Stochastic subgrid scale modeling and structure-preserving flux limiting for hyperbolic systems

该项目的目标是推导出物理兼容的随机参数化，用于微观行为由偏微分方程双曲系统控制的粗粒度流过程模拟。利用变分多尺度（VMS）方法的框架，对细网格有限元离散化进行平均，提取可压缩欧拉方程和浅水方程的随机子网格尺度模型。提出的子网格升级方法类似于reynolds -average Navier- Stokes （RANS）湍流模型。“慢”粗尺度分量（平均）的过滤方程包含依赖于“快”细尺度分量（波动）的非线性项。用随机过程代替这些项，就得到了均值演化方程的封闭形式简化系统。不能保证这种多尺度近似是熵稳定的和不变域保持的。在这个项目中，将使用代数通量校正（AFC）工具，特别是由首席研究员及其合作者开发的用于双曲问题的标准有限元离散化的新颖凸极限技术，确保模拟结果的物理可接受性。单片AFC方法通过在临界区域切换到保持结构的低阶方案，不仅可以实现离散的极大值原理，而且可以实现熵稳定的充分条件。随机VMS方法的极限通量应是熵耗散的，并满足所有相关约束。现有的关系，熵/涡流粘度模型将研究理论和数值。

The objective of this project is the derivation of physics-compatible stochastic parametrizations for coarse-grained simulations of flow processes whose microscopic behavior is governed by hyperbolic systems of partial differential equations. Using the framework of variational multiscale (VMS) methods, a fine-grid finite element discretization will be averaged to extract stochastic subgrid scale models for the compressible Euler equations and for the shallow water equations. The proposed approach to subgrid upscaling resembles Reynolds-averaged Navier--Stokes (RANS) turbulence modeling. The filtered equations for "slow" coarse-scale components (averages) contain nonlinear terms that depend on "fast" fine-scale components (fluctuations). Replacing these terms with stochastic processes, one obtains a closed-form reduced system of evolution equations for the averages. There is no guarantee that such multiscale approximations are entropy stable and invariant domain preserving. In this project, physical admissibility of simulation results will be ensured using algebraic flux correction (AFC) tools and, in particular, novel convex limiting techniques developed by the principal investigator and his collaborators for standard finite element discretizations of hyperbolic problems. Monolithic AFC approaches make it possible to enforce not only discrete maximum principles but also sufficient conditions of entropy stability by switching to a structure-preserving low-order scheme in critical regions. The limited fluxes of a stochastic VMS method should be entropy dissipative and satisfy all relevant constraints. Existing relationships to entropy/eddy viscosity models will be investigated theoretically and numerically.