Efficient high-order structure-preserving numerical methods for nonlinear evolution equations

非线性演化方程的高效高阶保结构数值方法

• 批准号: 513301895
• 负责人: Professor Dr. Hendrik Ranocha
• 金额: --
• 依托单位: Computer, Electrical and Mathematical Science and Engineering Division (CEMSE)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/513301895?language=en
• 关键词: Efficient; order; structure; preserving; numerical

Many problems in science and engineering are modeled by evolution equations. Some examples of such nonlinear partial differential equations include large-scale astrophysical plasmas, the airflow around a car, water waves, biological systems, or semiconductor devices. Despite their broad range of applications, evolution equations often share some common structures. Indeed, the qualitative behavior of such systems is often determined by how the total energy or entropy changes in time. General entropy methods have also proven successful in the mathematical theory of evolution equations. Moreover, preserving such entropy structures discretely usually results in robust algorithms, improved qualitative properties, a correct asymptotic behavior, and reduced numerical errors. We will develop and analyze high-order structure-preserving numerical methods for a range of evolution equations, including compressible fluid dynamics, Hamiltonian systems, nonlinear wave equations, and other nonlinear equations with higher-order dispersive and/or dissipative terms. While the focus is on efficient time integration methods preserving the correct evolution of an entropy functional, spatial discretizations are also considered. In particular, we will build on our recent works on relaxation methods in time and summation by parts (SBP) operators in space, combine them, and extend them significantly. Efficient implementations of all methods using the modern programming language Julia will be published in open source packages. These are anticipated to result in significant gains of efficiency, robustness, and accuracy compared to existing technology. Hence, they enable the application of numerical methods to problems in science and engineering that have been intractable before. Moreover, we will advance the understanding of structure-preserving numerical methods, e.g., by analyzing the error-growth rates.

科学和工程中的许多问题都是通过进化方程来建模的。这种非线性部分微分方程的一些例子包括大规模的天体等离子体，汽车周围的气流，水波，生物系统或半导体设备。尽管它们的应用范围广泛，但演变方程经常共享一些共同的结构。实际上，这种系统的定性行为通常取决于总能量或熵的时间如何变化。一般的熵方法也已证明在进化方程的数学理论中成功。此外，保留此类熵结构通常会导致鲁棒算法，改善定性特性，正确的渐近行为以及减少的数值误差。我们将开发和分析针对一系列进化方程的高阶结构性数值方法，包括可压缩流体动力学，哈密顿系统，非线性波方程以及其他具有高阶分散性和/或散发性项的非线性方程。虽然重点是保留熵功能的正确演变的有效时间整合方法，但还考虑了空间离散化。特别是，我们将基于我们最近在太空中的零件（SBP）操作员的时间和总结的著作，将它们组合起来并大大扩展它们。使用现代编程语言朱莉娅的所有方法的有效实现将在开源软件包中发布。与现有技术相比，这些预计会导致效率，鲁棒性和准确性的显着提高。因此，它们使数值方法能够应用于以前很难过的科学和工程问题。此外，我们将通过分析误差速率来提高对结构保存数值方法的理解。