Kinetic solvers with adaptive mesh in phase space

Abstract

An adaptive mesh in phase space (AMPS) methodology has been developed for solving multidimensional 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 in each $r$ cell. Mappings between neighboring $v$-space trees is implemented for the advection operator in $r$ space. We have developed 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, multipoint 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. Our 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.