  In this paper, we develop a family of third order asymptotic-preserving (AP)
and asymptotically accurate (AA) diagonally implicit Runge-Kutta (DIRK) time
discretization methods for the stiff hyperbolic relaxation systems and kinetic
Bhatnagar-Gross-Krook (BGK) model in the semi-Lagrangian (SL) setting. The
methods are constructed based on an accuracy analysis of the SL scheme for
stiff hyperbolic relaxation systems and kinetic BGK model in the limiting fluid
regime when the Knudsen number approaches $0$. An extra order condition for the
asymptotic third order accuracy in the limiting regime is derived. Linear Von
Neumann stability analysis of the proposed third order DIRK methods are
performed to a simplified two-velocity linear kinetic model. Extensive
numerical tests are presented to demonstrate the AA, AP and stability
properties of our proposed schemes.
