Application of the operator splitting to the Maxwell equations with the source term

Mikhail A. Bochev, I. Faragó, R. Horváth

Abstract

Motivated by numerical solution of the time-dependent Maxwell equations, we consider splitting methods for a linear system of differential equations $w'(t)=Aw(t)+f(t),$ $A\in\mathbb{R}^{n\times n}$ split into two subproblems $w_1'(t)=A_1w_1(t)+f_1(t)$ and $w_2'(t)=A_2w_2(t)+f_2(t),$ $A=A_1+A_2,$ $f=f_1+f_2.$ First, expressions for the leading term of the local error are derived for the Strang-Marchuk and the symmetrically weighted sequential splitting methods. The analysis, done in assumption that the subproblems are solved exactly, confirms the expected second order global accuracy of both schemes. Second, several relevant numerical tests are performed for the Maxwell equations discretized in space either by finite differences or by finite elements. An interesting case is the splitting into the subproblems $w_1'=Aw_1$ and $w_2'=f$ (with the split-off source term $f$). For the central finite difference staggered discretization, we consider second order splitting schemes and compare them to the classical Yee scheme on a test problem with loss and source terms. For the vector Nédélec finite element discretizations, we test the Gautschi-Krylov time integration scheme. Applied in combination with the split-off source term, it leads to splitting schemes that are exact per split step. Thus, the time integration error of the schemes consists solely of the splitting error.
Original languageUndefined
Place of PublicationEnschede
PublisherNumerical Analysis and Computational Mechanics (NACM)
Number of pages25
StatePublished - Jan 2007

Publication series

Name
PublisherDepartment of Applied Mathematics, University of Twente
No.2/1818
ISSN (Print)1874-4850
ISSN (Electronic)1874-4850

Fingerprint

Source terms
Splitting method
Time integration
Maxwell's equations
Finite difference
Sequential methods
Finite element discretization
System of differential equations
Test problems
Strings
Discretization
Linear systems
Numerical solution
Finite element
Term

Keywords

  • EWI-9206
  • IR-66919
  • METIS-242037

Cite this

Bochev, M. A., Faragó, I., & Horváth, R. (2007). Application of the operator splitting to the Maxwell equations with the source term. Enschede: Numerical Analysis and Computational Mechanics (NACM).

Bochev, Mikhail A.; Faragó, I.; Horváth, R. / Application of the operator splitting to the Maxwell equations with the source term.

Enschede : Numerical Analysis and Computational Mechanics (NACM), 2007. 25 p.

Research output: ProfessionalReport

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Bochev, MA, Faragó, I & Horváth, R 2007, Application of the operator splitting to the Maxwell equations with the source term. Numerical Analysis and Computational Mechanics (NACM), Enschede.

Application of the operator splitting to the Maxwell equations with the source term. / Bochev, Mikhail A.; Faragó, I.; Horváth, R.

Enschede : Numerical Analysis and Computational Mechanics (NACM), 2007. 25 p.

Research output: ProfessionalReport

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T1 - Application of the operator splitting to the Maxwell equations with the source term

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AU - Faragó,I.

AU - Horváth,R.

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N2 - Motivated by numerical solution of the time-dependent Maxwell equations, we consider splitting methods for a linear system of differential equations $w'(t)=Aw(t)+f(t),$ $A\in\mathbb{R}^{n\times n}$ split into two subproblems $w_1'(t)=A_1w_1(t)+f_1(t)$ and $w_2'(t)=A_2w_2(t)+f_2(t),$ $A=A_1+A_2,$ $f=f_1+f_2.$ First, expressions for the leading term of the local error are derived for the Strang-Marchuk and the symmetrically weighted sequential splitting methods. The analysis, done in assumption that the subproblems are solved exactly, confirms the expected second order global accuracy of both schemes. Second, several relevant numerical tests are performed for the Maxwell equations discretized in space either by finite differences or by finite elements. An interesting case is the splitting into the subproblems $w_1'=Aw_1$ and $w_2'=f$ (with the split-off source term $f$). For the central finite difference staggered discretization, we consider second order splitting schemes and compare them to the classical Yee scheme on a test problem with loss and source terms. For the vector Nédélec finite element discretizations, we test the Gautschi-Krylov time integration scheme. Applied in combination with the split-off source term, it leads to splitting schemes that are exact per split step. Thus, the time integration error of the schemes consists solely of the splitting error.

AB - Motivated by numerical solution of the time-dependent Maxwell equations, we consider splitting methods for a linear system of differential equations $w'(t)=Aw(t)+f(t),$ $A\in\mathbb{R}^{n\times n}$ split into two subproblems $w_1'(t)=A_1w_1(t)+f_1(t)$ and $w_2'(t)=A_2w_2(t)+f_2(t),$ $A=A_1+A_2,$ $f=f_1+f_2.$ First, expressions for the leading term of the local error are derived for the Strang-Marchuk and the symmetrically weighted sequential splitting methods. The analysis, done in assumption that the subproblems are solved exactly, confirms the expected second order global accuracy of both schemes. Second, several relevant numerical tests are performed for the Maxwell equations discretized in space either by finite differences or by finite elements. An interesting case is the splitting into the subproblems $w_1'=Aw_1$ and $w_2'=f$ (with the split-off source term $f$). For the central finite difference staggered discretization, we consider second order splitting schemes and compare them to the classical Yee scheme on a test problem with loss and source terms. For the vector Nédélec finite element discretizations, we test the Gautschi-Krylov time integration scheme. Applied in combination with the split-off source term, it leads to splitting schemes that are exact per split step. Thus, the time integration error of the schemes consists solely of the splitting error.

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Bochev MA, Faragó I, Horváth R. Application of the operator splitting to the Maxwell equations with the source term. Enschede: Numerical Analysis and Computational Mechanics (NACM), 2007. 25 p.