Adaptive Splitting for Nonlinear Schrödinger Equations

The project aim is to investigate the numerical solution of nonlinear Schrödinger equations. For the full discretization of initial value problems for these evolution equations we analyse the method of lines approach, which is suitable for problems where theoretical insight suggests a suitable space discretization and mesh. In this approach the evolutionary PDE is reduced to a (generally large) system of initial value problems for ordinary differential equations. For the resulting problems, splitting methods of high order will be investigated with respect to the structure of the local error and implications for the adaptive choice of time steps. Furthermore, new a priori and a posteriori error estimates based on the defect correction principle are put forward for the approximation of the matrix exponential function by splitting methods. This task occurs as a subproblem in the time integration of evolution equations in the context of the method of lines. A realization of this approach for nonlinear evolution equations will represent a nontrivial extension. Application of the project results to nonlinear single-particle Schrödinger equations associated with model reductions of the high-dimensional linear multi-particle Schrödinger equation like time-dependent density functional theory or the multi-configuration time-dependent Hartree-Fock method shall put the project results in an application-oriented context.

The project was devoted to the numerical solution of nonlinear time-dependent Schrödinger equations. These are a type of partial differential equations describing quantum-mechanical systems. Numerical integration on a digital computer is based on discretization in space and time leading to a finite-dimensional system. For the integration of the resulting discrete systems, highly accurate splitting approximations were investigated. This approach is based on a division of the problem at hand into two sub-problems which can be efficiently integrated using standard techniques. In detail, the following results were obtained: Design and analysis of splitting methods of higher order (the order of a method is a measure for its local accuracy), design and analysis of computable approximations for the local error of such methods, enabling a precise control of time steps, an extension to more generally applicable splitting techniques, where the full problem is separated into more than two subproblems, analysis, implementation and testing of methods for different types of Schrödinger equations.