Preprint 1998-007

Pointwise Error Estimates for Scalar Conservation Laws with Piecewise Smooth Solutions

Eitan Tadmor and Tao Tang


Abstract:

We introduce a new approach to obtain sharp pointwise error estimates for viscosity approximation (and in fact --- more general approximations) to scalar conservation laws with piecewise smooth solutions. To this end, we derive a transport inequality for an appropriately weighted error function. The key ingredient in our approach is a one-sided interpolation inequality between classical $L^1$ error estimates and $Lip^+$ stability bounds. The one-sided interpolation, interesting for its own sake, enables us to convert a global $L^1$ result into a (non-optimal) local estimate. This, in turn, provides the necessary bounds on the coefficients of the above mentioned transport inequality. Estimates on the weighted error then follow from the maximum principal, and a bootstrap argument yields optimal pointwise error bound for the viscosity approximation.

Unlike previous works in this direction, our method can deal with finitely many waves with possible collisions. Moreover, in our approach one does not follow the characteristics but instead makes use of the energy method, and hence this approach could be extended to other types of approximate solutions.



Paper:
Available as PostScript
Title:
Pointwise error estimates for scalar conservation laws with piecewise smooth solutions
Author(s):
Eitan Tadmor , mailto:tadmor@math.ucla.edu
Tao Tang, mailto:ttang@math.ucla.edu
Publishing information:
Comments:
Submitted by:
mailto:tadmor@math.ucla.edu February 4 1998.


[ 1996 Preprints | 1997 Preprints | 1998 Preprints | All Preprints | Preprint Server Homepage ]
(c) The copyright for the following documents lies with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use.

mailto:conservation@math.ntnu.no
Last modified: Thu Feb 5 11:31:45 1998
Сайт управляется системой uCoz