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Abstract. An Adaptive Mesh in Phase Space (AMPS) methodology has been developed for solving multi-dimensional kinetic equations by the discrete velocity method. A Cartesian mesh for both configuration (r) and velocity (v) spaces is produced using a "Tree of Trees" (ToT) data structure. The r-mesh is automatically generated around embedded boundaries, and is dynamically adapted to local solution properties. The v-mesh is created on-the-fly for each r-cell. Mappings between neighboring v-space trees is implemented for the advection operator in rspace. We have developed new algorithms for solving the full Boltzmann and linear Boltzmann equations with AMPS. Several recent innovations were used to calculate the discrete Boltzmann collision integral with dynamically adaptive v-mesh: the importance sampling, multi-point projection, and variance reduction methods. We have developed an efficient algorithm for calculating the linear Boltzmann collision integral for elastic and inelastic collisions of hot light particles in a Lorentz gas. The new AMPS technique has been demonstrated for simulations of hypersonic rarefied gas flows, ion and electron kinetics in weakly ionized plasma, radiation and light particle transport through thin films, and electron streaming in semiconductors. We have shown that AMPS allows minimizing the number of cells in phase space to reduce the computational cost and memory usage for solving challenging kinetic problems.

We describe our progress toward the development of a unified flow solver (UFS) that can automatically separate nonequilibrium and near-equilibrium domains and switch between continuum and kinetic solvers to combine the efficiency of continuum models with the accuracy of kinetic models. Direct numerical solution of the Boltzmann transport equation is used in kinetic regions, whereas kinetic schemes of gas dynamics are used elsewhere. The efficiency and numerical stability of the UFS is attained by using similar computational techniques for the kinetic and continuum solvers and by employing intelligent domain decomposition algorithms. Different criteria for identifying kinetic and continuum areas and two different mechanisms of coupling Boltzmann and Euler solvers are explored. Solutions of test problems with small Knudsen number are presented to illustrate the capabilities of the UFS for different conditions. It is shown that the UFS can automatically introduce and remove kinetic patches to maximize the accuracy and efficiency of simulations. To our knowledge, this is the first attempt to use direct Boltzmann and continuum flow solvers for developing a hybrid code with solution adaptive domain decomposition.