · Zbl 0541.65082 · doi:10.1016/0021-9991(84)90003-2
[27] Thom{\'e}e, Vidar, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics 25, xii+370 pp. (2006), Springer-Verlag, Berlin · Zbl 1105.65102
[28] Tolstykh94 A. I. Tolstykh, High Accuracy Non-centered Compact Difference Schemes for Fluid Dynamics Applications, World Scientific, Singapore, 1994. · Zbl 0852.76003
[29] Tolstykh, Andrei I.; Lipavskii, Michael V., On performance of methods with third- and fifth-order compact upwind differencing, J. Comput. Phys., 140, 2, 205-232 (1998) · Zbl 0936.76059 · doi:10.1006/jcph.1998.5887
[30] Wang, Tingchun, Convergence of an eighth-order compact difference scheme for the nonlinear Schr\"odinger equation, Adv. Numer. Anal., Art. ID 913429, 24 pp. (2012) · Zbl 1268.65119
[31] Wang, Tingchun; Guo, Boling; Xu, Qiubin, Fourth-order compact and energy conservative difference schemes for the nonlinear Schr\"odinger equation in two dimensions, J. Comput. Phys., 243, 382-399 (2013) · Zbl 1349.65347 · doi:10.1016/j.jcp.2013.03.007
[32] Wang, C.; Wise, S. M., An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., 49, 3, 945-969 (2011) · Zbl 1230.82005 · doi:10.1137/090752675
[33] Wise, S. M.; Wang, C.; Lowengrub, J. S., An energy-stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., 47, 3, 2269-2288 (2009) · Zbl 1201.35027 · doi:10.1137/080738143
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.