Leapfrog Time-Stepping for Hermite Methods

Vargas, Arturo; Hagstrom, Thomas; Chan, Jesse; Warburton, Tim

doi:10.1007/s10915-019-00938-x

Title: Leapfrog Time-Stepping for Hermite Methods

Abstract

We introduce here Hermite-leapfrog methods for first order linear wave systems. The new Hermite-leapfrog methods pair leapfrog time-stepping with the Hermite methods of Goodrich and co-authors et al. (Math Comput 75(254):595–630, 2006). The new schemes stagger field variables in both time and space and are high-order accurate for equations with smooth solutions and coefficients. We provide a detailed description of the method and demonstrate that the method conserves variable quantities. Higher dimensional versions of the method are constructed via tensor products. Numerical evidence and rigorous analysis in one space dimension establish stability and high-order convergence. Experiments demonstrating efficient implementations on a graphics processing unit are also presented.

Authors:

^[1]; Hagstrom, Thomas ^[2]; Chan, Jesse ^[3]; Warburton, Tim ^[4]

Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Southern Methodist Univ., Dallas, TX (United States). Dept. of Mathematics
Rice Univ., Houston, TX (United States). Dept. of Computational and Applied Mathematics
Virginia Polytechnic Inst. and State Univ. (Virginia Tech), Blacksburg, VA (United States). Dept. of Mathematics

Publication Date:: Mon Mar 18 00:00:00 EDT 2019

Research Org.:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States); Southern Methodist Univ., Dallas, TX (United States); Rice Univ., Houston, TX (United States)

Sponsoring Org.:: USDOE; National Science Foundation (NSF)

OSTI Identifier:: 1525718

Report Number(s):: LLNL-JRNL-757049
Journal ID: ISSN 0885-7474; 944652

Grant/Contract Number:: AC52-07NA27344; DMS-1418871; DMS-1719818; DMS-1712639

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Scientific Computing

Additional Journal Information:: Journal Volume: 80; Journal Issue: 1; Journal ID: ISSN 0885-7474

Publisher:: Springer

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; high order; Hermite; leapfrog

Citation Formats


                    Vargas, Arturo, Hagstrom, Thomas, Chan, Jesse, and Warburton, Tim. Leapfrog Time-Stepping for Hermite Methods.  United States: N. p., 2019. 
Web.  doi:10.1007/s10915-019-00938-x.

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                    Vargas, Arturo, Hagstrom, Thomas, Chan, Jesse, & Warburton, Tim. Leapfrog Time-Stepping for Hermite Methods.  United States.  https://doi.org/10.1007/s10915-019-00938-x

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                    Vargas, Arturo, Hagstrom, Thomas, Chan, Jesse, and Warburton, Tim. Mon .  
"Leapfrog Time-Stepping for Hermite Methods".  United States.  https://doi.org/10.1007/s10915-019-00938-x.  https://www.osti.gov/servlets/purl/1525718.

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@article{osti_1525718,

  title        = {Leapfrog Time-Stepping for Hermite Methods},

  author       = {Vargas, Arturo and Hagstrom, Thomas and Chan, Jesse and Warburton, Tim},

  abstractNote = {We introduce here Hermite-leapfrog methods for first order linear wave systems. The new Hermite-leapfrog methods pair leapfrog time-stepping with the Hermite methods of Goodrich and co-authors et al. (Math Comput 75(254):595–630, 2006). The new schemes stagger field variables in both time and space and are high-order accurate for equations with smooth solutions and coefficients. We provide a detailed description of the method and demonstrate that the method conserves variable quantities. Higher dimensional versions of the method are constructed via tensor products. Numerical evidence and rigorous analysis in one space dimension establish stability and high-order convergence. Experiments demonstrating efficient implementations on a graphics processing unit are also presented.},

  doi          = {10.1007/s10915-019-00938-x},

  journal      = {Journal of Scientific Computing},

  number       = 1,

  volume       = 80,

  place        = {United States},

  year         = {Mon Mar 18 00:00:00 EDT 2019},

  month        = {Mon Mar 18 00:00:00 EDT 2019}

}

Copy to clipboard

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Works referenced in this record:

