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This work is aimed to develop a new class of methods for the BGK model of theBoltzmann equation. This technique allows to get high order of accuracy both inspace and time, theoretically without CFL stability limitation. It's based on aLagrangian formulation of the problem: information is stored on a fixed grid inspace and velocity, and the equation is integrated along the characteristics.The source term is treated implicitly by using a DIRK (Diagonally ImplicitRunge Kutta) scheme in order to avoid the time step restriction due to thestiff relaxation. In particular some L-stable schemes are tested by smooth andRiemann problems, both in rarefied and fully fluid regimes. Numerical resultsshow good accuracy and efficiency of the method.

A New Class of Conservative Large Time Step Methods for the BGK Models of the Boltzmann Equation