A new class of high-order accuracy numerical methods for the BGK model of the Boltzmann equation is presented. The schemes are based on a semi-lagrangian formulation of the BGK equation; time integration is dealt with DIRK (Diagonally Implicit Runge Kutta) and BDF methods; the latter turn out to be accurate and computationally less expensive than the former. Numerical results and examples show that the schemes are reliable and efficient for the investigation of both rarefied and fluid regimes in gasdynamics. arXiv:1411.7929v1 [math.NA] 28 Nov 2014 1 More precisely, from the BGK model, to zero-th order in ε, one obtains the compressible Euler equations in the fluid-dynamic limit, while to first order in ε, the moments satisfy equations of compressible Navier-Stokes type, but with the wrong value for the Prandtl number. This problem can be fixed by resorting to the so-called ES-BGK model [4], but in the present paper we shall restrict to the classical BGK model.

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