Numerical Schemes for Coupled Multi-Scale Problems

耦合多尺度问题的数值方案

• 批准号: 525842915
• 负责人: Professor Dr. Michael Herty
• 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: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/525842915?language=en
• 关键词: Numerical; Schemes; Coupled; Multi; Scale

Coupling of models on different scales has been ubiquitous in the mathematical modeling of real-world phenomena with a prominent example being the earth system models. Here models on different scales need to be coupled for a complete description of the process. The coupling procedure itself poses major challenges analytically as well as numerically. A suitable method should ensure that relevant properties are conserved across models and scales and an efficient numerical computation should be possible to allow for real-time simulations. Although the coupling of systems of balance laws has been successfully conducted in the past, they suffer from short-comings. A bulk of work has been performed in the one-dimensional case in the past decades in the context of networked problems for various fields, where a key concept in the development of analytical and numerical results has been the notion of Riemann solvers or half-Riemann problems at the coupling interface. The developed techniques require the analytical expression of the nonlinear wave curves which in many relevant models might not be explicitly available. This severely limits the use of those techniques. The developments have also been mirrored in numerical schemes that in almost all literature relies on the Riemann solver at the coupling point safe for some exceptions, they, however, do not translate to the relevant multi-scale case. Also, property preserving numerical methods for the coupling at the same scale are not available. Multi-dimensional extensions have so far been addressed by projection of the flux in normal direction leading to formally one-dimensional systems. Finally, the question of the coupling across scales has very recently gained interest but a general methodological approach and corresponding property preserving methods are still at large. The aim of this project is to develop numerical schemes for multi-scale coupled problems arising in nonlinear fluid dynamics. More specifically, we aim to develop schemes that do not rely on Riemann solvers at the coupling interface and allow for the extension to multi-scale and, with the perspective of turbulent flow, towards multi-dimensional settings. The development of the scheme is accompanied by theoretical analysis on its properties as well as an implementation for different scenarios described by (multi-dimensional) systems of hyperbolic balance laws and/or transport-dominated problems exhibiting multiple scales. We exemplify the developed, numerical method on systems of gas dynamics and a transpiration cooling problem. For this purpose, we first derive a relaxation approach for coupled transport problems relying on one relaxation parameter. This is followed by the investigation of extended properties of the relaxation systems with different relaxation velocities. Finally, the developed techniques are applied to different scenarios of multi-scale coupling.

在不同尺度上的模型耦合在现实现象的数学建模中无处不在，一个突出的例子是地球系统模型。这里需要将不同尺度上的模型耦合以完成该过程的完整描述。耦合过程本身在分析和数值上提出了重大挑战。一种合适的方法应确保相关属性在模型和尺度上保存，并且应进行有效的数值计算以进行实时模拟。尽管过去已经成功地进行了平衡法律系统的耦合，但它们遭受了短暂的影响。在过去几十年中，在各个领域的网络问题的背景下，已经在一维情况下进行了许多工作，在分析和数值结果中，关键概念是Riemann求解器或半riemann的概念耦合接口的问题。开发的技术需要非线性波曲线的分析表达，在许多相关模型中可能无法明确可用。这严重限制了这些技术的使用。这些发展也反映在数值方案中，几乎所有文献都依赖于耦合点的Riemann求解器，但对于某些例外，它们并未转化为相关的多规模案例。同样，在同一量表上耦合的数值方法的属性也不可用。到目前为止，通过在正常方向上的通量投影导致正式一维系统的通量预测来解决多维扩展。最后，跨量表的耦合问题最近引起了人们的兴趣，但是一种一般的方法论方法和相应的特性保存方法仍然很大。该项目的目的是开发用于在非线性流体动力学中产生的多尺度耦合问题的数值方案。更具体地说，我们旨在开发不依赖耦合界面上的Riemann求解器的方案，并允许扩展到多尺度，并且具有湍流的角度，朝着多维设置。该方案的开发伴随着有关其性质的理论分析，以及针对双波利平衡定律（多维）系统所描述的不同场景的实施，并且表现出多个量表的运输学主导问题。我们说明了有关气体动力学系统和蒸腾冷却问题的开发的数值方法。为此，我们首先得出了一种依靠一个放松参数的耦合传输问题的放松方法。随后，研究了具有不同弛豫速度的弛豫系统的扩展特性。最后，将开发的技术应用于多尺度耦合的不同情况。