Dispersion and dissipation error in high-order Runge-Kutta discontinuous Galerkin discretisations of the Maxwell equations

D. Sarmany, Mikhail A. Bochev, Jacobus J.W. van der Vegt

  • 30 Citations

Abstract

Different time-stepping methods for a nodal high-order discontinuous Galerkin discretisation of the Maxwell equations are discussed. A comparison between the most popular choices of Runge-Kutta (RK) methods is made from the point of view of accuracy and computational work. By choosing the strong-stability-preserving Runge-Kutta (SSP-RK) time-integration method of order consistent with the polynomial order of the spatial discretisation, better accuracy can be attained compared with fixed-order schemes. Moreover, this comes without a significant increase in the computational work. A numerical Fourier analysis is performed for this Runge-Kutta discontinuous Galerkin (RKDG) discretisation to gain insight into the dispersion and dissipation properties of the fully discrete scheme. The analysis is carried out on both the one-dimensional and the two-dimensional fully discrete schemes and, in the latter case, on uniform as well as on non-uniform meshes. It also provides practical information on the convergence of the dissipation and dispersion error up to polynomial order 10 for the one-dimensional fully discrete scheme.
Original languageUndefined
Pages (from-to)47-74
Number of pages35
JournalJournal of scientific computing
Volume33
Issue numberLNCS4549/1
DOIs
StatePublished - Jul 2007

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Polynomials
Fourier analysis
Runge Kutta methods
Maxwell equations

Keywords

  • Maxwell equations
  • High-order nodal discontinuous Galerkin methods · Maxwell equations · Numerical dispersion and dissipation · Strong-stability-preserving Runge-Kutta methods
  • EWI-10903
  • high-order nodal discontinuous Galerkin methods
  • numerical dispersion and dissipation strong-stability-preserving Runge-Kutta methods
  • MSC-65L06
  • MSC-65M20
  • MSC-65M60
  • METIS-241848
  • IR-64298

Cite this

Sarmany, D.; Bochev, Mikhail A.; van der Vegt, Jacobus J.W. / Dispersion and dissipation error in high-order Runge-Kutta discontinuous Galerkin discretisations of the Maxwell equations.

Vol. 33, No. LNCS4549/1, 07.2007, p. 47-74.

Research output: Scientific - peer-reviewArticle

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abstract = "Different time-stepping methods for a nodal high-order discontinuous Galerkin discretisation of the Maxwell equations are discussed. A comparison between the most popular choices of Runge-Kutta (RK) methods is made from the point of view of accuracy and computational work. By choosing the strong-stability-preserving Runge-Kutta (SSP-RK) time-integration method of order consistent with the polynomial order of the spatial discretisation, better accuracy can be attained compared with fixed-order schemes. Moreover, this comes without a significant increase in the computational work. A numerical Fourier analysis is performed for this Runge-Kutta discontinuous Galerkin (RKDG) discretisation to gain insight into the dispersion and dissipation properties of the fully discrete scheme. The analysis is carried out on both the one-dimensional and the two-dimensional fully discrete schemes and, in the latter case, on uniform as well as on non-uniform meshes. It also provides practical information on the convergence of the dissipation and dispersion error up to polynomial order 10 for the one-dimensional fully discrete scheme.",
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Dispersion and dissipation error in high-order Runge-Kutta discontinuous Galerkin discretisations of the Maxwell equations. / Sarmany, D.; Bochev, Mikhail A.; van der Vegt, Jacobus J.W.

Vol. 33, No. LNCS4549/1, 07.2007, p. 47-74.

Research output: Scientific - peer-reviewArticle

TY - JOUR

T1 - Dispersion and dissipation error in high-order Runge-Kutta discontinuous Galerkin discretisations of the Maxwell equations

AU - Sarmany,D.

AU - Bochev,Mikhail A.

AU - van der Vegt,Jacobus J.W.

N1 - Please note different possible spellings of second author's name: "Botchev" or "Bochev".

PY - 2007/7

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N2 - Different time-stepping methods for a nodal high-order discontinuous Galerkin discretisation of the Maxwell equations are discussed. A comparison between the most popular choices of Runge-Kutta (RK) methods is made from the point of view of accuracy and computational work. By choosing the strong-stability-preserving Runge-Kutta (SSP-RK) time-integration method of order consistent with the polynomial order of the spatial discretisation, better accuracy can be attained compared with fixed-order schemes. Moreover, this comes without a significant increase in the computational work. A numerical Fourier analysis is performed for this Runge-Kutta discontinuous Galerkin (RKDG) discretisation to gain insight into the dispersion and dissipation properties of the fully discrete scheme. The analysis is carried out on both the one-dimensional and the two-dimensional fully discrete schemes and, in the latter case, on uniform as well as on non-uniform meshes. It also provides practical information on the convergence of the dissipation and dispersion error up to polynomial order 10 for the one-dimensional fully discrete scheme.

AB - Different time-stepping methods for a nodal high-order discontinuous Galerkin discretisation of the Maxwell equations are discussed. A comparison between the most popular choices of Runge-Kutta (RK) methods is made from the point of view of accuracy and computational work. By choosing the strong-stability-preserving Runge-Kutta (SSP-RK) time-integration method of order consistent with the polynomial order of the spatial discretisation, better accuracy can be attained compared with fixed-order schemes. Moreover, this comes without a significant increase in the computational work. A numerical Fourier analysis is performed for this Runge-Kutta discontinuous Galerkin (RKDG) discretisation to gain insight into the dispersion and dissipation properties of the fully discrete scheme. The analysis is carried out on both the one-dimensional and the two-dimensional fully discrete schemes and, in the latter case, on uniform as well as on non-uniform meshes. It also provides practical information on the convergence of the dissipation and dispersion error up to polynomial order 10 for the one-dimensional fully discrete scheme.

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KW - high-order nodal discontinuous Galerkin methods

KW - numerical dispersion and dissipation strong-stability-preserving Runge-Kutta methods

KW - MSC-65L06

KW - MSC-65M20

KW - MSC-65M60

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