Extending Discrete Exterior Calculus to a Fractional Derivative

Justin Crum, Joshua A. Levine, Andrew Gillette

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

Fractional partial differential equations (FDEs) are used to describe phenomena that involve a “non-local” or “long-range” interaction of some kind. Accurate and practical numerical approximation of their solutions is challenging due to the dense matrices arising from standard discretization procedures. In this paper, we begin to extend the well-established computational toolkit of Discrete Exterior Calculus (DEC) to the fractional setting, focusing on proper discretization of the fractional derivative. We define a Caputo-like fractional discrete derivative, in terms of the standard discrete exterior derivative operator from DEC, weighted by a measure of distance between p-simplices in a simplicial complex. We discuss key theoretical properties of the fractional discrete derivative and compare it to the continuous fractional derivative via a series of numerical experiments.

Original languageEnglish (US)
Pages (from-to)64-72
Number of pages9
JournalCAD Computer Aided Design
Volume114
DOIs
StatePublished - Sep 2019

Keywords

  • Discrete exterior calculus (DEC)
  • Fractional derivative
  • Fractional differential equations

ASJC Scopus subject areas

  • Computer Science Applications
  • Computer Graphics and Computer-Aided Design
  • Industrial and Manufacturing Engineering

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