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Self-adaptive moving mesh schemes for short pulse type equations and their Lax pairs
Integrable self-adaptive moving mesh schemes for short pulse type equations
(the short pulse equation, the coupled short pulse equation, and the complex
short pulse equation) are investigated. Two systematic methods, one is based on
bilinear equations and another is based on Lax pairs, are shown. Self-adaptive
moving mesh schemes consist of two semi-discrete equations in which the time is
continuous and the space is discrete. In self-adaptive moving mesh schemes, one
of two equations is an evolution equation of mesh intervals which is deeply
related to a discrete analogue of a reciprocal (hodograph) transformation. An
evolution equations of mesh intervals is a discrete analogue of a conservation
law of an original equation, and a set of mesh intervals corresponds to a
conserved density which play an important role in generation of adaptive moving
mesh. Lax pairs of self-adaptive moving mesh schemes for short pulse type
equations are obtained by discretization of Lax pairs of short pulse type
equations, thus the existence of Lax pairs guarantees the integrability of
self-adaptive moving mesh schemes for short pulse type equations. It is also
shown that self-adaptive moving mesh schemes for short pulse type equations
provide good numerical results by using standard time-marching methods such as
the improved Euler's method.Comment: 13 pages, 6 figures, To be appeared in Journal of Math-for-Industr
On the indefinite Helmholtz equation: complex stretched absorbing boundary layers, iterative analysis, and preconditioning
This paper studies and analyzes a preconditioned Krylov solver for Helmholtz
problems that are formulated with absorbing boundary layers based on complex
coordinate stretching. The preconditioner problem is a Helmholtz problem where
not only the coordinates in the absorbing layer have an imaginary part, but
also the coordinates in the interior region. This results into a preconditioner
problem that is invertible with a multigrid cycle. We give a numerical analysis
based on the eigenvalues and evaluate the performance with several numerical
experiments. The method is an alternative to the complex shifted Laplacian and
it gives a comparable performance for the studied model problems
Factorization of quantum charge transport for non-interacting fermions
We show that the statistics of the charge transfer of non-interacting
fermions through a two-lead contact is generalized binomial, at any temperature
and for any form of the scattering matrix: an arbitrary charge-transfer process
can be decomposed into independent single-particle events. This result
generalizes previous studies of adiabatic pumping at zero temperature and of
transport induced by bias voltage.Comment: 13 pages, 3 figures, typos corrected, references adde
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