A stable, efficient and accurate high-order discretization for low Mach flows

我们提出，进一步发展，并分析了一个特别有效的方法来计算奇摄动问题，如可压缩Navier-Stokes方程在低马赫数。一个关键的贡献是一个新的分裂的通量快速和缓慢的组件，这是基于一个合适的参考解决方案（RS）。这与众所周知的隐式-显式（IMEX）范式相结合。我们称之为RS-IMEX方案。最近的研究表明，原型双曲型方程组的低阶有限体积格式，以及刚性常微分方程组的高阶IMEX Runge-Kutta和IMEX BDF格式的精度都有所提高，本项目将RS-IMEX方法推广到高阶IMEX间断Galerkin方法。为了构建一个高效的求解器，我们需要重新开发几个基本成分，以便它们适合RS-IMEX上下文：基于参考解的预处理器，适应极端刚度的线性求解器，减少隐式问题维数的混合IMEX DG离散化，以及可压缩和不可压缩求解器的有效耦合，后者对于有效地使用参比溶液特别重要。在整个发展过程中，我们分析了每个新的组件的渐近一致性和稳定性，并比较所产生的方法，最新的最先进的IMEX计划。我们的目标是建立基于DG的RS-IMEX计划作为一个通用的，系统的，特别是准确的，稳定的和有效的方法渐近刚性问题。

We propose, further develop, and analyze a particularly efficient method to compute singularly perturbed problems such as the compressible Navier-Stokes equations at low Mach number. A key contribution is a new splitting of the flux into fast and slow components, which is based on a suitable reference solution (RS). This is combined with the well-known implicit-explicit (IMEX) paradigm. We call the resulting scheme RS-IMEX. Recent studies of low order finite volume schemes for prototype hyperbolic systems, and high order IMEX Runge-Kutta and IMEX BDF schemes for stiff ODEs have shown increased accuracy.In this project, we generalize the RS-IMEX approach to high order IMEX discontinuous Galerkin methods. In order to build an efficient solver, we need to re-develop several essential ingredients in such a way that they are tailored to the RS-IMEX context: preconditioners based on the reference solution, linear solvers adapted to the extreme stiffness, hybrid IMEX DG discretizations reducing the dimension of the implicit problem, and an efficient coupling of compressible and incompressible solvers, the latter one being particularly important to efficiently employ the reference solution. Throughout the development we analyze the asymptotic consistency and stability of each new component, and compare the resulting methods to recent state of the art IMEX schemes.Our goal is to establish the DG-based RS-IMEX scheme as a general, systematic, particularly accurate, stable and efficient method for asymptotically stiff problems.