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Titel: On the Efficient Computation of Large Scale Singular Sums with Applications to Long-Range Forces in Crystal Lattices
VerfasserIn: Buchheit, Andreas A.
Keßler, Torsten
Sprache: Englisch
Titel: Journal of Scientific Computing
Bandnummer: 90
Heft: 1
Verlag/Plattform: Springer Nature
Erscheinungsjahr: 2021
Freie Schlagwörter: Euler–Maclaurin expansion
Quadrature
Long-range interactions
Condensed matter physics
Solitons
DDC-Sachgruppe: 500 Naturwissenschaften
Dokumenttyp: Journalartikel / Zeitschriftenartikel
Abstract: We develop a new expansion for representing singular sums in terms of integrals and vice versa. This method provides a powerful tool for the efficient computation of large singular sums that appear in long-range interacting systems in condensed matter and quantum physics. It also offers a generalised trapezoidal rule for the precise computation of singular integrals. In both cases, the difference between sum and integral is approximated by derivatives of the non-singular factor of the summand function, where the coefficients in turn depend on the singularity. We show that for a physically meaningful set of functions, the error decays exponentially with the expansion order. For a fixed expansion order, the error decays alge braically both with the grid size, if the method is used for quadrature, or the characteristic length scale of the summand function in case the sum over a fixed grid is approximated by an integral. In absence of a singularity, the method reduces to the Euler–Maclaurin summation formula. We demonstrate the numerical performance of our new expansion by applying it to the computation of the full nonlinear long-range forces inside a domain wall in a macro scopic one-dimensional crystal with 2 × 1010 particles. The code of our implementation in Mathematica is provided online. For particles that interact via the Coulomb repulsion, we demonstrate that finite size effects remain relevant even in the thermodynamic limit of macro scopic particle numbers. Our results show that widely-used continuum limits in condensed matter physics are not applicable for quantitative predictions in this case.
DOI der Erstveröffentlichung: 10.1007/s10915-021-01731-5
Link zu diesem Datensatz: urn:nbn:de:bsz:291--ds-355824
hdl:20.500.11880/32459
http://dx.doi.org/10.22028/D291-35582
ISSN: 1573-7691
0885-7474
Datum des Eintrags: 24-Feb-2022
Fakultät: MI - Fakultät für Mathematik und Informatik
Fachrichtung: MI - Mathematik
Professur: MI - Prof. Dr. Sergej Rjasanow
Sammlung:SciDok - Der Wissenschaftsserver der Universität des Saarlandes

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