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A note on (2K+1)-point conservative monotone schemes

Published online by Cambridge University Press:  15 March 2004

Huazhong Tang  and
Gerald Warnecke
Huazhong Tang
Affiliation:
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, PR China, [email protected].
Gerald Warnecke
Affiliation:
Institüt für Analysis und Numerik, Otto–von–Guericke Universität Magdeburg, 39106 Magdeburg, Germany, [email protected].
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Abstract

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.

Keywords

Hyperbolic conservation lawsfinite difference schememonotone schemeconvergenceoscillation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 2 , March 2004 , pp. 345 - 357
Copyright
© EDP Sciences, SMAI, 2004

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References

Crandall, M.G. and Majda, A., Monotone difference approximations for scalar conservation laws. Math. Comput. 34 (1980) 1–21. CrossRef
Harten, A., High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49 (1983) 357–393. CrossRef
Harten, A. and Osher, S., Uniformly high order accurate non-oscillatory schemes I. SIAM J. Numer. Anal. 24 (1987) 229–309. CrossRef
Harten, A., Hyman, J.M. and Lax, P.D., On finite difference approximations and entropy conditions for shocks. Comm. Pure Appl. Math. 29 (1976) 297–322. CrossRef
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/
Kuznetsov, N.N., 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. CrossRef
Liu, X.D. and Tadmor, E., Third order nonoscillatory central scheme for hyperbolic conservation laws. Numer. Math. 79 (1998) 397–425. CrossRef
Sabac, F., The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws. SIAM J. Numer. Anal. 34 (1997) 2306–2318 CrossRef
Sanders, R., On the convergence of monotone finite difference schemes with variable spatial differencing. Math. Comput. 40 (1983) 91–106. CrossRef
Tadmor, E., The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs schemes. Math. Comput. 43 (1984) 353–368. CrossRef
Tang, T. and Teng, Z.-H., The sharpness of Kuznetsov's $O(\sqrt{\Delta x})L^1$ -error estimate for monotone difference schemes. Math. Comput. 64 (1995) 581–589.
Tang, T. and Teng, Z.-H., Viscosity methods for piecewise smooth solutions to scalar conservation laws. Math. Comput. 66 (1997) 495–526. CrossRef