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

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

Research output: Book/ReportReport

### 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.
Language Undefined Enschede Numerical Analysis and Computational Mechanics (NACM) 25 Published - Jan 2007

### Publication series

Name Department of Applied Mathematics, University of Twente 2/1818 1874-4850 1874-4850

• 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.
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author = "Bochev, {Mikhail A.} and I. Farag{\'o} and R. Horv{\'a}th",
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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: Book/ReportReport

TY - BOOK

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

AU - Bochev,Mikhail A.

AU - Faragó,I.

AU - Horváth,R.

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

PY - 2007/1

Y1 - 2007/1

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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KW - IR-66919

KW - METIS-242037

M3 - Report

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

PB - Numerical Analysis and Computational Mechanics (NACM)

CY - Enschede

ER -

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.