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Title: 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 L2-norm. However, for non-smooth problems, convergence is significantly worse—the L2-norm of the error for a discontinuous problem will converge at a sub-linear rate and the infinity norm will not converge at all. We explore and improve upon a post-processing technique—optimally convergent mollifiers—to recover exponential convergence from a poorly-converging spectral reconstruction of non-smooth data. Furthermore this is an important first step towards using these techniques for simulations of realistic systems.

Authors:
ORCiD logo [1]; ORCiD logo [2]; ORCiD logo [3]
  1. Univ. of Cambridge, Cambridge (United Kingdom); Perimeter Inst. for Theoretical Physics, Waterloo, ON (Canada)
  2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  3. 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):
LA-UR-17-31492
Journal ID: ISSN 0021-9991
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 0021-9991
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 L2-norm. However, for non-smooth problems, convergence is significantly worse—the L2-norm of the error for a discontinuous problem will converge at a sub-linear rate and the infinity norm will not converge at all. We explore and improve upon a post-processing technique—optimally convergent mollifiers—to recover exponential convergence from a poorly-converging spectral reconstruction of non-smooth 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}
}

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