A note on $\left(\mathsf{2}𝖪+\mathsf{1}\right)$-point conservative monotone schemes
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) no. 2, pp. 345-357.

First-order accurate monotone conservative schemes have good convergence and stability properties, and thus play a very important role in designing modern high resolution shock-capturing schemes. Do the monotone difference approximations always give a good numerical solution in sense of monotonicity preservation or suppression of oscillations? This note will investigate this problem from a numerical point of view and show that a $\left(2K+1\right)$-point monotone scheme may give an oscillatory solution even though the approximate solution is total variation diminishing, and satisfies maximum principle as well as discrete entropy inequality.

DOI : https://doi.org/10.1051/m2an:2004016
Classification : 35L65,  65M06,  65M10
Mots clés : hyperbolic conservation laws, finite difference scheme, monotone scheme, convergence, oscillation
@article{M2AN_2004__38_2_345_0,
author = {Tang, Huazhong and Warnecke, Gerald},
title = {A note on $\sf (2K+1)$-point conservative monotone schemes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {345--357},
publisher = {EDP-Sciences},
volume = {38},
number = {2},
year = {2004},
doi = {10.1051/m2an:2004016},
zbl = {1075.65113},
mrnumber = {2069150},
language = {en},
url = {http://www.numdam.org/item/M2AN_2004__38_2_345_0/}
}
Tang, Huazhong; Warnecke, Gerald. A note on $\sf (2K+1)$-point conservative monotone schemes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) no. 2, pp. 345-357. doi : 10.1051/m2an:2004016. http://www.numdam.org/item/M2AN_2004__38_2_345_0/

[1] M.G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws. Math. Comput. 34 (1980) 1-21. | Zbl 0423.65052

[2] A. Harten, High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983) 357-393. | Zbl 0565.65050

[3] A. Harten and S. Osher, Uniformly high order accurate non-oscillatory schemes I. SIAM J. Numer. Anal. 24 (1987) 229-309. | Zbl 0627.65102

[4] A. Harten, J.M. Hyman and P.D. Lax, On finite difference approximations and entropy conditions for shocks. Comm. Pure Appl. Math. 29 (1976) 297-322. | Zbl 0351.76070

[5] C. Helzel and G. Warnecke, Unconditionally stable explicit schemes for the approximation of conservation laws, in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, B. Fiedler Ed., Springer (2001). Also available at http://www.math.fu-berlin.de/$\stackrel{˜}{\phantom{\rule{4pt}{0ex}}}$danse/bookpapers/ | MR 1850329 | Zbl 0999.65093

[6] N.N. Kuznetsov, Accuracy of some approximate methods for computing the weaks solutions of a first-order quasi-linear equation. USSR. Comput. Math. Phys. 16 (1976) 105-119. | Zbl 0381.35015

[7] X.D. Liu and E. Tadmor, Third order nonoscillatory central scheme for hyperbolic conservation laws. Numer. Math. 79 (1998) 397-425. | Zbl 0906.65093

[8] F. Sabac, The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws. SIAM J. Numer. Anal. 34 (1997) 2306-2318 | Zbl 0992.65099

[9] R. Sanders, On the convergence of monotone finite difference schemes with variable spatial differencing. Math. Comput. 40 (1983) 91-106. | Zbl 0533.65061

[10] E. Tadmor, The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs schemes. Math. Comput. 43 (1984) 353-368. | MR 758188 | Zbl 0598.65067

[11] T. Tang and Z.-H. Teng, The sharpness of Kuznetsov’s $O\left(\sqrt{\Delta x}\right){L}^{1}$-error estimate for monotone difference schemes. Math. Comput. 64 (1995) 581-589. | MR 1270625 | Zbl 0845.65053

[12] T. Tang and Z.-H. Teng, Viscosity methods for piecewise smooth solutions to scalar conservation laws. Math. Comput. 66 (1997) 495-526. | MR 1397446 | Zbl 0864.65060