Numerical analysis of Hamiltonian partial differential equations and high dimensional problems

该项目将分析哈密顿偏微分方程和高维微分方程的数值离散。一方面，将研究时间离散的数值方法的定性性质，如分裂方法和龙格-库塔方法。特别是，我们将追问数值方法是否能够以及在哪些时间间隔上能够再现波的稳定性，这一问题在方程的数学分析中得到了详细的研究。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。分析张量流形上的逼近，不仅要考虑它们的逼近性质，还要分析它们的长期行为。这种近似在高维线性薛定谔方程的量子动力学中得到了成功的应用。此外，还将根据最近的正则性结果，研究生物和化学中的重要方程--化学主方程的数值方法的收敛问题。

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.