Spectral methods in the presence of discontinuities

Piotrowska, Joanna; Miller, Jonah Maxwell; Schnetter, Erik

doi:10.1016/j.jcp.2019.03.048

Title: Spectral methods in the presence of discontinuities

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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:: 2019-04-11

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.

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                    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

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                    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.

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@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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