Extending Discrete Exterior Calculus to a Fractional Derivative

Crum, Justin; Levine, Joshua A.; Gillette, Andrew

doi:10.1016/j.cad.2019.05.018

Extending Discrete Exterior Calculus to a Fractional Derivative

Journal Article · Thu May 16 00:00:00 EDT 2019 · Computer Aided Design

DOI:https://doi.org/10.1016/j.cad.2019.05.018· OSTI ID:1544812

Crum, Justin ^[1]; ^[2]; Gillette, Andrew ^[2]

Univ. of Arizona, Tucson, AZ (United States); Department of Computer Science, University of Arizona
Univ. of Arizona, Tucson, AZ (United States)

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.

View Accepted Manuscript (DOE)

Research Organization:: Univ. of Arizona, Tucson, AZ (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

Grant/Contract Number:: SC0019039

OSTI ID:: 1544812

Journal Information:: Computer Aided Design, Journal Name: Computer Aided Design Journal Issue: C Vol. 114; ISSN 0010-4485

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (16)

Texture Mapping via Optimal Mass Transport Dominitz, A.; Tannenbaum, A. IEEE Transactions on Visualization and Computer Graphics, Vol. 16, Issue 3 https://doi.org/10.1109/tvcg.2009.64	journal	May 2010
Directional Field Synthesis, Design, and Processing Vaxman, Amir; Campen, Marcel; Diamanti, Olga Computer Graphics Forum, Vol. 35, Issue 2 https://doi.org/10.1111/cgf.12864	journal	May 2016
Foundations of Computational Geometric Mechanics Leok, Melvin California Institute of Technology https://doi.org/10.7907/kdv0-wr34	text	January 2004
On the geometric character of stress in continuum mechanics Kanso, Eva; Arroyo, Marino; Tong, Yiying Zeitschrift für angewandte Mathematik und Physik, Vol. 58, Issue 5 https://doi.org/10.1007/s00033-007-6141-8	journal	June 2007
Discrete Lie Advection of Differential Forms Mullen, P.; McKenzie, A.; Pavlov, D. Foundations of Computational Mathematics, Vol. 11, Issue 2 https://doi.org/10.1007/s10208-010-9076-y	journal	September 2010
Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2 Roop, John Paul Journal of Computational and Applied Mathematics, Vol. 193, Issue 1 https://doi.org/10.1016/j.cam.2005.06.005	journal	August 2006
Fractional differential forms Cottrill-Shepherd, Kathleen; Naber, Mark Journal of Mathematical Physics, Vol. 42, Issue 5 https://doi.org/10.1063/1.1364688	journal	January 2001
Numerical methods for Hamiltonian PDEs Bridges, Thomas J.; Reich, Sebastian Journal of Physics A: Mathematical and General, Vol. 39, Issue 19 https://doi.org/10.1088/0305-4470/39/19/S02	journal	April 2006
Texture Mapping via Optimal Mass Transport Dominitz, A.; Tannenbaum, A. IEEE Transactions on Visualization and Computer Graphics, Vol. 16, Issue 3 https://doi.org/10.1109/TVCG.2009.64	journal	May 2010
Linear Models of Dissipation whose Q is almost Frequency Independent--II Caputo, M. Geophysical Journal International, Vol. 13, Issue 5 https://doi.org/10.1111/j.1365-246X.1967.tb02303.x	journal	November 1967
Spectral Mesh Processing Zhang, H.; Van Kaick, O.; Dyer, R. Computer Graphics Forum, Vol. 29, Issue 6 https://doi.org/10.1111/j.1467-8659.2010.01655.x	journal	September 2010
On Discrete Killing Vector Fields and Patterns on Surfaces Ben-Chen, Mirela; Butscher, Adrian; Solomon, Justin Computer Graphics Forum, Vol. 29, Issue 5 https://doi.org/10.1111/j.1467-8659.2010.01779.x	journal	July 2010
Stable, circulation-preserving, simplicial fluids Elcott, Sharif; Tong, Yiying; Kanso, Eva ACM Transactions on Graphics, Vol. 26, Issue 1 https://doi.org/10.1145/1189762.1189766	journal	January 2007
Mesh parameterization: theory and practice Hormann, Kai; Lévy, Bruno; Sheffer, Alla ACM SIGGRAPH 2007 courses on - SIGGRAPH '07 https://doi.org/10.1145/1281500.1281510	conference	January 2007
Underwater cloth simulation with fractional derivatives Ozgen, Oktar; Kallmann, Marcelo; Ramirez, Lynnette Es ACM Transactions on Graphics, Vol. 29, Issue 3 https://doi.org/10.1145/1805964.1805967	journal	June 2010
HOT: Hodge-optimized triangulations Mullen, Patrick; Memari, Pooran; de Goes, Fernando ACM Transactions on Graphics, Vol. 30, Issue 4 https://doi.org/10.1145/2010324.1964998	journal	July 2011

Similar Records

Meshes optimized for discrete exterior calculus (DEC).

Technical Report · Thu Nov 30 23:00:00 EST 2017 · OSTI ID:1414659

Numerical electromagnetic frequency domain analysis with discrete exterior calculus

Journal Article · Thu Nov 30 23:00:00 EST 2017 · Journal of Computational Physics · OSTI ID:22701641

Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

Journal Article · Sun Dec 03 19:00:00 EST 2017 · Journal of Scientific Computing · OSTI ID:1633907

Related Subjects

97 MATHEMATICS AND COMPUTING

Extending Discrete Exterior Calculus to a Fractional Derivative

Citation Formats

References (16)

Similar Records

Related Subjects