# Time-integration methods for finite element discretisations of the second-order Maxwell equation

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

Research output: Book/ReportReport

### Abstract

This article deals with time integration for the second-order Maxwell equations with possibly non-zero conductivity in the context of the discontinuous Galerkin finite element method DG-FEM) and the $H(\mathrm{curl})$-conforming FEM. For the spatial discretisation, hierarchic $H(\mathrm{curl})$-conforming basis functions are used up to polynomial order $p=3$ over tetrahedral meshes, meaning fourth-order convergence rate. A high-order polynomial basis often warrants the use of high-order time-integration schemes, but many well-known high-order schemes may suffer from a severe time-step stability restriction owing to the conductivity term. We investigate several possible time-integration methods from the point of view of accuracy, stability and computational work. We also carry out a numerical Fourier analysis to study the dispersion and dissipation properties of the semi-discrete DG-FEM scheme as well as the fully-discrete schemes with several of the time-integration methods. The dispersion and dissipation properties of the spatial discretisation and those of the time-integration methods are investigated separately, providing additional insight into the two discretisation steps.
Language Undefined Enschede Department of Applied Mathematics, University of Twente 25 Published - Feb 2012

### Publication series

Name Memorandum / Department of Applied Mathematics University of Twente, Department of Applied Mathematics 1975 1874-4850 1874-4850

### Keywords

• Second-order damped Maxwell wave equation
• High-order numerical time integration
• H(curl)-conforming finite element method
• IR-79682
• METIS-285242
• Discontinuous Galerkin finite element method
• MSC-65M60
• MSC-65M20
• MSC-65L20
• MSC-65L06
• EWI-21520

### Cite this

Sarmany, D., Bochev, M. A., & van der Vegt, J. J. W. (2012). Time-integration methods for finite element discretisations of the second-order Maxwell equation. (Memorandum / Department of Applied Mathematics; No. 1975). Enschede: Department of Applied Mathematics, University of Twente.
Sarmany, D. ; Bochev, Mikhail A. ; van der Vegt, Jacobus J.W./ Time-integration methods for finite element discretisations of the second-order Maxwell equation. Enschede : Department of Applied Mathematics, University of Twente, 2012. 25 p. (Memorandum / Department of Applied Mathematics; 1975).
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abstract = "This article deals with time integration for the second-order Maxwell equations with possibly non-zero conductivity in the context of the discontinuous Galerkin finite element method DG-FEM) and the $H(\mathrm{curl})$-conforming FEM. For the spatial discretisation, hierarchic $H(\mathrm{curl})$-conforming basis functions are used up to polynomial order $p=3$ over tetrahedral meshes, meaning fourth-order convergence rate. A high-order polynomial basis often warrants the use of high-order time-integration schemes, but many well-known high-order schemes may suffer from a severe time-step stability restriction owing to the conductivity term. We investigate several possible time-integration methods from the point of view of accuracy, stability and computational work. We also carry out a numerical Fourier analysis to study the dispersion and dissipation properties of the semi-discrete DG-FEM scheme as well as the fully-discrete schemes with several of the time-integration methods. The dispersion and dissipation properties of the spatial discretisation and those of the time-integration methods are investigated separately, providing additional insight into the two discretisation steps.",
keywords = "Second-order damped Maxwell wave equation, High-order numerical time integration, H(curl)-conforming finite element method, IR-79682, METIS-285242, Discontinuous Galerkin finite element method, MSC-65M60, MSC-65M20, MSC-65L20, MSC-65L06, EWI-21520",
author = "D. Sarmany and Bochev, {Mikhail A.} and {van der Vegt}, {Jacobus J.W.}",
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series = "Memorandum / Department of Applied Mathematics",
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Sarmany, D, Bochev, MA & van der Vegt, JJW 2012, Time-integration methods for finite element discretisations of the second-order Maxwell equation. Memorandum / Department of Applied Mathematics, no. 1975, Department of Applied Mathematics, University of Twente, Enschede.

Time-integration methods for finite element discretisations of the second-order Maxwell equation. / Sarmany, D.; Bochev, Mikhail A.; van der Vegt, Jacobus J.W.

Enschede : Department of Applied Mathematics, University of Twente, 2012. 25 p. (Memorandum / Department of Applied Mathematics; No. 1975).

Research output: Book/ReportReport

TY - BOOK

T1 - Time-integration methods for finite element discretisations of the second-order Maxwell equation

AU - Sarmany,D.

AU - Bochev,Mikhail A.

AU - van der Vegt,Jacobus J.W.

N1 - Devoted to the memory of Jan Verwer. Second author's surname can also be spelled as "Bochev".

PY - 2012/2

Y1 - 2012/2

N2 - This article deals with time integration for the second-order Maxwell equations with possibly non-zero conductivity in the context of the discontinuous Galerkin finite element method DG-FEM) and the $H(\mathrm{curl})$-conforming FEM. For the spatial discretisation, hierarchic $H(\mathrm{curl})$-conforming basis functions are used up to polynomial order $p=3$ over tetrahedral meshes, meaning fourth-order convergence rate. A high-order polynomial basis often warrants the use of high-order time-integration schemes, but many well-known high-order schemes may suffer from a severe time-step stability restriction owing to the conductivity term. We investigate several possible time-integration methods from the point of view of accuracy, stability and computational work. We also carry out a numerical Fourier analysis to study the dispersion and dissipation properties of the semi-discrete DG-FEM scheme as well as the fully-discrete schemes with several of the time-integration methods. The dispersion and dissipation properties of the spatial discretisation and those of the time-integration methods are investigated separately, providing additional insight into the two discretisation steps.

AB - This article deals with time integration for the second-order Maxwell equations with possibly non-zero conductivity in the context of the discontinuous Galerkin finite element method DG-FEM) and the $H(\mathrm{curl})$-conforming FEM. For the spatial discretisation, hierarchic $H(\mathrm{curl})$-conforming basis functions are used up to polynomial order $p=3$ over tetrahedral meshes, meaning fourth-order convergence rate. A high-order polynomial basis often warrants the use of high-order time-integration schemes, but many well-known high-order schemes may suffer from a severe time-step stability restriction owing to the conductivity term. We investigate several possible time-integration methods from the point of view of accuracy, stability and computational work. We also carry out a numerical Fourier analysis to study the dispersion and dissipation properties of the semi-discrete DG-FEM scheme as well as the fully-discrete schemes with several of the time-integration methods. The dispersion and dissipation properties of the spatial discretisation and those of the time-integration methods are investigated separately, providing additional insight into the two discretisation steps.

KW - Second-order damped Maxwell wave equation

KW - High-order numerical time integration

KW - H(curl)-conforming finite element method

KW - IR-79682

KW - METIS-285242

KW - Discontinuous Galerkin finite element method

KW - MSC-65M60

KW - MSC-65M20

KW - MSC-65L20

KW - MSC-65L06

KW - EWI-21520

M3 - Report

T3 - Memorandum / Department of Applied Mathematics

BT - Time-integration methods for finite element discretisations of the second-order Maxwell equation

PB - Department of Applied Mathematics, University of Twente

CY - Enschede

ER -

Sarmany D, Bochev MA, van der Vegt JJW. Time-integration methods for finite element discretisations of the second-order Maxwell equation. Enschede: Department of Applied Mathematics, University of Twente, 2012. 25 p. (Memorandum / Department of Applied Mathematics; 1975).