Alternating Direction Implicit Methods for FDTD Using the Dey-Mittra Embedded Boundary Method

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1 Te Open Plasma Pysics Journal, 2010, 3, Open Access Alternating Direction Implicit Metods for FDTD Using te Dey-Mittra Embedded Boundary Metod T.M. Austin *, J.R. Cary, D.N. Smite C. Nieter Tec-X Corporation 5621 Arapaoe Ave. Suite A Boulder, CO 80303, USA Abstract: Te alternating direction implicit (ADI) metod is an attractive option to use in avoiding te Courant- Friedrics-Lewy (CFL) condition tat limits te size of te time step required by explicit finite-difference time-domain (FDTD) metods for stability. Implicit metods like Crank-Nicolson offer te same advantages as ADI metods but tey do not rely on simple, one-dimensional, tridiagonal system solves for wic tere are well-known fast solution metods. To date, te ADI metod applied to te FDTD metod for curved domains as been used witin te context of subgridding (i.e., local refinement) or for stairstepped boundaries tat are only first-order accurate. A popular secondorder accurate approac to representing smoot domains wit te FDTD metod is te Dey-Mittra embedded boundary metod. However, to be useful in a realistic setting, te cells wit only a small fraction of teir volume inside te domain need to be discarded from simulations for stability considerations or else te time step size will be proibitively small. Using te ADI metod instead of te explicit metod implies tat time step can be cosen to depend on accuracy no cells need discarding. We sow in tis paper te ability to maintain stability beyond te CFL limit for te Dey-Mittra metod witout discarding any cells. We also consider convergence of te ADI metod as compared to te stard explicit metod tat is limited by te CFL condition. 1. INTRODUCTION Te alternating direction implicit (ADI) metod is a powerful implicit metod for solving a finite-difference time-domain (FDTD) discretization of Maxwell's equations. Tis metod (ADI-FDTD) consists of a series of simple, one-dimensional, tridiagonal system solves in contrast to a single large system solve as is required by te Crank- Nicolson metod. Te majority of te work on te ADI- FDTD metod as focused on simple, rectangular domains (cf. [1-4]) avoided modeling curved domains like tose found in complex accelerator structures. One approac to modeling curved domains is to use a subgridding sceme on te boundary [5]. For suc a subgridding sceme, a ybrid ADI-FDTD metod was proposed in Ref. [6] tat uses an ADI metod on te iger resolved Yee cells an explicit metod on te coarser cells. In tis work, instead of using subgridding, we address te boundary wit scemes tat alter te discrete equations in te cells cut by te boundaries, i.e., te cut-cells. Te original approac proposed by Yee in Ref. [7] for addressing te cut-cells was te stairstepping approac. In stairstepping, a cut-cell is labelled eiter interior or exterior depending on te location of te cut-cell center. See Fig. (1a). A more recent accurate approac was proposed by Dey Mittra in Ref. [8]. Te Dey-Mittra approac accurately includes te fractional face areas edge lengts of te cut-cells in te explicit metod wit te condition tat small fractional face areas can cause a significant reduction in te time step to maintain stability. In tis paper, we consider *Address correspondence to tis autor at te Tec-X Corporation 5621 Arapaoe Ave. Suite A Boulder, CO 80303, USA; austin@txcorp.com te FDTD metod using a Dey-Mittra approac for te cutcells, wic we refer to as te EXP-FDTD approac. As described in Ref. [8], cut-cells wit restrictively small fractional areas can be discarded from te simulation to maintain a reasonable time step. If too many cells are discarded, ten te metod can become first-order. Tus, a metod is needed tat employs te Dey-Mittra second-order approac but does not lead to restrictively small time steps te reduction in convergence order. For tis reason, we propose te ADI-FDTD metod to be used in combination wit te Dey-Mittra embedded boundary metod. We will sow tat te ADI-FDTD metod is stable at any time step yields accurate frequency calculations