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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

A priori error estimates for numerical
methods for scalar conservation laws.
Part I: The general approach


Authors: Bernardo Cockburn and Pierre-Alain Gremaud
Journal: Math. Comp. 65 (1996), 533-573
MSC (1991): Primary 65M60, 65N30, 35L65
MathSciNet review: 1333308
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Abstract: In this paper, we construct a general theory of a priori error estimates for scalar conservation laws by suitably modifying the original Kuznetsov approximation theory. As a first application of this general technique, we show that error estimates for conservation laws can be obtained without having to use explicitly any regularity properties of the approximate solution. Thus, we obtain optimal error estimates for the Engquist-Osher scheme without using the fact (i) that the solution is uniformly bounded, (ii) that the scheme is total variation diminishing, and (iii) that the discrete semigroup associated with the scheme has the $\mathrm{L}^{1}$-contraction property, which guarantees an upper bound for the modulus of continuity in time of the approximate solution.


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Additional Information

Bernardo Cockburn
Affiliation: School of Mathematics, University of Minnesota, 127 Vincent Hall, Minneapolis, Minnesota 55455
Email: cockburn@math.umn.edu

Pierre-Alain Gremaud
Affiliation: Department of Mathematics, North Carolina State University, Box 8205, Raleigh, North Carolina 27695-8205
Email: gremaud@dali.math.ncsu.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-96-00701-6
PII: S 0025-5718(96)00701-6
Keywords: A priori error estimates, monotone schemes, conservation laws
Received by editor(s): August 22, 1994
Received by editor(s) in revised form: February 22, 1995
Additional Notes: First author partially supported by the National Science Foundation (Grant DMS-9407952) and by the University of Minnesota Supercomputer Institute.
Second author partially supported by the University of Minnesota Supercomputer Institute.
Article copyright: © Copyright 1996 American Mathematical Society