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Numerical methods for nonlocal and fractional models

Journal Article · · Acta Numerica
 [1];  [2];  [1];  [3];  [4];  [5]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  2. Columbia Univ., New York, NY (United States)
  3. Florida State Univ., Tallahassee, FL (United States)
  4. Univ. of Texas, Austin, TX (United States)
  5. Hong Kong Polytechnic Univ. (Hong Kong)
+ Show Author Affiliations

Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. Here, we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling.

Research Organization:
Sandia National Lab. (SNL-CA), Livermore, CA (United States); Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF); US Army Research Office (ARO); Hong Kong Polytechnic Univ.
Grant/Contract Number:
AC04-94AL85000; NA-0003525; DMS-1719699; CCF-1704833; DMS-1819233; W911NF-15-1-0562; 25300818
OSTI ID:
1769927
Report Number(s):
SAND-2021-1962J; 694060
Journal Information:
Acta Numerica, Vol. 29; ISSN 0962-4929
Publisher:
Cambridge University PressCopyright Statement
Country of Publication:
United States
Language:
English

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