Kinetic Solvers with Adaptive Mesh in Phase Space

TL;DR

This paper introduces an Adaptive Mesh in Phase Space (AMPS) methodology that dynamically adapts meshes in both configuration and velocity spaces for efficient multi-dimensional kinetic equation solutions, demonstrated across various physical applications.

Contribution

The paper presents a novel AMPS approach with new algorithms for solving Boltzmann equations using adaptive meshing, importance sampling, and variance reduction techniques.

Findings

Reduces computational cost and memory usage in kinetic simulations.

Successfully applied to hypersonic flows, plasma kinetics, and semiconductor electron transport.

Demonstrates improved efficiency over traditional fixed-mesh methods.

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 data structure. The mesh in r-space is automatically generated around embedded boundaries and dynamically adapted to local solution properties. The mesh in v-space is created on-the-fly for each cell in r-space. Mappings between neighboring v-space trees implemented for the advection operator in configuration space. 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 mesh in velocity space: importance sampling, multi-point projection method, and the variance reduction method. We have developed an efficient algorithm for calculating the linear Boltzmann collision integral for elastic and inelastic collisions in a Lorentz gas. 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 computational cost and memory usage for solving challenging kinetic problems.