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An implicit high-order discontinuous Galerkin (DG) method is developed tofind steady-state solution of rarefied gas flow described by the Boltzmannequation with full collision operator. In the physical space, velocitydistribution function is approximated by the piecewise polynomials of degree upto 4, while in the velocity space the fast spectral method is incorporated intothe DG to evaluate the collision operator. A specific polynomial approximationfor the collision operator is proposed to reduce the computational complexityof the fast spectral method by $K$ times, where for two-dimensional problems$K$ is 15 when DG with 4th-order polynomials are used on triangular mesh. Basedon the first-order upwind scheme, a sweeping technique is employed to solve thelocal linear equations resulting from the DG discretization sequentially overspatial elements. This technique can preserve stability of the scheme andrequires no nonlinear limiter in solving hypersonic rarefied gas flow when theflow is fully resolved. Moreover, without assembling large sparse linearsystem, the computational cost in terms of memory and CPU time can besignificantly reduced. Five different one/two-dimensional tests includinglow-speed microscale flows and hypersonic rarefied gas flows are used to verifythe proposed approach. Our results show that, DG schemes of different order ofapproximating polynomial require the same number of iterative steps to obtainthe steady-state solution with the same order of accuracy; and the higher orderthe scheme, fewer spatial elements thus less CPU time is needed. Besides, ourmethod can be faster than the finite difference solver by about one order ofmagnitude. The produced solutions can be used as benchmark data for assessingthe accuracy of other gas kinetic solvers for the Boltzmann equation and gaskinetic models that simplify the Boltzmann collision operator.

Implicit Discontinuous Galerkin Method for the Boltzmann Equation