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。此类非线性偏微分方程的一些示例包括大规模天体物理等离子体、汽车周围的气流、水波、生物系统或半导体设备。尽管应用范围广泛，但演化方程通常具有一些共同的结构。事实上，此类系统的定性行为通常取决于总能量或熵随时间的变化。一般熵方法在进化方程的数学理论中也被证明是成功的。此外，离散地保留这种熵结构通常会产生鲁棒的算法、改进的定性特性、正确的渐近行为并减少数值误差。我们将为一系列演化方程开发和分析高阶结构保持数值方法，包括可压缩流体动力学、哈密顿系统、非线性波动方程以及其他具有高阶色散和/或耗散项的非线性方程。虽然重点是保持熵函数正确演化的有效时间积分方法，但也考虑了空间离散化。特别是，我们将在最近关于时间松弛方法和空间分部求和（SBP）算子的工作的基础上，将它们结合起来，并显着扩展它们。使用现代编程语言 Julia 的所有方法的高效实现将在开源包中发布。与现有技术相比，预计这些技术将显着提高效率、稳健性和准确性。因此，它们使得数值方法能够应用于以前难以解决的科学和工程问题。此外，我们将增进对结构保持数值方法的理解，例如通过分析误差增长率。