for te 2D Palevsky-Bekefi A6 magnetron [9]. Te outline of te paper is as follows. In te next section, we introduce te Dey-Mittra embedded boundary metod tat is used to model curved domains wit te FDTD metod ten we introduce te ADI-FDTD algoritm. We ten briefly introduce te frequency extraction metod of Ref. [10] present results for te 2D A6 magnetron tat is modeled using te FDTD metod combined wit te Dey- Mittra boundary algoritm. We consider a frequency extraction algoritm tat requires time domain simulations using eiter te EXP-FDTD metod or te ADI-FDTD metod. We explore te accuracy of te various approaces teir stability. We end wit a discussion of tese results a conclusion. 2. METHODOLOGY 2.1. Dey-Mittra Embedded Boundary Metod Te classical FDTD metod is te Yee metod [7] used for solving Maxwell's equations on a lattice grid. In tree dimensions, electric field values are located at te centers of / Bentam Open

2 30 Te Open Plasma Pysics Journal, 2010, Volume 3 Austin et al. (a) Stairstepping (b) Dey-Mittra Fig. (1). (a) Approximating a curved boundary wit a stairstepping approac, were te stard Yee algoritm is performed on te wite cells, wile te dark gray cells are in te metal boundary left out of te computations. (b) Approximating wit te Dey- Mittra embedded boundary metod were te classical Yee metod is used on all wite cells, a Dey-Mittra embedded boundary metod on te ligt gray cells, te dark gray cells are in te metal boundary left out of te computations. Te X'd cells ave only a fractional area in te domain are assigned to be a part of te metal boundary. te edges of a cubic cell magnetic field values are located at te centers of te faces of a cubic cell. Te Yee metod is second-order accurate in time space preserves divergence-free quantities. Te original approac for modeling embedded boundaries proposed by Yee in Ref. [7] stairsteps te boundary as in Fig. (1a). Because tis is only a first-order metod, te development of iger-order metods as been a ig priority. In 1997 Dey Mittra presented an embedded boundary algoritm tat wit some restrictions yields an overall second-order metod [8]. Te Dey-Mittra algoritm is te boundary approac used in te VORPAL computational framework [11], tus as been extensively used tested in Refs. [12, 13]. To describe te Dey-Mittra embedded boundary metod, we consider te x-component of te equation for Faraday's Law in Maxwell's equation as B x / t = (E) x. Te finite difference version of tis equation is given by n+1/2 B x;i, j+1/2,k+1/2 n1/2 = B x;i, j+1/2,k+1/2 + t A yz (1) ( E n z;i, j+1,k+1/2 l z + E n z;i, j,k+1/2 l z + E n y;i, j+1/2,k+1 l y E n y;i, j+1/2,k l y ) n+1/2 were B x,i, j+1/2,k+1/2 is te z-component of te magnetic field in cell (i, j, k) at time t =(n +1/2)t. Additionally, l y l z are te lengts of te cell edges on wic E y E z are located, A yz is te cell area of te face centered at B x. In Eq. (1) te electric field values are located on te edges of a face of a cell wit te magnetic field value centered on te face. Te Dey-Mittra embedded boundary metod redefines Eq. (1) wen a metal boundary cuts troug te cell face as in Fig. (2). A more accurate approac tan stairstepping is needed to acieve a second-order metod. Te Dey-Mittra embedded boundary metod sets te lengts ( l y l z ) te areas ( A yz ) in Eq. (1) to account for te cut-cell lengts areas. See Fig. (2) for reference. Te equivalent equation to Eq. (1) for te electric field update is advanced as wit te Yee metod but setting te electric field to zero wen te corresponding edge is contained entirely witin te metal boundary. In Ref. [14], te autors describe te advantages disadvantages of te Dey-Mittra embedded boundary algoritm. Of cief concern is te effect of te cut-cells on te maximum stable time step permitted. For te Yee algoritm, te time step for stability is limited by te CFL condition. As detailed in Ref. [14], te stability condition derived from te Gerscgorin Circle Teorem states tat t is related to x, y, z by t c 1 x y z 2 were c is te speed-of-ligt. As a result, te time step is determined by te stability of te numerical metod, not te desired accuracy. Te effect of a cut-cell on stability is to furter restrict te time step because te cell size is being reduced wen a cell is cut. Again, tis is detailed in Ref. [14] were te autors introduce a factor, f DM [0,1], tat acts as (2)

