Recovery-Based Error Estimator for the Discontinuous Galerkin Method for Nonlinear Scalar Conservation Laws in One Space Dimension

2015 ◽  
Vol 66 (2) ◽  
pp. 459-476 ◽  
Author(s):  
Mahboub Baccouch
Keyword(s):  
Conservation Laws ◽  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Space Dimension ◽  
Error Estimator ◽  
Scalar Conservation Laws ◽  
Scalar Conservation ◽  
Nonlinear Scalar Conservation Laws ◽  
One Space Dimension

scholarly journals Regularity estimates for scalar conservation laws in one space dimension

2018 ◽  
Vol 15 (04) ◽  
pp. 623-691 ◽  
Author(s):  
Elio Marconi
Keyword(s):  
Conservation Laws ◽  
Space Dimension ◽  
Scalar Conservation Laws ◽  
Kinetic Formulation ◽  
Flux Function ◽  
Regularity Estimates ◽  
Degeneracy Condition ◽  
Inflection Points ◽  
Scalar Conservation ◽  
One Space Dimension

We deal with the regularizing effect that, in scalar conservation laws in one space dimension, the nonlinearity of the flux function [Formula: see text] has on the entropy solution. More precisely, if the set [Formula: see text] is dense, the regularity of the solution can be expressed in terms of [Formula: see text] spaces, where [Formula: see text] depends on the nonlinearity of [Formula: see text]. If moreover the set [Formula: see text] is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that [Formula: see text] for every [Formula: see text] and that this can be improved to [Formula: see text] regularity except an at most countable set of singular times. Finally, we present some examples that show the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces.

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