Error analysis of high-order splitting methods for nonlinear evolutionary Schrödinger equations and application to the MCTDHF equations in electron dynamics

Othmar Koch, Christof Neuhauser, Mechthild Thalhammer
2013 Mathematical Modelling and Numerical Analysis  
In this work, the error behaviour of high-order exponential operator splitting methods for the time integration of nonlinear evolutionary Schrödinger equations is investigated. The theoretical analysis utilises the framework of abstract evolution equations on Banach spaces and the formal calculus of Lie derivatives. The general approach is substantiated on the basis of a convergence result for exponential operator splitting methods of (nonstiff) order p applied to the multi-configuration
more » ... endent Hartree-Fock (MCTDHF) equations, which are associated with a model reduction for highdimensional linear Schrödinger equations describing free electrons that interact by Coulomb force. Provided that the analytical solution of the MCTDHF equations constituting a system of coupled linear ordinary differential equations and low-dimensional nonlinear partial differential equations satisfies suitable regularity requirements, convergence of order p − 1 in the H 1 Sobolev norm and convergence of order p in the L 2 norm is proven. An analogous result follows for the cubic nonlinear Schrödinger equation, which is also illustrated by a numerical experiment. Mathematics Subject Classification. 65L05, 65M12, 65J15.
doi:10.1051/m2an/2013067 fatcat:mjvkb2sw2zg7rjjvunz5w6ayua