Numerical analysis of Hamiltonian partial differential equations and high dimensional problems

哈密​​顿偏微分方程和高维问题的数值分析

• 批准号: 255990239
• 负责人: Dr. Ludwig Gauckler
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2014
• 资助国家: 德国
• 起止时间: 2013-12-31 至 2016-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/255990239?language=en
• 关键词: Numerical; analysis; Hamiltonian; partial; differential

Numerical discretizations of Hamiltonian partial differential equations and differential equations in high dimensions shall be analysed in the project.On the one hand, qualitative properties of numerical methods for the discretization in time such as splitting and Runge-Kutta methods will be investigated. In particular, we will pursue the question if and on which time intervals a numerical method is able to reproduce the stability of waves, which is studied in detail in the mathematical analysis of the equations.On the other hand, the analysis of approximations in high spatial dimensions will be the second key activity in the project. Approximations on tensor manifolds shall be analysed with respect to their approximation properties, but also their long-time behaviour. Such approximations are used successfully in quantum dynamics in the case of the high dimensional linear Schrödinger equation. In addition, the convergence of numerical methods for the chemical master equation, an important equation in biology and chemistry, will be studied on the basis of recent regularity results.

本课题将对哈密顿偏微分方程和高维微分方程的数值离散化进行分析。一方面，研究了离散化数值方法的定性性质，如分裂法和龙格-库塔法。特别是，我们将探讨数值方法是否以及在哪个时间间隔上能够再现波的稳定性的问题，这在方程的数学分析中进行了详细的研究。另一方面，高空间维度的近似分析将是该项目的第二个关键活动。对张量流形的近似应分析其近似性质，以及它们的长时间行为。这种近似已成功地应用于量子动力学中的高维线性Schrödinger方程。此外，将在最近的正则性结果的基础上，研究化学主方程（生物学和化学中的一个重要方程）数值方法的收敛性。