Istituto di Matematica Applicata e Tecnologie Informatiche     
Vicario P. Diffusive relaxation to scalar viscous conservation laws. Technical report ercim.cnr.ian//1997-1056, 1997.
In this paper we are concerned with the diffusive scaling of kinetic models with two velocities of Ruijgrok-Wu type towards scalar viscous conservation laws. The interest is mainly related to the fact that the knowledge of a physical kinetic system that relaxes gives at the same time a new way of constructing numerical schemes for the target equation. Recently, this idea has been developed for some linear and non linear diffusion equations by Jin, Pareschi and Toscani. Their work was motivated by the results of Lions and Toscani in the diffusive relaxation of Carleman-like systems towards the porous media equation. While the relaxation of hyperbolic systems towards scalar conservation laws is well studied from the pionering papers of Liu and Chen,Levermore and Liu, the study of diffusive relaxation of kinetic equations towards scalar viscous conservation laws is at the beginning. Very recently, Gabetta and Perthame studied the diffusive relaxation of the Ruijgrok-Wu model. In this paper we will study the diffusive limit of a model which generalizes system, in that the collisional frequency is taken to be function of the $v$ density.
Subject Conservation laws, diffusive relaxation

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