Approximate solutions of scalar conservation laws

We study compactness properties of time-discrete and continuous time BGK-type schemes for scalar conservation laws, in which microscopic interactions occur only when the state of a system deviates significantly from an equilibrium distribution. The threshold deviation, $ε,$ is a parameter of the problem. In the vanishing relaxation time limit we obtain solutions of a conservation law in which flux is pointwisely close (of order $ε$) to the flux of the original equation and derive several other properties of such solutions, including an example of approximate solution to a shock for Burger's equation.

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