955 research outputs found
Magnus-Lanczos methods with simplified commutators for the Schr\"odinger equation with a time-dependent potential
The computation of the Schr\"odinger equation featuring time-dependent
potentials is of great importance in quantum control of atomic and molecular
processes. These applications often involve highly oscillatory potentials and
require inexpensive but accurate solutions over large spatio-temporal windows.
In this work we develop Magnus expansions where commutators have been
simplified. Consequently, the exponentiation of these Magnus expansions via
Lanczos iterations is significantly cheaper than that for traditional Magnus
expansions. At the same time, and unlike most competing methods, we simplify
integrals instead of discretising them via quadrature at the outset -- this
gives us the flexibility to handle a variety of potentials, being particularly
effective in the case of highly oscillatory potentials, where this strategy
allows us to consider significantly larger time steps.Comment: 27 pages, 5 figure
Solution of second kind Fredholm integral equations by means of Gauss and anti-Gauss quadrature rules
This paper is concerned with the numerical approximation of Fredholm integral equa-
tions of the second kind. A Nyström method based on the anti-Gauss quadrature
formula is developed and investigated in terms of stability and convergence in appro-
priate weighted spaces. The Nyström interpolants corresponding to the Gauss and
the anti-Gauss quadrature rules are proved to furnish upper and lower bounds for the
solution of the equation, under suitable assumptions which are easily verified for a
particular weight function. Hence, an error estimate is available, and the accuracy of
the solution can be improved by approximating it by an averaged Nyström interpolant.
The effectiveness of the proposed approach is illustrated through different numerical
tests
Fast and spectrally accurate summation of 2-periodic Stokes potentials
We derive a Ewald decomposition for the Stokeslet in planar periodicity and a
novel PME-type O(N log N) method for the fast evaluation of the resulting sums.
The decomposition is the natural 2P counterpart to the classical 3P
decomposition by Hasimoto, and is given in an explicit form not found in the
literature. Truncation error estimates are provided to aid in selecting
parameters. The fast, PME-type, method appears to be the first fast method for
computing Stokeslet Ewald sums in planar periodicity, and has three attractive
properties: it is spectrally accurate; it uses the minimal amount of memory
that a gridded Ewald method can use; and provides clarity regarding numerical
errors and how to choose parameters. Analytical and numerical results are give
to support this. We explore the practicalities of the proposed method, and
survey the computational issues involved in applying it to 2-periodic boundary
integral Stokes problems
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