Dispersion and dissipation errors of two fully discrete discontinuous galerkin methods

He Yang, Fengyan Li, Jianxian Qiu

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

The dispersion and dissipation properties of numerical methods are very important in wave simulations. In this paper, such properties are analyzed for Runge-Kutta discontinuous Galerkin methods and Lax-Wendroff discontinuous Galerkin methods when solving the linear advection equation. With the standard analysis, the asymptotic formulations are derived analytically for the discrete dispersion relation in the limit of K = kh → 0 (k is the wavenumber and h is the meshsize) as a function of the CFL number, and the results are compared quantitatively between these two fully discrete numerical methods. For Lax-Wendroff discontinuous Galerkin methods, we further introduce an alternative approach which is advantageous in dispersion analysis when the methods are of arbitrary order of accuracy. Based on the analytical formulations of the dispersion and dissipation errors, we also investigate the role of the spatial and temporal discretizations in the dispersion analysis. Numerical experiments are presented to validate some of the theoretical findings. This work provides the first analysis for Lax-Wendroff discontinuous Galerkin methods.

Original languageEnglish (US)
Pages (from-to)552-574
Number of pages23
JournalJournal of Scientific Computing
Volume55
Issue number3
DOIs
StatePublished - Jan 1 2013

Fingerprint

Discontinuous Galerkin Method
Galerkin methods
Dissipation
Numerical Methods
Numerical methods
Advection Equation
Formulation
Runge-Kutta
Dispersion Relation
Advection
Linear equation
Discretization
Numerical Experiment
Alternatives
Arbitrary
Simulation
Experiments

Keywords

  • Discrete dispersion relation
  • Lax-Wendroff discontinuous Galerkin method
  • Runge-Kutta discontinuous Galerkin method

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • Engineering(all)
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics

Cite this

Dispersion and dissipation errors of two fully discrete discontinuous galerkin methods. / Yang, He; Li, Fengyan; Qiu, Jianxian.

In: Journal of Scientific Computing, Vol. 55, No. 3, 01.01.2013, p. 552-574.

Research output: Contribution to journalArticle

@article{32a6e00656c04484adc764456f2953e0,
title = "Dispersion and dissipation errors of two fully discrete discontinuous galerkin methods",
abstract = "The dispersion and dissipation properties of numerical methods are very important in wave simulations. In this paper, such properties are analyzed for Runge-Kutta discontinuous Galerkin methods and Lax-Wendroff discontinuous Galerkin methods when solving the linear advection equation. With the standard analysis, the asymptotic formulations are derived analytically for the discrete dispersion relation in the limit of K = kh → 0 (k is the wavenumber and h is the meshsize) as a function of the CFL number, and the results are compared quantitatively between these two fully discrete numerical methods. For Lax-Wendroff discontinuous Galerkin methods, we further introduce an alternative approach which is advantageous in dispersion analysis when the methods are of arbitrary order of accuracy. Based on the analytical formulations of the dispersion and dissipation errors, we also investigate the role of the spatial and temporal discretizations in the dispersion analysis. Numerical experiments are presented to validate some of the theoretical findings. This work provides the first analysis for Lax-Wendroff discontinuous Galerkin methods.",
keywords = "Discrete dispersion relation, Lax-Wendroff discontinuous Galerkin method, Runge-Kutta discontinuous Galerkin method",
author = "He Yang and Fengyan Li and Jianxian Qiu",
year = "2013",
month = "1",
day = "1",
doi = "10.1007/s10915-012-9647-y",
language = "English (US)",
volume = "55",
pages = "552--574",
journal = "Journal of Scientific Computing",
issn = "0885-7474",
publisher = "Springer New York",
number = "3",

}

TY - JOUR

T1 - Dispersion and dissipation errors of two fully discrete discontinuous galerkin methods

AU - Yang, He

AU - Li, Fengyan

AU - Qiu, Jianxian

PY - 2013/1/1

Y1 - 2013/1/1

N2 - The dispersion and dissipation properties of numerical methods are very important in wave simulations. In this paper, such properties are analyzed for Runge-Kutta discontinuous Galerkin methods and Lax-Wendroff discontinuous Galerkin methods when solving the linear advection equation. With the standard analysis, the asymptotic formulations are derived analytically for the discrete dispersion relation in the limit of K = kh → 0 (k is the wavenumber and h is the meshsize) as a function of the CFL number, and the results are compared quantitatively between these two fully discrete numerical methods. For Lax-Wendroff discontinuous Galerkin methods, we further introduce an alternative approach which is advantageous in dispersion analysis when the methods are of arbitrary order of accuracy. Based on the analytical formulations of the dispersion and dissipation errors, we also investigate the role of the spatial and temporal discretizations in the dispersion analysis. Numerical experiments are presented to validate some of the theoretical findings. This work provides the first analysis for Lax-Wendroff discontinuous Galerkin methods.

AB - The dispersion and dissipation properties of numerical methods are very important in wave simulations. In this paper, such properties are analyzed for Runge-Kutta discontinuous Galerkin methods and Lax-Wendroff discontinuous Galerkin methods when solving the linear advection equation. With the standard analysis, the asymptotic formulations are derived analytically for the discrete dispersion relation in the limit of K = kh → 0 (k is the wavenumber and h is the meshsize) as a function of the CFL number, and the results are compared quantitatively between these two fully discrete numerical methods. For Lax-Wendroff discontinuous Galerkin methods, we further introduce an alternative approach which is advantageous in dispersion analysis when the methods are of arbitrary order of accuracy. Based on the analytical formulations of the dispersion and dissipation errors, we also investigate the role of the spatial and temporal discretizations in the dispersion analysis. Numerical experiments are presented to validate some of the theoretical findings. This work provides the first analysis for Lax-Wendroff discontinuous Galerkin methods.

KW - Discrete dispersion relation

KW - Lax-Wendroff discontinuous Galerkin method

KW - Runge-Kutta discontinuous Galerkin method

UR - http://www.scopus.com/inward/record.url?scp=84893676539&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893676539&partnerID=8YFLogxK

U2 - 10.1007/s10915-012-9647-y

DO - 10.1007/s10915-012-9647-y

M3 - Article

AN - SCOPUS:84893676539

VL - 55

SP - 552

EP - 574

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

IS - 3

ER -