Hermite Methods for the Scalar Wave Equation
journal, January 2018

Appelö, Daniel; Hagstrom, Thomas; Vargas, Arturo
SIAM Journal on Scientific Computing, Vol. 40, Issue 6
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Hermite Methods for Aeroacoustics: Recent Progress
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Appelo, Daniel; Inkman, Matthew; Hagstrom, Thomas
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P -adaptive Hermite methods for initial value problems
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Chen, Ronald; Hagstrom, Thomas
ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 46, Issue 3
DOI: 10.1051/m2an/2011050

A hybrid Hermite–discontinuous Galerkin method for hyperbolic systems with application to Maxwellʼs equations
journal, January 2014

Chen, Xi (Ronald); Appelö, Daniel; Hagstrom, Thomas
Journal of Computational Physics, Vol. 257
DOI: 10.1016/j.jcp.2013.09.046

Two‐dimensional nonlinear inversion of seismic waveforms: Numerical results
journal, July 1986

Gauthier, Odile; Virieux, Jean; Tarantola, Albert
GEOPHYSICS, Vol. 51, Issue 7
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Hermite methods for hyperbolic initial-boundary value problems
journal, December 2005

Goodrich, John; Hagstrom, Thomas; Lorenz, Jens
Mathematics of Computation, Vol. 75, Issue 254
DOI: 10.1090/S0025-5718-05-01808-9

Flux-Conservative Hermite Methods for Simulation of Nonlinear Conservation Laws
journal, November 2017

Kornelus, Adeline; Appelö, Daniel
Journal of Scientific Computing, Vol. 76, Issue 1
DOI: 10.1007/s10915-017-0613-6

A comparison of nonstaggered compact FDTD schemes for the 3D wave equation
conference, March 2010

Kowalczyk, Konrad; van Walstijn, Maarten
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Fourth‐order finite‐difference P-SV seismograms
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Levander, Alan R.
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Variations on Hermite Methods for Wave Propagation
journal, June 2017

Vargas, Arturo; Chan, Jesse; Hagstrom, Thomas
Communications in Computational Physics, Vol. 22, Issue 2
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Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media
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A Staggered Fourth-Order Accurate Explicit Finite Difference Scheme for the Time-Domain Maxwell's Equations
journal, April 2001

Yefet, Amir; Petropoulos, Peter G.
Journal of Computational Physics, Vol. 168, Issue 2
DOI: 10.1006/jcph.2001.6691

Flux-conservative Hermite methods for simulation of nonlinear conservation laws
preprint, January 2017

Kornelus, Adeline; Appelö, Daniel
arXiv
DOI: 10.48550/arxiv.1703.06848

An explicit fourth-order staggered finite-difference time-domain method for Maxwell's equations
journal, October 2002

Xie, Zhongqiang; Chan, Chi-Hou; Zhang, Bo
Journal of Computational and Applied Mathematics, Vol. 147, Issue 1
DOI: 10.1016/s0377-0427(02)00394-1

P -adaptive Hermite methods for initial value problems
journal, January 2012

Chen, Ronald; Hagstrom, Thomas
ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 46, Issue 3
DOI: 10.1051/m2an/2011050

Fourth‐order finite‐difference P-SV seismograms
journal, November 1988

Levander, Alan R.
GEOPHYSICS, Vol. 53, Issue 11
DOI: 10.1190/1.1442422

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Title: Leapfrog Time-Stepping for Hermite Methods

Abstract

Citation Formats

Hermite Methods for the Scalar Wave Equation journal, January 2018

Hermite Methods for Aeroacoustics: Recent Progress conference, November 2012

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Two‐dimensional nonlinear inversion of seismic waveforms: Numerical results journal, July 1986

Hermite methods for hyperbolic initial-boundary value problems journal, December 2005

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A comparison of nonstaggered compact FDTD schemes for the 3D wave equation conference, March 2010

Fourth‐order finite‐difference P-SV seismograms journal, November 1988

Finite Difference Methods for Ordinary and Partial Differential Equations book, January 2007

Variations on Hermite Methods for Wave Propagation journal, June 2017

Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media journal, May 1966

A Staggered Fourth-Order Accurate Explicit Finite Difference Scheme for the Time-Domain Maxwell's Equations journal, April 2001

Flux-conservative Hermite methods for simulation of nonlinear conservation laws preprint, January 2017

An explicit fourth-order staggered finite-difference time-domain method for Maxwell's equations journal, October 2002

P -adaptive Hermite methods for initial value problems journal, January 2012

Fourth‐order finite‐difference P-SV seismograms journal, November 1988

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