3 Alternating Direction Implicit Metods for FDTD Te Open Plasma Pysics Journal, 2010, Volume 3 31 a tresold on te size of a cut-cell. For example, setting f DM = 0.5 ensures tat a cell wit a local Courant evaluation tat is less tan 50% of te nominal value is excluded from computations. Tis implies te time step, t, as te new stability condition given by t 0.5 t CFL 0.5 c 1 x y + 1. (3) 2 z 2 Oter disadvantages suc as a reduction in order of convergence from second-order to first-order due to te deletion of too many cut-cells is discussed in Ref. [14]. In conclusion, an approac tat uses te same set of equations but does not suffer from te effect of cut-cells on time step size convergence is needed. Fig. (2). A cut-cell in te Dey-Mittra embedded boundary algoritm is led in te Faraday update (above) by adjusting te lengt (l) area values (A) according to teir fraction witin te domain. Te equivalent Ampere update for te electric field (not sown) is altered by setting te electric field value to zero for cells contained entirely witin te metal boundary Alternating Direction Implicit Metod Te alternating direction implicit metod is an implicit metod tat solves a set of simple, one-dimensional, tridiagonal systems as part of te time step update. Tere is no CFL condition tat limits te time step tus no effect of fractional cut-cells from te Dey-Mittra metod. Hence, it is an ideal cidate for use wit te Dey-Mittra embedded boundary metod. Furtermore, recently in Ref. [15], te autors presented a new version of te ADI-FDTD metod. We briefly summarize te metod ere ten describe te translation to ling te embedded boundary equations from te Dey-Mittra approac. To present te ADI-FDTD metod, we express te Maxwell's equations as B t = E (4) E t = c2 B J 0, (5) note tat tese equations can be written as W t = S + (P + M)W (6) were W is te six-component field, (E,cB), S is te source term, ( J 0,0), te operators P M are defined by E x E y E z PW = P cb x cb y cb z E x E y E z M W = M cb x cb y cb z c cb z / y cb x / z cb y / x E y / z E z / x E x / y cb y / z cb z / x cb x / y c E z / y E x / z E y / x As in Ref. [15] we use te mnemonic tat P as a plus sign on te rigt, wile te operator M as a minus sign. Te discrete representation of Eq. (7) is obtained by assuming te electric magnetic fields are laid out as described previously. Tat is, te electric field values are located at te centers of cell edges (wit index (i, j, k)) wile te magnetic fields are located at te centers of cell faces (wit index (i, j, k)). Terefore, te discrete representation of Eq. (7) is given by c l z (B z,i, j,k B z,i, j1,k )/A yz c l x (B x,i, j,k B x,i, j,k1 )/A xz c l y (B y,i, j,k B y,i1, j,k )/A xy P W c l y (E y,i, j,k+1 E y,i, j,k )/A yz l z (E z,i+1, j,k E z,i, j,k )/A xz l x (E x,i, j+1,k E x,i, j,k )/A xy c l y (B y,i, j,k B y,i, j,k1 )/A yz c l x (B z,i, j,k B z,i1, j,k )/A xz c l x (B x,i, j,k B x,i, j1,k )/A xy M W c l z (E z,i, j+1,k E z,i, j,k )/A yz l x (E x,i, j,k+1 E x,i, j,k )/A xz l y (E y,i+1, j,k E y,i, j,k )/A xy were W is te discrete representation of W P M are te finite difference versions of te continuum operators (7) (8)

4 32 Te Open Plasma Pysics Journal, 2010, Volume 3 Austin et al. P M from above. Subsequently, te discrete Maxwell's equations witout sources is given by W t =(P + M )W. (9) In Ref. [15] te autors compared te various secondorder ADI operators tat ave been proposed by Zeng et al. in Ref. [2] Lee Fornberg in Ref. [1]. Instead of using tese operators, te autors proposed an update operator tat updates W according to W n+1 = I + t 2 M I t 2 P 1 I + t 2 P I t 2 M W n + ts n1/2 (10) wit te property tat B =0 E n+1 E n =( n+1 n )/ 0 to macine precision. We focus on tis form for te ADI update of Maxwell's equations but note noting ere is specific to tis form. Te extension of Eq. (10) to embedded boundaries is a direct application of te Dey-Mittra boundary algoritm presented in Sec For te Ampere update of