Positivity and Stability Analysis of LSKUM in One and Two Dimensions.

Numerical schemes for fluid flows must preserve the positivity of density and pressure. This is a weak stability condition and is the first step towards establishing an entropy inequality. It also allows us to derive a rigorous CFL condition. Positivity of finite volume methods based on Kinetic Theory has been established by Villedieu and Estivalezes and some general results have been given for finite volume methods in Perthame. It must be remarked that very few numerical methods for Euler equations are positivity preserving. In the present work, the positivity and stability of first order LSKUM in one- and two-dimensions is established under a CFL-like condition, though the stability results are not very strong. The analysis leads to the discovery of many length scales which are integral averages of the node spacings, and are used in the definition of the CFL number. In 2-D, the concept of a "genuine connectivity" is defined which can be used to obtain a "good" connectivity for LSKUM