Geometric integrators: Open questions - novel perspectives

Evolution equations are essential means for the mathematical description of dynamical processes and thus of high relevance in various areas of application. Hamiltonian systems and time-dependent Schrödinger equations constitute prevalent models for classical and quantum physical systems. The complexity of these classes of ordinary and partial differential equations makes computer-based simulations indispensable, and their diverse structural peculiarities necessitate the use of advanced numerical integration methods. Elementary requirements on numerical integrators for evolution equations are reflected in the terms stability and accuracy. In connection with simulations of physical systems, additional aspects have to be taken into account to ensure robust and reliable computations. Intrinsic characteristics of Hamiltonian systems and Schrödinger equations are that fundamental quantities such as the total particle number and the total energy are conserved. Accordingly, it is desirable to design so-called geometric numerical integrators that preserve these quantities in an adequate manner over long time frames. Further relevant qualities of numerical integrators linked to practical implementations for large-scale and multi-scale applications are their feasibility for given computing and memory capacities as well as their efficiency for prescribed tolerance ranges. Our research project is devoted to the development and theoretical investigation of numerical integrators for various kinds of evolution equations. Principal objectives are the design and implementation of reliable and efficient geometric integrators for ordinary and partial differential equations arising in quantum physics and beyond. This purpose will go hand in hand with theoretical investigations, the development of freely available research codes, and thorough numerical tests. Concerning the scope of applications, our focus will be on time-dependent Schrödinger equations and related partial differential equations. This in particular includes the study of linear as well as nonlinear and autonomous as well as nonautonomous evolution equations. Concerning the considered classes of numerical integrators, our focus will be on modifications of exponential operator splitting methods and Magnus-type exponential integrators, which feature intrinsic benefits for large-scale applications. Our main scientific hypotheses are that these classes of geometric integrators permit the design of novel schemes that are favourable in general and superior to standard numerical integrators as well as established geometric integrators in specific situations. Our core project team comprises two young researchers, two international cooperation partners, and the project applicant.