E, it was discussed in Sec. 2.1 tat te electric field is set to zero for edges contained entirely witin te conductor; oterwise, te update step from te Yee algoritm was not canged for electric field values on edges partially in te conductor. In terms of Eq. (7), te electric field update for edges entirely in te conductor affects P M by setting te lengt coefficients on te B { x,y,z},i, j,k coefficients to be zero suc tat, in Eq. (10), eac of te four operators becomes te identity operator, I, for tose values. Tus, for all electric field values on edges in te conductor, Eq. (10) becomes W n+1 = W n since we ave no sources in te conductor. For te Faraday update, we ave to adjust te coefficients on te E { x,y,z},i, j,k terms in Eq. (10) according to te fraction of te corresponding edge witin te domain as was done in Eq. (1). Similarly, we ave to adjust te areas for te faces according to te fraction witin te domain. See Fig. (2) for reference. Tus, for magnetic field values on faces tat are partially in te conductor, te contribution to P M in Eq. (7) is altered by te fractions of te lengts of eac edge te fraction of te area of eac face associated wit te update. Once tese canges are made to P M te ADI update proceeds accordingly. As we will see in te next section tere is no effect on te stability of te metod by incorporating te Dey-Mittra boundary contributions to te update step. Furtermore, te fraction of te edges areas witin te conductor as no effect on te time step due to te implicit nature of te ADI metod. 3. RESULTS 3.1. Background To establis te stability accuracy of te ADI-FDTD metod tat employs te Dey-Mittra embedded boundary approximation, we consider te extraction of frequencies from te well-studied 2D Palevsky-Bekefi A6 magnetron device [9] pictured in Fig. (3). In particular, we use te frequency extraction metod presented in Refs. [10, 12] to determine te frequencies between GHz. Previous work [16] as found te frequencies between GHz to be tose given in Table 1. We consider te accuracy of te extracted frequencies as it depends on spatial resolution temporal resolution. We apply te frequency extraction algoritm to simulations performed wit te ADI-FDTD metod presented also for reference we use te EXP-FDTD metod. We sow for te A6 magnetron tat te factor, f DM, can be made arbitrarily small ensuring tat all cells are kept in te simulation, wile maintaining stability at time steps beyond te CFL condition. Fig. (3). Te 2D Palevsky-Bekefi A6 magnetron device for wic te ADI metod was used for extracting frequencies wit te broadlyfiltered diagonalization metod of Ref. [10]. Note tat te magnetron is 8 m by 8 m wile te domain is 9 m by 9 m. Before presenting any results, we briefly describe te metod presented in Ref. [10, 12] to give te reader a context for te simulations. Te frequency extraction metod as two pases. Te first pase is te ring-up pase during wic a Gaussian-modulated signal is applied to te Maxwell's equations (troug a current source) suc tat only te frequencies between GHz are excited. Table 1. Frequencies (in GHz) of te 2D Palevsky-Bekefi A6 Magnetron Between GHz as Presented in Ref. [16] f 0 f 1 f 2 f 3 f 4 f 5 f

5 Alternating Direction Implicit Metods for FDTD Te Open Plasma Pysics Journal, 2010, Volume 3 33 Te current, J(x, y, z,t), is given by J(x, y, z,t)= f (t)ĵ(x, y, z) were 2 sin( 1(t T /2)) sin( 2(t T /2)) 2 (tt /2) 2 /2 f (t)= t T /2 t T /2 exp 0 t T, 0. oterwise (11) were 1 = 2.0e9 2 = 16.0e9. Ĵ(x, y, z) as a pattern tat encourages excitation of te desired modes in te frequency range [ 1, 2 ]. Te parameter is determined by te separation of te frequencies in [ 1, 2 ] from te next nearest frequency value. If ˆ < 1 is te nearest frequency, ten < 1 ˆ 5.68 T > 11.4 ensures tat ˆ all oter outside modes are suppressed by at least O( 1e -7). Te second pase of te frequency extraction approac is were te fields are sampled small scale linear algebra is performed to determine te frequencies of te modes found between GHz. At tis time, te corresponding mode patterns can also be constructed during te determination of frequencies. Additional details of te tecniques are found in