Spectral methods in the presence of discontinuities
Abstract
Spectral methods provide an elegant and efficient way of numerically solving differential equations of all kinds. For smooth problems, truncation error for spectral methods vanishes exponentially in the infinity norm and L_{2}norm. However, for nonsmooth problems, convergence is significantly worse—the L_{2}norm of the error for a discontinuous problem will converge at a sublinear rate and the infinity norm will not converge at all. We explore and improve upon a postprocessing technique—optimally convergent mollifiers—to recover exponential convergence from a poorlyconverging spectral reconstruction of nonsmooth data. Furthermore this is an important first step towards using these techniques for simulations of realistic systems.
 Authors:

 Univ. of Cambridge, Cambridge (United Kingdom); Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada); Univ. of Waterloo, Waterloo, ON (Canada); Louisiana State Univ., Baton Rouge, LA (United States)
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC)
 OSTI Identifier:
 1508545
 Alternate Identifier(s):
 OSTI ID: 1547523
 Report Number(s):
 LAUR1731492
Journal ID: ISSN 00219991
 Grant/Contract Number:
 89233218CNA000001; 20170508DR; SC0018297
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Volume: 390; Journal ID: ISSN 00219991
 Publisher:
 Elsevier
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Spectral method; Numerical analysis; Gibbs' phenomenon; Shock handling; Discontinuous functions; Partial differential equations
Citation Formats
Piotrowska, Joanna, Miller, Jonah Maxwell, and Schnetter, Erik. Spectral methods in the presence of discontinuities. United States: N. p., 2019.
Web. doi:10.1016/j.jcp.2019.03.048.
Piotrowska, Joanna, Miller, Jonah Maxwell, & Schnetter, Erik. Spectral methods in the presence of discontinuities. United States. doi:10.1016/j.jcp.2019.03.048.
Piotrowska, Joanna, Miller, Jonah Maxwell, and Schnetter, Erik. Thu .
"Spectral methods in the presence of discontinuities". United States. doi:10.1016/j.jcp.2019.03.048. https://www.osti.gov/servlets/purl/1508545.
@article{osti_1508545,
title = {Spectral methods in the presence of discontinuities},
author = {Piotrowska, Joanna and Miller, Jonah Maxwell and Schnetter, Erik},
abstractNote = {Spectral methods provide an elegant and efficient way of numerically solving differential equations of all kinds. For smooth problems, truncation error for spectral methods vanishes exponentially in the infinity norm and L2norm. However, for nonsmooth problems, convergence is significantly worse—the L2norm of the error for a discontinuous problem will converge at a sublinear rate and the infinity norm will not converge at all. We explore and improve upon a postprocessing technique—optimally convergent mollifiers—to recover exponential convergence from a poorlyconverging spectral reconstruction of nonsmooth data. Furthermore this is an important first step towards using these techniques for simulations of realistic systems.},
doi = {10.1016/j.jcp.2019.03.048},
journal = {Journal of Computational Physics},
number = ,
volume = 390,
place = {United States},
year = {2019},
month = {4}
}