Continuous finite element methods for under resolved turbulence in compressible flow

在这里，我们将概述本项目的主要研究问题以及我们打算如何解决这些问题。三维可压缩流动问题将湍流等不可压缩流动中已知的不稳定性与亚音速区域的声学效应以及超音速区域的激波、接触不连续性和稀疏性等非线性波动的影响相结合。如何设计一种方法，在保持计算效率的同时，以统一的方式处理所有这些现象？2.可压缩流动的数值方法的分析通常限于对标量问题的渐近估计。部分原因是缺乏对连续方程的理论理解，但即使对于线性模型问题，抑制局部振荡所需的非线性激波捕获型格式仍然知之甚少。如果对精确解作额外的假设，能进行更完整的数值分析吗？例如，解可以分解成有限数量的光滑部分，由间断隔开，这些间断的支持具有某些有利的性质？3.在实践中，计算效率对于大规模计算是至关重要的。数值稳定性和使用显式时间步长的可能性使间断Galerkin方法在可压缩流动计算中很受欢迎。然而，在二维空间中，分段仿射间断逼近的自由度是标准连续有限元的6倍。这个数字在三维空间里增加到了20倍。不连续Galerkin方法还依赖于昂贵的Riemann求解器，这些求解器可能并不总是健壮的，见[Abg17a]。因此，我们问是否可以设计一种连续有限元方法，结合不连续Galerkin方法的优点，使用更少的DOF[Gue16a，Abg17b]？本项目的主要目标是利用我们最近的结果来解决这些问题，这些结果基于有限元方法在不可压缩的Navier-Stokes方程[Mou22]的近似下近似湍流流动，用于标量线性输运问题的稳定化有限元的局部估计[Bu22a]和具有非线性稳定化的标量线性传输问题的全局估计[Bu22b]。本文的基础是：1.发展非线性波的不变保激波捕捉方法；2.发展控制二次振荡的线性稳定化方法；3.显式和隐式-显式时间离散格式的发展和数值分析；4.质量矩阵的负载平衡区域分解方法，用于显式时间推进；5.可压缩湍流流动的高性能三维计算。《双曲问题数值方法手册》，2017年第18期。[Abg17b]Abgrall，R.使用全局连续逼近和避免质量矩阵的双曲型问题的高阶格式。J.Sci。电脑。(2017)[Bu22a]Burman，E.用连续有限元离散的梯度跳跃上具有内部惩罚稳定化的瞬时运输问题的加权误差估计。越南数学杂志。(2022年)。[Bu22b]Burman，E.关于应用于双曲型守恒律的连续有限元方法的线性和非线性镇定之间相互作用的一些观察。出现在SIAM J.Sci上。电脑。[Gue16a]Guermond，J.-L.；Popov，B.非线性标量守恒方程一阶Lagrange有限元方法的误差估计.暹罗·J·努默。肛门。(2016).项目[Mou22]Moura，R.C.；Cassinelli，A.；da Silva，A.；Burman，E.；Sherwin，S.用于欠分辨湍流模拟的谱/HP元素离散的梯度跳跃惩罚稳定化。CMAME(2022)

Here we will give a summary of the main research questions of the present project and how we intend to address them.1. Three dimensional compressible flow problems combine the instabilities known from incompressible flows such as turbulence with the effects of acoustics in the subsonic regime and nonlinear waves such as shocks, contact discontinuities and rarefactions in the supersonic regime. How can a method be designed that handles all these phenomena in a unified way while remaining computationally efficient?2. The analysis of numerical methods for compressible flow is typically restricted to asymptotic estimates for scalar problems. In part due to the lack of theoretical understanding of the continuous equations, but even for linear model problems the non-linear shock capturing type schemes necessary to suppress local oscillations, remain poorly understood. Can a more complete numerical analysis be carried out if additional assumptions are made on the exact solution, for instance that the solution can be decomposed in a finite number of smooth parts, separated by discontinuities, where the support of these discontinuities has some favourable properties?3. In practice computational efficiency is of essence for large-scale computations. Aspects of numerical stability and the possibility to use explicit time-stepping have made the discontinuous Galerkin method popular for compressible flow computations. However, in two space dimensions piecewise affine discontinuous approximation has six times as many degrees of freedom (dofs) as standard continuous FEM. This number increases to 20 times in three dimensions. The discontinuous Galerkin method also relies on expensive Riemann solvers that may not always be robust, see [Abg17a].Therefore, we ask if a continuous finite element method can be designed that incorporates the advantages of the discontinuous Galerkin using substantially fewer dofs [Gue16a,Abg17b]?The main aim of the present project is to address these questions drawing on our recent results on finite element methods (FEM) for approximating of turbulent flows in the approximation of the incompressible Navier-Stokes' equations [Mou22], local estimates for stabilized FEM for scalar linear transport problems [Bu22a] and global estimates for scalar linear transport problems discretized with nonlinear stabilization [Bu22b]. The cornerstones of the present proposal are:1. development of invariant preserving shock capturing methods for nonlinear waves;2. development of linear stabilisation methods for the control of secondary oscillations;3. development and numerical analysis of explicit and implicit-explicit time discretisation schemes;4. load balanced domain decomposition methods for the mass matrix, for explicit time-stepping;5. High performance three dimensional computations of compressible turbulent flows.References:[Abg17a] Abgrall, R. Some failures of Riemann solvers. Handbook of numerical methods for hyperbolic problems, 18 2017. [Abg17b] Abgrall, R. High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices. J. Sci. Comput. (2017).[Bu22a] Burman, E. Weighted Error Estimates for Transient Transport Problems Discretized Using Continuous Finite Elements with Interior Penalty Stabilization on the Gradient Jumps. Vietnam J. Math. (2022). [Bu22b] Burman, E. Some observations on the interaction between linear and nonlinear stabilization for continuous finite element methods applied to hyperbolic conservation laws. To appear in SIAM J. Sci. Comput. [Gue16a] Guermond, J.-L.; Popov, B. Error estimates of a first-order Lagrange finite element technique for nonlinear scalar conservation equations. SIAM J. Numer. Anal. (2016).\item[Mou22] Moura, R. C.; Cassinelli, A.; da Silva, A.; Burman, E.; Sherwin, S. Gradient jump penalty stabilisation of spectral/hp element discretisation for under-resolved turbulence simulations. CMAME (2022)