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的物理过程相关的自然和工程科学的双曲平衡定律的水平，其特征在于高维和碰撞算子模拟粒子相互作用。相对于从空间和时间的平均量的偏微分方程建立的宏观流体模型，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，例如对直接数值模拟有极端要求的稀薄气体或可压缩湍流。解决小尺度需要进一步的数学建模，其中动力学模型构成可行的候选人建立。他们的桥梁规模的多尺度性质的比例，平均自由程和特征长度。对于消失比，合理的动力学方程收敛到宏观模型，如可压缩欧拉方程。接近极限，宏观模型在降低的成本下足够准确。然而，动力学模型的多尺度性质可能在空间和时间上局部变化，与相应宏观模型的可行性一致。需要完整的动力学模型，主要的数值挑战是高维，物理上有趣的碰撞算子的非线性，渐近极限的离散保存，和多尺度引起的刚度。该建议旨在推进动力学模型的数值方案，以促进对多尺度行为和欠分辨流体流动的理解。总体目标是为动力学模型设计新的尖端数值技术，这些模型在稳定性，准确性和渐近性方面具有坚实的数学基础。为此，我们遵循渐进保持计划的路径，从动力学到宏观模型的离散水平，并利用结构保持，离散化独立的部分求和（SBP）框架，具有可证明的精度和稳定性，以及一个新的途径隐式显式（IMEX）时间积分和适当的分裂的空间离散方程。基于微观-宏观分解，我们确定了隐式时间步长的具体刚性项。理解空间和时间离散化之间的相互作用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、稳定和渐近保持的方案至关重要。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。我们努力的预测稳定性的线性和非线性的稳定性和渐近保存的新开发的完全离散的计划，包括渐近保存和熵稳定的非线性情况下之间的相互作用的统一分析。潜在地，我们将因此能够更深入地理解欠分辨流体流动现象。

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.