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

渐近保持高阶广义迎风 SBP 方案与应用于动力学输运模型的 IMEX 时间积分

• 批准号: 526073189
• 负责人: Dr. Sigrun Ortleb
• 金额: --
• 依托单位: Department of Applied Mathematics
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/526073189?language=en
• 关键词: Asymptotic; preserving; order; generalized; upwind

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.

动力学模型普遍地描述了与自然科学和工程科学相关的物理过程，其特征是高维和碰撞算子模拟粒子相互作用的双曲平衡定律。与由时间和空间上的偏微分方程组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较少了解的现象的更多洞察，例如稀薄气体或具有对直接数值模拟的极端要求的可压缩湍流。解决小尺度问题需要进一步的数学建模，从而使动力学模型成为可行的备选方案。它们通过平均自由程与特征长度之比的多尺度性质来连接尺度。对于消失率，合理的动力学方程收敛到宏观模型，如可压缩欧拉方程。在接近极限的情况下，宏观模型以较低的成本获得足够的精度。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而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、物理上有趣的碰撞算子的非线性、渐近极限的离散保持以及多尺度引起的刚度。这项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未完全分辨的流体流动的理解。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建立在坚实的数学基础上。为此，我们遵循渐近保持格式的路径，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、离散化独立部分求和(SBP)框架以及隐式-显式(IMEX)时间积分和空间离散方程的适当分裂的新途径。基于微观-宏观分解，我们识别了隐式时间推进的特定刚性项。了解空间和时间离散化之间的相互作用以及不同特征的离散化项之间的相互作用对于设计精确、稳定和渐近的保持格式是至关重要的。新的高阶渐近守恒IMEX迎风SBP格式将被设计用于与中子输运、稀薄气体和湍流有关的动力学方程。我们通过统一分析新发展的全离散格式的线性和非线性稳定性和渐近保持性，包括非线性情形下渐近保持性和熵稳定性之间的相互作用，来争取预测稳定性。潜在地，我们将使我们能够更深入地了解未解决的流体流动现象。