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

Huazhong Tang; Gerald Warnecke

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 38, Issue: 2, page 345-357
- ISSN: 0764-583X

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topTang, Huazhong, and Warnecke, Gerald. "A note on (2K+1)-point conservative monotone schemes." ESAIM: Mathematical Modelling and Numerical Analysis 38.2 (2010): 345-357. <http://eudml.org/doc/194217>.

@article{Tang2010,

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.
},

author = {Tang, Huazhong, Warnecke, Gerald},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Hyperbolic conservation laws; finite difference scheme;
monotone scheme; convergence; oscillation.; hyperbolic conservation laws; numerical examples; monotone conservative schemes; stability properties; shock-capturing schemes; monotonicity preservation; suppression of oscillations; oscillatory solution; total variation diminishing; satisfies maximum principle},

language = {eng},

month = {3},

number = {2},

pages = {345-357},

publisher = {EDP Sciences},

title = {A note on (2K+1)-point conservative monotone schemes},

url = {http://eudml.org/doc/194217},

volume = {38},

year = {2010},

}

TY - JOUR

AU - Tang, Huazhong

AU - Warnecke, Gerald

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 38

IS - 2

SP - 345

EP - 357

AB -
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.

LA - eng

KW - Hyperbolic conservation laws; finite difference scheme;
monotone scheme; convergence; oscillation.; hyperbolic conservation laws; numerical examples; monotone conservative schemes; stability properties; shock-capturing schemes; monotonicity preservation; suppression of oscillations; oscillatory solution; total variation diminishing; satisfies maximum principle

UR - http://eudml.org/doc/194217

ER -

## References

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- R. Sanders, On the convergence of monotone finite difference schemes with variable spatial differencing. Math. Comput.40 (1983) 91–106. Zbl0533.65061
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- 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. Zbl0845.65053
- T. Tang and Z.-H. Teng, Viscosity methods for piecewise smooth solutions to scalar conservation laws. Math. Comput.66 (1997) 495–526. Zbl0864.65060

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