Abstract
We adapt ideas from image processing, namely the concept of anisotropic diffusion and integrate them in the concept of stabilizing numerical schemes for conservation laws by adding some nonlinear artificial dissipation as introduced by Harten [5]. We resume this concept and supplement it with a multidimensional nonlinear anisotropic diffusion filters, to allow different dissipation directions. We combine this with investigations concerning the production of numerical entropy inside the scheme, taking entropy production as a measure for the dose of artificial dissipation necessary to stabilize the algorithm.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. Godlewski and P.-A. Raviart, “Hyperbolic Systems of Conservation laws”, Ellipses 1991.
Th. Grahs, A. Meister and Th. Sonar, “Image processing for numerical approximation of conservation laws: nonlinear anisotropic artificial diffusion”, Reihe F, Computational Fluid Dynamics and Data Analysis 8, Hamburger Beiträge zur Angewandten Mathematik, 1998 (accepted for publication in SIAM J. Sci. Comput.).
Th. Grabs, A. Meister and Th. Sonar, “Nonlinear anisotropic artificial dissipation - Characteristic filters for computation of the Euler equations”, in Finite Volumes for Complex Applications II, Eds.: R. Vilsmeier, F. Benkhaldoun, D. Hänel, HERMES Science Publication 1999
Th. Grabs and Th. Sonar, “Entropy controlled artificial anisotropic diffusion for the numerical solution of conservation laws based on algorithms from image processing” accepted for publication in J.Vis.Commspecial issue on PDE in image processing computer vision and computer graphicsto appear.
A. Harten, “High resolution schemes for hyperbolic conservation laws”, J. Comp. Phy. 49, 357–393, 1983.
R.J. LeVeque, “Numerical Methods for Conservation Laws”, Birkhäuser Verlag, 1992.
A. Majda and S. Osher, “Numerical viscosity and the entropy condition”, Comm Pure Appl. Math., 32, 797–838, 1979.
M.L. Merriam, “Smoothing and the second law”, Comp. Meth. App. Mech, Eng. 64(1), 177–193, 1987.
M.L. Merriam, “An entropy-based approach to nonlinear stability”, NASA Technical Memorandum 10 1086, 1989.
P. Perona, J. Malik, “Scale space and edge detection using anisotropic diffusion”, IEEE Trans. Pattern Anal. Mach. Intell., Vol. 12, 629–639, 1990.
R.D. Richtmyer and K.W. Morton, “Difference Methods for Initial-value Problems” Wiley-Interscience, 1967.
Th. Sonar, “Entropy production in second-order three-point schemes”, Num. Math. 62, 371–390, 1992.
E. Tadmor, “Numerical viscosity and the entropy condition for conservative difference schemes”, Math. Comp. 43, 369–381, 1984.
E. Tadmor, “The numerical viscosity of entropy stable schemes for systems of conservation laws P”, Math. Comp. 49, 91–103, 1987.
Shokin, Yu.I., “The Method of Differential Approximation”, Springer Series in Computational Physics, 1983
J. Weickert, “Anisotropic Diffusion in Image Processing”, Teubner 1998.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Basel AG
About this paper
Cite this paper
Grahs, T., Sonar, T. (2001). Multidimensional Artificial Dissipation for the Numerical Approximation of Conservation Laws. In: Freistühler, H., Warnecke, G. (eds) Hyperbolic Problems: Theory, Numerics, Applications. International Series of Numerical Mathematics, vol 140. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8370-2_49
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8370-2_49
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9537-8
Online ISBN: 978-3-0348-8370-2
eBook Packages: Springer Book Archive