Ref. [10] Frequency Convergence Results Te number of grid points considered is defined by Nx = Ny =50*k were k =1,2,4,8 yielding resolutions from 18 m to 0225 m. Te factor, f DM, is cosen to be 0.1, 0.3, 0.5 wic defines te time step necessary for stability. We compare computed frequency values for te EXP-FDTD metod wit tese values for te ADI- FDTD metod. At eac f DM value, a certain number of cutcells are discarded due to teir size relative to te size of te typical cell of te domain. See Table 2 for te percentage of cells trown away te corresponding time step of various grid resolutions f DM values. Te important case is wen f DM 0 so tat no cut-cells are discarded as would be required by te EXP-FDTD metod. Using suc a small f DM for te EXP-FDTD metod would imply tat te time step would be effectively zero. In Fig. (4) we consider two cases (last two in legend) were f DM 0. One of tese uses a time step tat is equivalent to setting f DM 0.5 ( ADI-0.5) te oter uses a time step tat is 2x te CFL limit (ADI-2.0). In Table 2 we state te time steps generated by tese metods te corresponding percentage of cells trown out wit f DM 0. Te convergence of te first four modes is plotted in Fig. (4). Te last tree modes exibited similar convergence beavior. We ave also included in Fig. (5a) color contour plot of te z- component of te magnetic field for te first four modes computed wit ADI DISCUSSION 4.1. Stability As is illustrated in Fig. (4), we maintain stability can extract accurate frequencies at 2x te CFL limit. To furter illustrate stability, we ave performed simulations wit te ADI-FDTD metod from 1x to 8x te CFL limit. Fig. (6) sows te results of convergence after running te simulation at tese time steps beyond te CFL limit. All simulations are stable permit extraction of frequencies. In tis figure, we ave also included te convergence for te EXP-FDTD metod at f DM = 0.1. Tese results are te most accurate due to te igest temporal resolution. Wat is clearly observed from te results in Fig. (6) is tat te time step wit te ADI-FDTD metod can be cosen for accuracy considerations instead of stability considerations. Performing simulations wit te EXP-FDTD metod at 1x to 8x te CFL limit would lead to unstable calculations. Tus, te ADI- FDTD metod yields stable simulations well beyond te CFL limit for a geometry wit a curved domain as in Fig. (3) Accuracy As as been noted previously, te focus of tis work is not on te accuracy of te frequency extraction approac wit te ADI-FDTD approac. Te accuracy is determined by bot te discretization approac te spatial temporal resolution. Tis as been studied in Ref. [17] were te autors use te EXP-FDTD metod to run te Table 2. Time Step in ps for Eac Nx f DM Value (left) te Percentage of cut-cells Discarded from te Simulation Given te f DM Value (Rigt) Note tat for CFL2 te Time Stepping is Performed via te ADI-FDTD Metod so we can Include All Cells in te Simulation since we are Not Limited by te CFL Condition Nx f DM = 0.1 f DM = 0.3 f DM = 0.5 f DM = /% /5.7% /11% /% / % / 4.0% / 11% / % / 0.5% / 4.5% / 14% / % / 0.6% / 4.8% 6004 / 12% / %

6 34 Te Open Plasma Pysics Journal, 2010, Volume 3 Austin et al. (a) E+09 Hz (a) E+09 Hz (b) E+09 Hz (b) E+09 Hz (c) E+09 Hz (c) E+09 Hz (d) E+09 Hz (d) E+09 Hz Fig. (4). Convergence of magnetron frequencies for an explicit FDTD metod an alternating direction implicit FDTD metods wit Dey-Mittra cut-cells [8]. Te frequency extraction approac of Werner Cary in Ref. [10] is used to obtain te frequencies from te time domain simulations. Te values 0.1, 0.3, 0.5 refer to te f DM values discussed previously tat are used to determine te time step te percentage of cells kept in te simulation (see Table 2). Te final two in te legend in bold ave time steps determined by f DM but keep all of te cells in te simulation. Tis is only possible because of te implicit nature of te time stepping. Fig. (5). Color contour plots of B z for te first four modes calculated using te frequency extraction metod wit simulations performed by te ADI-FDTD metod. Results illustrate our ability to use ADI-FDTD simulations to also reconstruct spatial mode patterns.

