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

Bernardo Cockburn, Pierre Alain Gremaud

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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 L1-contraction property, which guarantees an upper bound for the modulus of continuity in time of the approximate solution.

Original languageEnglish (US)
Pages (from-to)533-573
Number of pages41
JournalMathematics of Computation
Volume65
Issue number214
DOIs
StatePublished - Apr 1996

Keywords

  • A priori error estimates
  • Conservation laws
  • Monotone schemes

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