ERROR ANALYSIS OF HIGH-ORDER SPLITTING METHODS FOR NONLINEAR EVOLUTIONARY SCHRODINGER EQUATIONS AND APPLICATION TO THE MCTDHF EQUATIONS IN ELECTRON DYNAMICS

作者：Koch Othmar^*; Neuhauser Christof; Thalhammer Mechthild

来源：ESAIM: Mathematical Modelling and Numerical Analysis , 2013, 47(5): 1265-1286.

DOI：10.1051/m2an/2013067

• Nonlinear evolution equations
• time-dependent nonlinear Schrodinger equations
• multi-configuration time-dependent Hartree-Fock (MCTDHF) equations
• high-order exponential operator splitting methods
• local error expansion
• convergence

In this work, the error behaviour of high-order exponential operator splitting methods for the time integration of nonlinear evolutionary Schrodinger 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 time-dependent Hartree-Fock (MCTDHF) equations, which are associated with a model reduction for high-dimensional linear Schrodinger 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 Schrodinger equation, which is also illustrated by a numerical experiment.

• 出版日期2013-9