# Application of operator splitting to the Maxwell equations including a source term

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

12 Citations (Scopus)

### 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 language Undefined 10.1016/j.apnum.2008.03.031 522-541 25 Applied numerical mathematics 59 3-4 https://doi.org/10.1016/j.apnum.2008.03.031 Published - Mar 2009

### Keywords

• MSC-65M06
• Gautschi cosine scheme
• MSC-65M20
• MSC-65M60
• MSC-78M10
• MSC-78M20
• MSC-35Q60
• Maxwell equations
• Whitney finite elements
• Yee method
• Krylov subspace methods
• Staggered finite differences
• Splitting methods
• Strang splitting
• EWI-14897
• METIS-263720
• IR-62695
• MSC-65M12

### Cite this

Bochev, M. A., Faragó, I., & Horváth, R. (2009). Application of operator splitting to the Maxwell equations including a source term. Applied numerical mathematics, 59(3-4), 522-541. [10.1016/j.apnum.2008.03.031]. https://doi.org/10.1016/j.apnum.2008.03.031
Bochev, Mikhail A. ; Faragó, I. ; Horváth, R. / Application of operator splitting to the Maxwell equations including a source term. In: Applied numerical mathematics. 2009 ; Vol. 59, No. 3-4. pp. 522-541.
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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 N{\'e}d{\'e}lec 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 2009, 'Application of operator splitting to the Maxwell equations including a source term' Applied numerical mathematics, vol. 59, no. 3-4, 10.1016/j.apnum.2008.03.031, pp. 522-541. https://doi.org/10.1016/j.apnum.2008.03.031

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

In: Applied numerical mathematics, Vol. 59, No. 3-4, 10.1016/j.apnum.2008.03.031, 03.2009, p. 522-541.

TY - JOUR

T1 - Application of operator splitting to the Maxwell equations including a 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 - 2009/3

Y1 - 2009/3

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.

KW - MSC-65M06

KW - Gautschi cosine scheme

KW - MSC-65M20

KW - MSC-65M60

KW - MSC-78M10

KW - MSC-78M20

KW - MSC-35Q60

KW - Maxwell equations

KW - Whitney finite elements

KW - Yee method

KW - Krylov subspace methods

KW - Staggered finite differences

KW - Splitting methods

KW - Strang splitting

KW - EWI-14897

KW - METIS-263720

KW - IR-62695

KW - MSC-65M12

U2 - 10.1016/j.apnum.2008.03.031

DO - 10.1016/j.apnum.2008.03.031

M3 - Article

VL - 59

SP - 522

EP - 541

JO - Applied numerical mathematics

JF - Applied numerical mathematics

SN - 0168-9274

IS - 3-4

M1 - 10.1016/j.apnum.2008.03.031

ER -