Asymptotic preserving high order generalized upwind SBP schemes with IMEX time integration applied to kinetic transport models

动力学模型普遍描述了与自然科学和工程科学相关的物理过程，其特征是高维双曲平衡定律和模拟粒子相互作用的碰撞算子。与在空间和时间上平均量的偏微分方程建立的宏观流体模型相比，动力学流体模型更接近于粒子描述。它们的更深层次提供了对不太了解的现象的更多见解，例如稀薄气体或可压缩湍流对直接数值模拟的极端要求。解决小尺度问题需要进一步的数学建模，因此动力学模型构成了可行的候选基础。它们通过平均自由程和特征长度之比的多尺度特性来桥接尺度。对于消失比，合理的动力学方程收敛于宏观模型，如可压缩欧拉方程。在接近极限的情况下，宏观模型在降低成本的情况下足够精确。然而，动力学模型的多尺度性质可能会随着相应宏观模型的可行性在空间和时间上发生局部变化。需要完整的动力学模型，主要的数值挑战是高维性、物理上有趣的碰撞算子的非线性、渐近极限的离散保存以及多尺度引起的刚度。本提案旨在提出动力学模型的数值格式，以促进对多尺度行为和欠分解流体流动的理解。总体目标是为动力学模型设计新颖的前沿数值技术，这些模型在稳定性，准确性和渐近性方面建立了坚实的数学基础。为此，我们沿着渐近保持方案的路径，在离散水平上从动力学模型过渡到宏观模型，并利用具有可证明的精度和稳定性的结构保持，离散化独立的分部求和（SBP）框架，以及隐式-显式（IMEX）时间积分的新途径和空间离散方程的适当分裂。基于微观-宏观分解，我们确定了隐式时间步进的特定刚性项。了解空间离散化与时间离散化之间以及具有不同特征的离散化项之间的相互作用对于设计精确、稳定和渐近保持方案至关重要。针对中子输运、稀薄气体和湍流等动力学方程，设计了新的高阶渐近保持IMEX迎风SBP格式。我们通过统一分析新开发的完全离散格式的线性和非线性稳定性性质以及渐近保持性，包括非线性情况下渐近保持性和熵稳定性之间的相互作用，努力实现预测稳定性。因此，我们有可能对未解流体流动现象有更深入的了解。

Kinetic models universally describe physical processes relevant to natural and engineering sciences at the level of hyperbolic balance laws characterized by high dimensionality and collision operators modeling particle interaction. Compared to macroscopic fluid models built from PDEs in space and time for averaged quantities, kinetic fluid models are closer to particle descriptions. Their deeper level offers more insight into less understood phenomena, e.g. rarefied gases or compressible turbulence with extreme demands on direct numerical simulation. Resolving small scales requires further mathematical modeling, whereby kinetic models constitute viable candidates to build upon. They bridge scales by their multiscale nature with respect to the ratio of mean free path and characteristic length. For vanishing ratios, reasonable kinetic equations converge towards a macroscopic model such as the compressible Euler equations. Close to the limit, the macroscopic model is sufficiently accurate at reduced cost. However, the multiscale nature of kinetic models may vary locally in space and time in concert with the viability of the corresponding macroscopic model. Requiring the full kinetic model, major numerical challenges are high dimensionality, nonlinearity of physically interesting collision operators, discrete preservation of the asymptotic limit, and stiffness caused by multiple scales. This proposal intends to advance numerical schemes for kinetic models in order to forward the understanding of multiscale behavior and underresolved fluid flow. The overall goal is to devise novel cutting-edge numerical techniques for kinetic models which stand on firm mathematical ground regarding stability, accuracy and asymptotics. To this end, we follow the path of asymptotic preserving schemes to pass from kinetic to macroscopic models on the discrete level and utilize the structure preserving, discretization independent summation-by-parts (SBP) framework owning provable accuracy and stability properties together with a new avenue to implicit-explicit (IMEX) time integration and suitable splittings of the space-discretized equations. Based on micro-macro decompositions, we identify specific stiff terms for implicit time stepping. Understanding the interplay between space and time discretization and between discretized terms with different characteristics is crucial to the design of accurate, stable and asymptotic preserving schemes. New high order asymptotic preserving IMEX upwind SBP schemes will be designed for kinetic equations related to neutron transport, rarefied gases and turbulence. We strive for predictive stability by a unified analysis of the linear and nonlinear stability properties and asymptotic preservation of the newly developed fully discrete schemes, including the interplay between asymptotic preservation and entropy stability in the nonlinear case. Potentially, we will thus enable a deeper understanding of underresolved fluid flow phenomena.