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

稳定、高效、准确的低马赫流高阶离散化

基本信息

批准号：
318864836
负责人：
Professor Dr. Sebastian Noelle
金额：
--
依托单位：
Institut für Geometrie und Praktische Mathematik (IGPM)
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2016
资助国家：
德国
起止时间：
2015-12-31 至 2020-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/318864836?language=en
关键词：
stable efficient accurate order discretization

项目摘要

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.

我们提出，进一步发展，并分析了一种特别有效的方法来计算奇摄动问题，如低马赫数下的可压缩Navier-Stokes方程。一个关键的贡献是基于合适的参考解决方案（RS）将通量划分为快速和慢速组分。这与众所周知的隐式显式（IMEX）范式相结合。我们将生成的方案称为RS-IMEX。最近对双曲原型系统的低阶有限体积格式和刚性ode的高阶IMEX Runge-Kutta和IMEX BDF格式的研究表明，精度有所提高。在这个项目中，我们将RS-IMEX方法推广到高阶IMEX不连续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，以使它们适合RS-IMEX环境：基于参考解的预调节器，适应极端刚度的线性求解器，混合IMEX DG离散化，减少隐式问题的维度，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考解尤为重要。在整个开发过程中，我们分析了每个新分量的渐近一致性和稳定性，并将所得方法与最新的最先进的IMEX方案进行了比较。我们的目标是建立基于dg的RS-IMEX格式，使其成为求解渐近刚性问题的一种通用的、系统的、特别精确的、稳定的和有效的方法。