scholarly journals An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime

2010 ◽  
Vol 50 (4) ◽  
pp. 729-749 ◽  
Author(s):  
Stéphane Descombes ◽  
Mechthild Thalhammer
Keyword(s):  
Operator Splitting ◽  
Splitting Methods ◽  
Schrodinger Equations ◽  
Local Error ◽  
Schrödinger Equations ◽  
Operator Splitting Methods ◽  
Exponential Operator

scholarly journals Adaptive Time Propagation for Time-dependent Schrödinger equations

2020 ◽  
Vol 7 (1) ◽  
Author(s):  
Winfried Auzinger ◽  
Harald Hofstätter ◽  
Othmar Koch ◽  
Michael Quell
Keyword(s):  
Splitting Methods ◽  
Time Variable ◽  
Time Dependent ◽  
Schrodinger Equations ◽  
Local Error ◽  
Schrödinger Equations ◽  
Time Integrators ◽  
Potential Part ◽  
Adaptive Time ◽  
Time Propagation

AbstractWe compare adaptive time integrators for the numerical solution of linear Schrödinger equations where the Hamiltonian explicitly depends on time. The approximation methods considered are splitting methods, where the time variable is split off and advanced separately, and commutator-free Magnus-type methods. The time-steps are chosen adaptively based on asymptotically correct estimators of the local error in both cases. It is found that splitting methods are more efficient when the Hamiltonian naturally suggests a separation into kinetic and potential part, whereas Magnus-type integrators excel when the structure of the problem only allows to advance the time variable separately.

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