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:
-
[1];
[2];
[3]
- 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 Laboratory (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. https://doi.org/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. https://doi.org/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 = {Thu Apr 11 00:00:00 EDT 2019},
month = {Thu Apr 11 00:00:00 EDT 2019}
}
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