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New approaches to the construction of efficient high order time integration methods in the context of DG space discretisations for viscous and inviscid fluid flow

在粘性和非粘性流体流动的 DG 空间离散化背景下构建高效高阶时间积分方法的新方法

基本信息

批准号：
288967378
负责人：
Professor Dr. Andreas Meister
金额：
--
依托单位：
Institut für Partielle Differentialgleichungen
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 2019-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/288967378?language=en
关键词：
New approaches construction efficient order

项目摘要

Discontinuous space discretizations, especially the Discontinuous Galerkin(DG) methods, are a modern and popular class of numerical methods especially for computationally intensive fluid dynamics calculations. Their popularity is due to the fact that DG methods allow for high order approximations in combination with high flexibility - e.g. in choosing different polynomial degrees on neighbouring elements. Furthermore, the challenge of the future in order to enable reliable simulations of complex real life problems is the design of methods for parallel applications. Here. DG methods are perfectly suitable and thus it is necessary to develop and analyse them especially with respect to the time integration employed. In the context of practically relevant problems, semi-discrete DG equations are often extremely stiff. In the case of complex geometries, e.g. for fluid flow around obstacles, the DG mesh is locally refined with elements very different in size. In addition, for high Reynods numbers, applications require a considerable grid refinement in boundary layer zones. In this context, the time integration methods applied so far are yet far from being efficient. Especially with regard to the skillful coupling of explicit and implicit methods, as well as the use of local time steps as in multirate strategies, considerably more research is needed. In preliminary work, a robust, high order DG scheme with low numerical dissipation based on novel efficient filtering strategies has been developed. Based on this groundwork, the innovative contribution of this proposal is the development and analysis of novel IMEX time integration methods. In particular, for the first time we will incorporate hybrid approaches of the basic IMEX splitting combined with multirate methods in order to accelerate time integration of the semi-discrete DG equations. The main objective of this project is hence the development, analysis and the direct comparison of novel approaches to the construction of efficient, high order time integration schemes for viscous and inviscid fluid flow. These approaches will be studied in a uniform framework in order to develop suitable strategies to decide between IMEX or multirate method or to use a combination of both of them. In this context, stiffness detectors will be developed and analysed, we will assess concrete methods of implicit type within the IMEX approach and include multirate approaches as well. A further objective is to establish an analogy to IMEX and multirate approaches for exponential integrators which currently show considerable gain in efficiency. The The efficient time integration methods based on IMEX and multirate strategies which will be developed in this project will be highly suitable for practical applications. Hence they can be expected to set new standards both for the numerical calculation of fluid flow as for the simulation of phenomena based on fluid-structure-interaction which will be focussed on in the future.

非连续空间离散，特别是非连续Galerkin（DG）方法，是一种现代流行的数值方法，特别是对于计算密集型流体动力学计算。它们的流行是由于DG方法允许高阶近似与高灵活性相结合的事实-例如在相邻元素上选择不同的多项式次数。此外，为了使复杂的真实的生活问题的可靠模拟的未来的挑战是并行应用程序的方法的设计。这里. DG方法是完全合适的，因此有必要开发和分析它们，特别是在时间积分方面。在实际相关问题的背景下，半离散DG方程往往是非常刚性的。在复杂几何形状的情况下，例如，对于障碍物周围的流体流动，DG网格局部细化，其中元素的大小非常不同。此外，对于高雷诺数，在边界层区域应用时需要相当大的网格细化。在这种情况下，时间积分的方法，迄今为止应用还远远不够有效。特别是关于显式和隐式方法的巧妙耦合，以及在多速率策略中使用本地时间步长，需要进行更多的研究。在初步工作中，一个强大的，高阶DG计划与低数值耗散的基础上，新的有效的过滤策略已被开发。在此基础上，本文的创新性贡献是对IMEX时间积分方法的发展和分析。特别是，第一次，我们将采用混合方法的基本IMEX分裂相结合的多速率方法，以加速时间积分的半离散DG方程。因此，本项目的主要目标是开发，分析和直接比较新的方法来构建有效的，高阶的时间积分计划的粘性和无粘流体流动。这些方法将在一个统一的框架内进行研究，以制定适当的战略，以决定IMEX或多速率方法或使用两者的组合。在这种情况下，将开发和分析刚度检测器，我们将评估IMEX方法中隐式类型的具体方法，并且还包括多速率方法。另一个目标是建立一个类比IMEX和多速率的指数积分器，目前表现出相当大的增益效率的方法。本计画所发展的基于IMEX与多速率策略的有效时间积分方法，将非常适合于实际应用。因此，无论是对流体流动的数值计算，还是对基于流体-结构-相互作用的现象的模拟，都可以期望它们建立新的标准，这将是未来的重点。