Implicit kinetic schemes for the Euler equations

Recently, the kinetic schemes, namely the kinetic flux-vector split (KFVS) scheme and kinetic wave/particle split (KWPS) scheme, for Euler flows have gained wide recognition for their efficiency and robustness. However, to date, all computations performed with these schemes have employed a time-explicit formulation. The explicit kinetic schemes severely restrict the time-step allowed for stability. In this paper, an implicit formulation is derived for both the KFVS and KWPS schemes, and is applied to compute the shock tube and shock structure problems in one-dimension, and oblique shock reflection from a flat plate and supersonic flow past a blunt-body in two dimensions. Results are compared with analytical results where available and solutions from explicit formulations. It is shown that implicit formulations retain the efficiency and robustness of their explicit counterparts without the restrictive time step constraints. This results in an increase in computational speed for steady state computations. To the authors' knowledge, this is the first time that the implicit kinetic schemes have been formulated.