Gas-kinetic finite volume methods, flux-vector splitting, and artificial diffusion

In order to improve our understanding of Boltzmann-type schemes, which have been recently proposed for the Euler and Navier-Stokes equations by Prendergast and Xu, we have modified the gas-kinetic approach and applied it to simple hyperbolic conservation laws. We found that the gas-kinetic discretization can be explained in terms of standard central difference and upwind schemes. Artificial viscosity concepts are reviewed and linked to the grid size and the physical length scale of the discontinuity. Also, a new three-dimensional gas-kinetic scheme for the numerical Navier-Stokes equations, whose solution satisfies the entropy condition, is presented. Two numerical limits of the scheme are obtained. The first one is the one-step Lax-Wendroff scheme, and the second one is the kinetic flux vector splitting scheme. A new relaxation scheme for steady state calculations is also formulated and implemented in the multigrid time stepping technique of Jameson. When applied to the Euler equations, the resulting method yields high accuracy and fast convergence to a steady state.