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An efficient third-order discrete unified gas kinetic scheme (DUGKS) withefficiency is presented in this work for simulating continuum and rarefiedflows. By employing two-stage time-stepping scheme and the high-order DUGKSflux reconstruction strategy, third-order of accuracy in both time and spacecan be achieved in the present method. It is also analytically proven that thesecond-order DUGKS is a special case of the present method. Compared with thehigh-order lattice Boltzmann equation {LBE} based methods, the present methodis capable to deal with the rarefied flows by adopting the Newton-Cotesquadrature to approximate the integrals of moments. Instead of beingconstrained by the second-order (or lower-order) of accuracy in time splittingscheme as in the conventional high-order Runge-Kutta (RK) based kineticmethods, the present method solves the original BE, which overcomes thelimitation in time accuracy. Typical benchmark tests are carried out forcomprehensive evaluation of the present method. It is observed in the teststhat the present method is advantageous over the original DUGKS in accuracy andcapturing delicate flow structures. Moreover, the efficiency of the presentthird-order method is also shown in simulating rarefied flows.

A third-order discrete unified gas kinetic scheme for continuum and rarefied flows: low-speed isothermal case