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

Abstract

top
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.

How to cite

top

Tang, 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

top
  1. M.G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws. Math. Comput.34 (1980) 1–21.  Zbl0423.65052
  2. A. Harten, High resolution schemes for hyperbolic conservation laws. J. Comput. Phys.49 (1983) 357–393.  Zbl0565.65050
  3. A. Harten and S. Osher, Uniformly high order accurate non-oscillatory schemes I. SIAM J. Numer. Anal.24 (1987) 229–309.  Zbl0627.65102
  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.  Zbl0351.76070
  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 ∼danse/bookpapers/  Zbl0999.65093URIhttp://www.math.fu-berlin.de/
  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.  
  7. X.D. Liu and E. Tadmor, Third order nonoscillatory central scheme for hyperbolic conservation laws. Numer. Math.79 (1998) 397–425.  Zbl0906.65093
  8. F. Sabac, The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws. SIAM J. Numer. Anal.34 (1997) 2306–2318 Zbl0992.65099
  9. R. Sanders, On the convergence of monotone finite difference schemes with variable spatial differencing. Math. Comput.40 (1983) 91–106.  Zbl0533.65061
  10. E. Tadmor, The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs schemes. Math. Comput.43 (1984) 353–368.  Zbl0598.65067
  11. T. Tang and Z.-H. Teng, The sharpness of Kuznetsov's O ( Δ x ) L 1 -error estimate for monotone difference schemes. Math. Comput.64 (1995) 581–589.  Zbl0845.65053
  12. T. Tang and Z.-H. Teng, Viscosity methods for piecewise smooth solutions to scalar conservation laws. Math. Comput.66 (1997) 495–526.  Zbl0864.65060

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.