TY - JOUR

T1 - Quasimonotone schemes for scalar conservation laws. Part III

AU - Cockburn, Bernardo

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1990

Y1 - 1990

N2 - In this paper the definition and analysis of the quasimonotone numerical schemes is extended to the general case d > 1, where d is the number of space variables. These schemes are constructed using the simple but very important class of monotone schemes that are defined by two-point monotone fluxes. To enforce the compactness in L∞(Lloc1) of the sequence of approximate solutions, the case of meshes that are a Cartesian product of one-dimensional partitions is addressed. It is proved that the main stability and convergence results for one-dimensional quasimonotone schemes (of the first type) also hold in the general case. As a by-product of this theory, the theory of relaxed, quasimonotone schemes is developed. These schemes are L∞-stable, and they can be more accurate than the quasimonotone schemes; unfortunately, the compactness in L∞(Lloc1 is lost. Nevertheless, if they converge, they do so to the entropy solution.

AB - In this paper the definition and analysis of the quasimonotone numerical schemes is extended to the general case d > 1, where d is the number of space variables. These schemes are constructed using the simple but very important class of monotone schemes that are defined by two-point monotone fluxes. To enforce the compactness in L∞(Lloc1) of the sequence of approximate solutions, the case of meshes that are a Cartesian product of one-dimensional partitions is addressed. It is proved that the main stability and convergence results for one-dimensional quasimonotone schemes (of the first type) also hold in the general case. As a by-product of this theory, the theory of relaxed, quasimonotone schemes is developed. These schemes are L∞-stable, and they can be more accurate than the quasimonotone schemes; unfortunately, the compactness in L∞(Lloc1 is lost. Nevertheless, if they converge, they do so to the entropy solution.

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U2 - 10.1137/0727018

DO - 10.1137/0727018

M3 - Article

AN - SCOPUS:0025388515

VL - 27

SP - 259

EP - 276

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

IS - 1

ER -