In the project P 30819-N32 supported by the Austrian Science Fund (FWF)
we will study large time-dependent systems of ordinary differential equations of Schrödinger type.
The models considered are relevant, among others, for the simulation of a new type of transistor,
a new class of solar cells, and the control of quantum systems, i.e., the optimization of a system
with respect to a suitable criterion. The understanding of the underlying physical processes strongly
relies on numerical simulations by computer based methods. To advance the numerical approximation in time,
Magnus-type integrators are put forward, which approximate the solution as a sequence of exponentials of fixed matrices.
The main focus will be on the construction and analysis of error estimators for the adaptive choice of the time-steps,
as adaptivity is the key to the realization of large-scale simulations of real-world application problems.
We will construct and analyze high-order schemes equipped with a posteriori error estimators as the basis for the
adaptive selection of the time steps. In each step, matrix exponentials have to be computed. These will be approximated
in low-dimensional spaces by the Lanczos method. A defect-based error indicator will replace the common strategies to
control the exponentiation algorithm which are viable only when the approximation is already close to the true solution.
The new time integrators will hence enable a more accurate, efficient and reliable integration of the differential
equations and moreover provide a posteriori information on the error, thereby enabling solid predictions regarding
the properties of the investigated problems. The applications which will be investigated with the new numerical time
propagators arise in solid state physics and the control of quantum systems. A new type of transistor, a Mott transistor,
will be explored via numerical simulation, and a new class of solar cells, both based on transition metal oxide heterostructures.
The Mott transistor has an ideal switching characteristic between on (metal) and off (insulator). The solar cells on the other
hand are ideally suited to overcome the Schockley-Queisser limit of 33% efficiency, bearing good prospects for solar cells of
the highest efficiency. An important aspect in technical applications is the control of quantum systems by manipulating
electric and magnetic fields by laser pulses. This introduces the necessity to solve a large number of problems of the
type treated by the numerical approaches we will provide, making efficiency of numerical time propagators an important issue.
All the mentioned project goals cannot be realized with the required accuracy and efficiency by available numerical methods.
The adaptive high-order methods of this project will hence also enable us to reach new frontiers regarding the proposed applications.
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H. Hofstätter, and O. Koch,
An algorithm for computing coefficients of
words in expressions involving exponentials
and its application to the construction of
exponential integrators, submitted.