A note on (2𝖪+1)-point conservative monotone schemes
ESAIM: Modélisation mathématique et analyse numérique, Volume 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 (2K+1)-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: 10.1051/m2an:2004016
Classification: 35L65, 65M06, 65M10
Keywords: hyperbolic conservation laws, finite difference scheme, monotone scheme, convergence, oscillation
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     title = {A note on $\sf (2K+1)$-point conservative monotone schemes},
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Tang, Huazhong; Warnecke, Gerald. A note on $\sf (2K+1)$-point conservative monotone schemes. ESAIM: Modélisation mathématique et analyse numérique, Volume 38 (2004) no. 2, pp. 345-357. doi : 10.1051/m2an:2004016. http://www.numdam.org/articles/10.1051/m2an:2004016/

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

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

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

[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

[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/ ˜danse/bookpapers/ | MR | Zbl

[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

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

[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

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

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

[11] T. Tang and Z.-H. Teng, The sharpness of Kuznetsov’s O(Δx)L 1 -error estimate for monotone difference schemes. Math. Comput. 64 (1995) 581-589. | MR | Zbl

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

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