A relaxation scheme for the approximation of the pressureless Euler equations

In the present work, we consider the numerical approximation of pressureless gas dynamics in one and two spatial dimensions. Two particular phenomena are of special interest for us, namely δ-shocks and vacuum states. A relaxation scheme is developed which reliably captures these phenomena. In one space dimension, we prove the validity of several stability criteria, i.e., we show that a maximum principle as well as the TVD property for the discrete velocity component and the validity of discrete entropy inequalities hold. Several numerical tests considering not only the developed first-order scheme but also a classical MUSCL-type second-order extension confirm the reliability and robustness of the relaxation approach. The paper extends previous results on the topic: the stability conditions for relaxation methods for the pressureless case are refined, useful properties for the time stepping procedure are established and two-dimensional numerical results are presented. c © ??? John Wiley & Sons, Inc.

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