We introduce a coupled multiscale, multiphysics method (CM3) for solving for the behaviour of rarefied gas flows. The approach is to solve the kinetic equation for rarefied gases (the Boltzmann equation) over a very short interval of time in order to obtain accurate estimates of the components of the stress tensor and heat-flux vector. These estimates are used to close the conservation laws for mass, momentum and energy, which are subsequently used to advance continuum-level flow variables forward in time. After a finite time interval, the Boltzmann equation is solved again for the new continuum field, and the cycle is repeated. The target applications for this type of method are transition-regime gas flows for which standard continuum models (e.g. Navier–Stokes equations) cannot be used, but solution of Boltzmann's equation is prohibitively expensive. The use of molecular-level data to close the conservation laws significantly extends the range of applicability of the continuum conservation laws. In this study, the CM3 is used to perform two proof-of-principle calculations: a low-speed Rayleigh flow and a thermal Fourier flow. Velocity, temperature, shear-stress and heat-flux profiles compare well with direct-simulation Monte Carlo solutions for various Knudsen numbers ranging from the near-continuum regime to the transition regime. We discuss algorithmic problems and the solutions necessary to implement the CM3, building upon the conceptual framework of the heterogeneous multiscale methods.