Numerical analysis of Hamiltonian partial differential equations and high dimensional problems

• Applicant: Dr. Ludwig Gauckler
• Subject Area: Mathematics
• Term: from 2014 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 255990239

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.

• DFG Programme: Research Grants