Numerical Schemes for Coupled Multi-Scale Problems

在真实世界现象的数学建模中，不同尺度上的模型耦合是普遍存在的，地球系统模型就是一个突出的例子。在这里，需要将不同比例的模型耦合在一起，以完整地描述这一过程。耦合过程本身在分析和数值上都构成了重大挑战。一种合适的方法应该确保相关的性质在模型和尺度上是保守的，并且应该能够进行有效的数值计算以允许实时模拟。虽然平衡法律体系的耦合在过去取得了成功，但它们也存在不足。在过去的几十年里，在不同领域的网络问题的背景下，在一维情况下进行了大量的工作，其中在发展分析和数值结果中的一个关键概念是耦合界面上的黎曼解算器或半黎曼问题的概念。所发展的技术需要非线性波浪曲线的解析表达式，而这在许多相关模型中可能不是明确可用的。这严重限制了这些技术的使用。这些发展也反映在数值格式中，这些数值格式在几乎所有文献中都依赖于耦合点处的Riemann求解器，但在某些例外情况下，它们不能转化为相关的多尺度情况。此外，对于相同尺度下的耦合，也没有保持性质的数值方法。到目前为止，多维扩展已经通过在法线方向上投影导致形式上的一维系统的通量来解决。最后，跨尺度的耦合问题最近引起了人们的兴趣，但一般的方法论方法和相应的属性保存方法仍然很普遍。本项目的目的是为非线性流体力学中出现的多尺度耦合问题开发数值格式。更具体地说，我们的目标是开发不依赖于耦合界面上的Riemann解算器的方案，并允许扩展到多尺度，并从湍流的角度向多维环境扩展。该方案的发展伴随着对其性质的理论分析，以及对双曲平衡律的(多维)系统描述的不同场景的实现和/或呈现多尺度的交通支配问题的实现。我们以气体动力学系统和发汗冷却问题的数值方法为例进行了说明。为此，我们首先给出了依赖于一个松弛参数的耦合输运问题的松弛方法。然后研究了具有不同松弛速度的松弛系统的扩展性质。最后，将所开发的技术应用于不同的多尺度耦合场景。

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.