FOR 5409: Structure-Preserving Numerical Methods for Bulk- and Interface-Coupling of Heterogeneous Models

可以用偏微分方程（PDEs）建模的现象的数值模拟是许多科学学科中必不可少的工具。虽然发展各种偏微分方程系统的精确数值方法仍然是一个非常活跃的研究领域，但大多数复杂的应用都是由几个偏微分方程的耦合系统来描述的，即由异构模型来描述。本研究单元专注于磁化等离子体、复杂流体和电化学过程描述中耦合系统的建模和仿真。通常，对于这类复杂的、耦合的、非线性的系统，没有严格的数学解理论。因此，需要开发能够显示出保留底层模型的某些结构特性的数值方法。重要的结构性质的例子是质量、电荷、动量和能量守恒，以及与熵平衡方程的一致性，熵平衡方程可以从感兴趣的方程中推导出来。其他重要的结构性质是保持渐近行为和稳定状态的精确近似。在本研究单元中，我们区分了异质模型出现的两种不同情况。在其中一种情况下，在感兴趣的域的同一点或区域中考虑多个物理过程。我们把这种情况称为大容量耦合。一个典型的例子是动力学理论的弗拉索夫方程与电动力学的麦克斯韦方程耦合。在另一种情况下，在领域的不同部分使用不同的数学模型，并在公共边界粘合在一起。我们称这种情况为接口耦合。出现界面耦合的典型例子是非线性和线性化模型的组合，或者在域的不同部分使用具有不同数量矩的矩方程。对于体耦合和界面耦合的非均质模型，结构保持方法的发展是数理建模、数值分析和科学计算相结合的一个新的研究方向。在一些项目中，仍然必须确定结构要素。在其他情况下，我们可以在已建立的模型上建立新的数值方法。数值模拟将在所有项目中发挥至关重要的作用。为了从相对简单的测试问题转移到并行计算机上的自适应模拟，耦合算法对高性能计算的影响也将被研究。

The numerical simulation of phenomena that can be modeled by partial differential equations (PDEs) is an essential tool in numerous scientific disciplines. While the development of accurate numerical methods for various systems of PDEs is still a very active research field, most complex applications are described by coupled systems of several PDEs, i.e. by heterogeneous models. This research unit focuses on the modeling and simulation of coupled systems arising in the description of magnetised plasmas, complex fluids and electro-chemical processes.Typically, no rigorous mathematical solution theory is available for these kind of complex, coupled, nonlinear systems. Therefore, it is desirable to develop numerical methods that can be shown to preserve certain structural properties of the underlying model. Examples of important structural properties are conservation of mass, charge, momentum and energy but also the consistency with an entropy balance equation which can be derived from the equations of interest. Other important structural properties are the preservation of asymptotic behaviour and exact approximation of steady states.In this research unit we distinguish two different situations for the appearance of heterogeneous models. In one of these situations multiple physical processes are considered in the same point or region of the domain of interest. We refer to such a situation as bulk-coupling. A typical example is the Vlasov equation of kinetic theory coupled to Maxwell's equations of electrodynamics. In another situation different mathematical models are used in different parts of the domain and glued together at common boundaries. We call this situation interface-coupling. Typical examples where interface-coupling arise are combinations of nonlinear and linearised models or the use of moment equations with different numbers of moments in different parts of the domain. For bulk- as well as interface-coupled heterogeneous models the development of structure-preserving methods is a new research direction which we address in a joint effort combining mathematical and physical modeling, numerical analysis and scientific computing. In some projects the structural elements must still be identified. In other cases we can build new numerical methods on established models. Numerical simulations will play a crucial role in all projects. In order to move from relatively simple test problems to adaptive simulations on parallel computers, the implications of coupling algorithms for high-performing computing will also be studied.