Fast algorithms for nonlinear kinetic models

在多尺度模型体系中，玻尔兹曼方程和相关的动力学方程是连接原子模型和连续模型的基石。这些方程描述了由大量随机运动并不断相互碰撞的粒子组成的气体或系统的非平衡动力学，并在各种领域得到了应用，如稀薄气体/等离子体动力学、辐射传输、半导体建模等。数值逼近类Boltzmann方程的突出挑战是计算碰撞项-一个高维、非线性、非局部积分算子。该项目开发的快速算法将极大地推进碰撞动力学方程的最新模拟，并将使科学家和工程师能够有效地处理更复杂的系统，这些系统以前由于评估碰撞项的巨大成本而无法实现。新算法将基于谱近似，在现有的Boltzmann解算器中，谱近似以其卓越的精度而脱颖而出。为了减少传统谱方法的巨大计算量，其主要思想是开发和利用碰撞积分中的卷积和低阶结构。具体目标有四个：1)具有一般碰撞核的Boltzmann碰撞算符的快速算法，可以有效模拟硬球模型以外的粒子相互作用；2)多物种Boltzmann方程的快速算法，这对于描述气体混合物将是非常有价值的；3)非弹性Boltzmann方程的快速算法，在颗粒材料模拟中立即有应用；4)薛定谔-量子Boltzmann系统的快速确定性求解器，模拟有限温度下玻色-爱因斯坦凝聚的动力学。

In multiscale modeling hierarchy, the Boltzmann and related kinetic equations serve as a building block that bridges atomistic and continuum models. These equations describe the non-equilibrium dynamics of a gas or system comprised of a large number of particles in random motion and constantly colliding with each other, and have found applications in various fields such as rarefied gas/plasma dynamics, radiative transfer, semiconductor modeling, etc. The prominent challenges associated with numerically approximating the Boltzmann-like equations are the expense of evaluating the collision term - a high-dimensional, nonlinear, nonlocal integral operator. Fast algorithms developed in this project will greatly advance the state-of-the-art simulation of collisional kinetic equations, and will enable scientists and engineers to effectively handle more complex systems that have previously not been feasible, due to the enormous expense of evaluating the collision term.The new algorithms will be based on spectral approximation which stands out for its superior accuracy among the available Boltzmann solvers. To reduce the huge computational cost of conventional spectral methods, the main idea is to exploit and leverage the convolutional and low-rank structure in the collision integral. Four specific aims will be addressed: 1) fast algorithms for the Boltzmann collision operator with general collision kernel that will allow efficient simulation of particle interactions beyond hard sphere model; 2) fast algorithms for the multi-species Boltzmann equation, which would be invaluable for describing gaseous mixtures; 3)fast algorithms for the inelastic Boltzmann equation, which have immediate application in modeling granular materials; 4) a fast deterministic solver for the Schrodinger-quantum Boltzmann system modeling the kinetics of Bose-Einstein condensate at finite temperature.