Continuous finite element methods for under resolved turbulence in compressible flow

在这里，我们将总结当前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不可压缩流动（如湍流）的不稳定性与亚音速区域的声学效应和非线性波（如激波、接触不连续和超声速区域的稀疏性）相结合。如何设计一种方法，以统一的方式处理所有这些现象，同时保持计算效率？可压缩流动数值方法的分析通常局限于标量问题的渐近估计。部分原因是由于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冲击捕获型方案仍然知之甚少。如果对精确解作一些额外的假设，例如，假设解可以分解为有限数量的光滑部分，由不连续点分开，而这些不连续点的支持有一些有利的性质，那么是否可以进行更完整的数值分析？在实际应用中，计算效率对大规模计算至关重要。数值稳定性方面和使用显式时间步进的可能性使得不连续伽辽金方法在可压缩流动计算中很受欢迎。然而，在二维空间中，分段仿射不连续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中增加到20倍。不连续伽辽金方法也依赖于昂贵的黎曼解算器，可能并不总是鲁棒的，见[Abg17a]。因此，我们要问的是，是否可以设计一种连续有限元方法，以更少的次数结合不连续伽辽金的优点[Gue16a，Abg17b]？本项目的主要目的是利用我们最近在不可压缩的Navier-Stokes方程近似中近似湍流的有限元方法（FEM） [Mou22]、标量线性输运问题的稳定FEM的局部估计[Bu22a]和非线性稳定离散的标量线性输运问题的全局估计[Bu22b]方面的研究结果来解决这些问题。本建议的基础是：1。2.非线性波的不变性保持激波捕获方法的发展；2 .控制二次振荡的线性稳定方法的发展；3 .显式和隐显式时间离散化方案的发展与数值分析；4 .质量矩阵的负载平衡域分解方法，用于显式时间步进；可压缩湍流的高性能三维计算。参考文献：[Abg17a] Abgrall， R.一些Riemann解的失效。双曲问题数值方法手册，2018。[Abg17b]王志强，王志强。基于全局连续逼近的双曲型问题高阶格式。j .科学。第一版。(2017)。[Bu22a]李晓明，李晓明。基于离散有限元的瞬态运输问题的加权误差估计。越南；(2022)。[Bu22b]李建军，李建军，李建军，等。基于双曲型守恒律的连续有限元法的非线性与线性稳定性研究。出现在SIAM J. Sci。第一版。[Gue16a]刘志强；非线性标量守恒方程的一阶拉格朗日有限元技术的误差估计。SIAM J. number。分析的。(2016)。\item[Mou22]张晓明；Cassinelli, a;达席尔瓦，a；伯尔曼,大肠;欠分辨湍流模拟中谱/马力元离散化的梯度跳跃惩罚稳定性。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)