Higher-order unconditionally stable algorithms to solve the time-dependent Maxwell equations

Kole, J.S.; Figge, Marc Thilo; Raedt, de Hans

doi:10.1103/physreve.65.066705

Cited by 23 publications

(41 citation statements)

References 18 publications

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“…(25) For (25), if the order of electric field components have not been rearranged we only have [12,13]. Although it is the fact that exp(∆ t L d ) is an orthogonal matrix, the symplectiness of the TEMA seems not be hold.…”

Section: ∂ ∂Tmentioning

confidence: 99%

Maxwell's Equations, Symplectic Matrix, and Grid

Sha¹,

Wu²,

Huang³

et al. 2008

PIER B

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Abstract-The connections between Maxwell's equations and symplectic matrix are studied. First, we analyze the continuous-time Maxwell's differential equations in free space and verify its time evolution matrix (TEMA) is symplectic-unitary matrix for complex space or symplectic-orthogonal matrix for real space. Second, the spatial differential operators are discretized by pseudo-spectral (PS) approach with collocated grid and by finite-difference (FD) method with staggered grid. For the PS approach, the TEMA conserves the symplectic-unitary property. For the FD method, the TEMA conserves the symplectic-orthogonal property. Finally, symplectic integration scheme is used in the time direction. In particular, we find the symplectiness of the TEMA also can be conserved. The mathematical proofs presented are helpful for further numerical study of symplectic schemes.

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Section: ∂ ∂Tmentioning

confidence: 99%

Maxwell's Equations, Symplectic Matrix, and Grid

Sha¹,

Wu²,

Huang³

et al. 2008

PIER B

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show abstract

“…A particularly useful fourth-order approximation (applied in for example [18,19,[27][28][29][30][31][32][33][34][35][36][37][38][39]) is given by [28] …”

Section: A Unconditionally Stable Algorithmsmentioning

confidence: 99%

“…to approximate the matrix exponential exp(tH). Here, we closely follow the derivation of algorithms to solve the time-dependent Maxwell equations [18][19][20][21][22][23], where the problem to be solved is stated in a very similar form, although the underlying physics is different. The first algorithm is based on conserving the existing symmetries during the discretization of time, and is unconditionally stable.…”

mentioning

confidence: 99%

Solving seismic wave propagation in elastic media using the matrix exponential approach

Kole

2003

Wave Motion

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Three numerical algorithms are proposed to solve the time-dependent elastodynamic equations in elastic solids. All algorithms are based on approximating the solution of the equations, which can be written as a matrix exponential. By approximating the matrix exponential with a product formula, an unconditionally stable algorithm is derived that conserves the total elastic energy density. By expanding the matrix exponential in Chebyshev polynomials for a specific time instance, a so-called "one-step" algorithm is constructed that is very accurate with respect to the time integration. By formulating the conventional velocity-stress finite-difference time-domain algorithm (VS-FDTD) in matrix exponential form, the staggered-in-time nature can be removed by a small modification, and higher order in time algorithms can be easily derived. For two different seismic events the accuracy of the algorithms is studied and compared with the result obtained by using the conventional VS-FDTD algorithm.

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“…The need for better stability motivated the creation of a number of unconditionally stable schemes which proved successful in the finite element framework [12,24]. Stable time stepping schemes for the Maxwell equations have been also of importance in connection with finite difference spatial discretizations [22,23,5,19,20]. A scheme proposed by Gautschi [11] has recently received attention in the literature for the solution of second order highly oscillatory ODE's [18,17,15].…”

Section: Introductionmentioning

confidence: 99%

“…Many time stepping schemes exist for the time integration of the spacediscretized Maxwell equations [41,12,24,22,23,5,19,20]. Often the time step in these schemes is restricted either due to stability restrictions or accuracy requirements, e.g.…”

Section: Introductionmentioning

confidence: 99%

The Gautschi time stepping scheme for edge finite element discretizations of the Maxwell equations

Botchev

Harutyunyan

Vegt

2006

Journal of Computational Physics

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For the time integration of edge finite element discretizations of the three-dimensional Maxwell equations, we consider the Gautschi cosine scheme where the action of the matrix function is approximated by a Krylov subspace method. First, for the spacediscretized edge finite element Maxwell equations, the dispersion error of this scheme is analyzed in detail and compared to that of two conventional schemes. Second, we show that the scheme can be implemented in such a way that a higher accuracy can be achieved within less computational time (as compared to other implicit schemes). We also analyzed the error made in the Krylov subspace matrix function evaluations. Although the new scheme is unconditionally stable, it is explicit in structure: as an explicit scheme, it requires only the solution of linear systems with the mass matrix.

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Higher-order unconditionally stable algorithms to solve the time-dependent Maxwell equations

Cited by 23 publications

References 18 publications

Maxwell's Equations, Symplectic Matrix, and Grid

Maxwell's Equations, Symplectic Matrix, and Grid

Solving seismic wave propagation in elastic media using the matrix exponential approach

The Gautschi time stepping scheme for edge finite element discretizations of the Maxwell equations

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