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.

不连续的空间离散化，尤其是不连续的盖尔金（DG）方法，是一种现代而流行的数值方法，尤其是用于计算密集的流体动力学计算。它们的受欢迎程度是由于DG方法允许高阶近似值和高灵活性（例如在相邻元素上选择不同的多项式度。此外，为了对复杂的现实生活问题进行可靠的模拟，未来的挑战是设计并行应用方法的设计。这里。 DG方法是完全合适的，因此有必要开发和分析它们，尤其是在使用的时间集成方面。在实际相关的问题的背景下，半分化的DG方程通常非常僵硬。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网格的尺寸大小不同。此外，对于高雷诺数字，应用需要在边界层区域进行大量的网格细化。在这种情况下，到目前为止应用的时间集成方法尚未有效。尤其是关于明确和隐式方法的熟练耦合，以及使用当地时间步骤（如多级策略）的使用，需要更多的研究。在初步工作中，已经开发了一种基于新型有效过滤策略的稳健高阶DG方案，具有低数值耗散的较低数值。基于这一基础，该提案的创新贡献是对新型IMEX时间整合方法的开发和分析。特别是，我们首次将基本IMEX拆分的混合方法与多条方法结合在一起，以加速半差异DG方程的时间积分。该项目的主要目的是因此，开发，分析和直接比较新颖的方法，用于构建高效，高阶时间整合方案，用于粘性和无关流体流动。这些方法将以统一的框架进行研究，以制定合适的策略以在IMEX或多次方法之间做出决定或使用两者的组合。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，并还包括多条方法。一个进一步的目的是为指数集成剂建立与IMEX和多次多次方法的类比，目前显示出效率可观。将在本项目中开发的基于IMEX和多次策略的有效时间集成方法将非常适合实际应用。因此，可以预期它们将为流体流量的数值计算而设置新的标准，因为基于流体结构互动的模拟现象的模拟，这将在将来重点介绍。