TY - JOUR

T1 - A priori error estimates for numerical methods for scalar conservation laws. Part I

T2 - The general approach

AU - Cockburn, Bernardo

AU - Gremaud, Pierre Alain

N1 - Copyright:
Copyright 2005 Elsevier B.V., All rights reserved.

PY - 1996/4

Y1 - 1996/4

N2 - 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.

AB - 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.

KW - A priori error estimates

KW - Conservation laws

KW - Monotone schemes

UR - http://www.scopus.com/inward/record.url?scp=0030360067&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030360067&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-96-00701-6

DO - 10.1090/S0025-5718-96-00701-6

M3 - Article

AN - SCOPUS:0030360067

VL - 65

SP - 533

EP - 573

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 214

ER -