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doi:10.22028/D291-35582
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 |
Dateien zu diesem Datensatz:
Datei | Beschreibung | Größe | Format | |
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Buchheit-Keßler2021_Article_OnTheEfficientComputationOfLar.pdf | 553,96 kB | Adobe PDF | Öffnen/Anzeigen |
Diese Ressource wurde unter folgender Copyright-Bestimmung veröffentlicht: Lizenz von Creative Commons