Fast algorithms for nonlinear kinetic models

在多尺度建模层次中，玻尔兹曼和相关的动力学方程作为一个构建块，连接原子模型和连续模型。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，并且已经在各种领域中找到了应用，例如稀薄气体/等离子体动力学，辐射传递，半导体建模，与数值近似玻尔兹曼方程相关的突出挑战是评估碰撞项的费用-高-一维非线性非局部积分算子。在这个项目中开发的快速算法将大大推进国家的最先进的碰撞动力学方程的模拟，并将使科学家和工程师能够有效地处理更复杂的系统，以前是不可行的，由于巨大的费用评估的碰撞term.新的算法将基于光谱近似，其中脱颖而出的上级精度之间可用的玻尔兹曼求解器。为了减少传统谱方法的巨大计算成本，主要思想是利用和利用碰撞积分中的卷积和低秩结构。具体目标有四个：1）具有一般碰撞核的Boltzmann碰撞算子的快速算法，这将允许有效模拟硬球模型之外的粒子相互作用; 2）多组分Boltzmann方程的快速算法，这将对描述气体混合物非常有价值; 3）非弹性Boltzmann方程的快速算法，这将直接应用于模拟颗粒材料; 4）有限温度下玻色-爱因斯坦凝聚动力学的薛定谔-量子玻尔兹曼系统的快速确定性求解。

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.