scholarly journals The Convergence Rate of Approximate Solutions for Nonlinear Scalar Conservation Laws

1992 ◽  
Vol 29 (6) ◽  
pp. 1505-1519 ◽  
Author(s):  
Haim Nessyahu ◽  
Eitan Tadmor
Keyword(s):  
Convergence Rate ◽  
Conservation Laws ◽  
Approximate Solutions ◽  
Scalar Conservation Laws ◽  
Scalar Conservation ◽  
Nonlinear Scalar Conservation Laws

scholarly journals On a quadratic functional for scalar conservation laws

2014 ◽  
Vol 11 (02) ◽  
pp. 355-435 ◽  
Author(s):  
Stefano Bianchini ◽  
Stefano Modena
Keyword(s):  
Convergence Rate ◽  
Conservation Laws ◽  
Natural Extension ◽  
Approximate Solutions ◽  
Quadratic Functional ◽  
Scalar Conservation Laws ◽  
Nonlinear Case ◽  
Glimm Scheme ◽  
Quadratic Interaction ◽  
Scalar Conservation

We prove a quadratic interaction estimate for approximate solutions to scalar conservation laws obtained by the wavefront tracking approximation or the Glimm scheme. This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme. The proof is based on the introduction of a quadratic functional 𝔔(t), decreasing at every interaction, and such that its total variation in time is bounded. Differently from other interaction potentials present in the literature, the form of this functional is the natural extension of the original Glimm functional, and coincides with it in the genuinely nonlinear case.

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