Computational methods for wave-type problems with non-standard boundary conditions

本计画致力于具有动力学与声学边界条件之波动型初边值问题之数值逼近与数值分析。如果要正确反映表面的有效性质，这种非标准边界条件是必不可少的。一个代表性的例子是低音鼓的膜振动的建模，其中标准狄利克雷或诺依曼边界条件将失败。在边界上的强非线性或不均匀性的情况下，然后导致不同的特征长度尺度的散装和边界上，这样的问题的数值模拟描绘了一个具有挑战性的任务。在这个项目中，我们考虑一个替代的问题制定为偏微分代数方程，它允许量身定制的数值处理。特别地，可以在块体和表面上应用不同的网格尺寸或甚至不同的离散化方案。这种方法产生了一类新的非标准边界条件的波型系统的近似计划。相应的分析结合了混合有限元方案与已知的微分代数方程领域的技术。此外，该项目的目的是推广Gautter-type波积分器的约束系统。这将允许通过显式时间步进方案有效地整合所提出的替代模型制剂，而没有限制性步长限制。该项目的另一个主要部分涉及体表面分裂方案的设计和分析。其中，想法是以交替的方式求解体动力学和表面动力学。因此，该系统是解耦的，并且能够在空间和时间上实现更灵活和定制的近似方案。总之，这就产生了有效的模拟工具，波型系统与非标准的边界条件。

This project is devoted to the numerical approximation and numerical analysis of initial-boundary value problems of wave type with kinetic and acoustic boundary conditions. Such non-standard boundary conditions are indispensable if the effective properties on the surface need to be reflected properly. A representative example is the modeling of membrane vibrations of a bass drum, where standard Dirichlet or Neumann boundary conditions would fail. In situations of strong nonlinearities or heterogeneities on the boundary, which then lead to different characteristic length scales in the bulk and on the boundary, the numerical simulation of such problems depicts a challenging task. Within this project, we consider an alternative problem formulation as a partial differential-algebraic equation, which allows a tailored numerical treatment. In particular, it is possible to apply different mesh sizes or even different discretization schemes in the bulk and on the surface. This approach yields a novel class of approximation schemes for wave-type systems with non-standard boundary conditions. The corresponding analysis combines mixed finite element schemes with techniques known from the field of differential-algebraic equations. Moreover, the project aims to generalize Gautschi-type wave integrators to constrained systems. This would allow an efficient integration of the proposed alternative model formulation by an explicit time stepping scheme without a restrictive step size restriction. Another major part of this project deals with the design and analysis of bulk-surface splitting schemes. Therein, the idea is to solve the bulk and surface dynamics in an alternating manner. Hence, the system is decoupled and enables more flexible and tailored approximation schemes in space and in time. All together, this then yields efficient simulation tools for wave-type systems with non-standard boundary conditions.