An Entropic Kinetic Scheme with Compactly Supported Velocities

We introduce a new kinetic model where the equilibrium distribution function has a range of velocities centered about two velocities ( \(\pm \lambda \) ) in 1D. In 2D, the equilibrium distribution comprises of a range of velocities about four velocities, one in each quadrant. This essential feature of a range of velocities ( \(\lambda \pm \Delta \lambda \) ) offers a significant advantage to obtain both an exact shock capturing as well as an entropic scheme. The range of velocities (2 \(\Delta \lambda \) ) serves the purpose of augmenting the stability of the model. \(\Delta \lambda \) is used to provide additional diffusion in expansion regions. The average velocity \(\lambda \) is fixed to exactly capture grid-aligned steady discontinuities, by enforcing Rankine-Hugoniot jump conditions in the discretization process. Further, a novel kinetic relative entropy, appropriate to our kinetic model, is introduced. Along with an additional criterion, this kinetic relative entropy is used to identify expansions. Some standard 1D and 2D benchmark test cases are solved using this scheme.