7 Alternating Direction Implicit Metods for FDTD Te Open Plasma Pysics Journal, 2010, Volume 3 35 simulations wit f DM = 5 f DM = 0.5. In Fig. (4) we observe tat te EXP-FDTD approac te ADI-FDTD approac bot ave comparable accuracy at te same time steps. Furtermore, te example given by ADI-0.5 wit no cells discarded sows comparable accuracy to ADI-0.3 wit cells trown out according to f DM 0. Tis tells us tat we can obtain comparable accuracy wit nearly twice te time step size by using ADI-FDTD not discarding any cells. Finally, we note tat wit te ADI metod we can go beyond te CFL time step. Tis example denoted as ADI-0.5 sows tat te accuracy is poorer because of te lower resolution in temporal space. Furtermore, wit Fig. (6), we see tat reducing te time step size improves te accuracy of te computations. Fig. (6). Convergence of te ADI-FDTD metod at various temporal resolutions, from 1x te CFL limit to 8x te CFL limit. Also included on tis convergence plot is te convergence for EXP- FDTD at f DM = 0.1 te exact value, as obtained from [9], plotted as a flat line at GHz. Te ADI-FDTD metod remains stable at all time steps beyond te CFL limit only exibits lower accuracy tan at time steps below te CFL limit. 5. CONCLUSION We ave presented an implementation of te ADI-FDTD metod combined wit te Dey-Mittra embedded boundary metod. Tis approac can model te curved domains associated wit complex accelerator structures at time step sizes beyond te CFL limit. It depends on simple, onedimensional, tridiagonal solves instead of te large system solves associated wit implicit metods like te Crank- Nicolson metod. Te one-dimensional solves can be efficiently completed using te Tomas algoritm [18]. In tree-dimensions, te metod can be directly applied witout any canges. Clearly, for large tree-dimensional problems, te metod must be extended to a large scale parallel computing platform to le te larger number of unknowns. An efficient implementation tus depends on te efficient solution of a number of tridiagonal systems at eac time step. We leave tis to future work. REFERENCES [1] Lee J, Fornberg B. Some unconditionally stable time stepping metods for te 3D Maxwell's equations. J Comput Appl Mat 2004; 166(2): [2] Zeng F, Cen Z, Zang J. Toward te development of a treedimensional unconditionally stable finite-difference time-domain metod. IEEE Trans Microw Teory Tec 2000; 48(9): [3] Namiki T. 3-D ADI-FDTD metod-unconditionally stable timedomain algoritm for solving full vector Maxwell's equations. IEEE Trans Microw Teory Tec 2000; 48(10): [4] Zao AP. Two special notes on te implementation of te unconditionally stable ADI-FDTD metod. Microw Opt Tecnol Lett 2002; 33(4). [5] Zivanovic SS, Yee KS, Mei KK. 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Accurately e_ciently studying te RF structures using a conformal _nitedi_erence time-domain particle-in-cell metod; Submitted to Open Plasma Pys J Special Issue on Recent Advances in Finite Difference Time Domain Electromagnetic Simulations. [18] Tomas LH. Elliptic problems in linear di_erence equations over a network. Watson Sci Comput Lab Rept, New York: Columbia University Received: September 8, 2009 Revised: October 15, 2009 Accepted: October 28, 2009 Austin et al.; Licensee Bentam Open. Tis is an open access article licensed under te terms of te Creative Commons Attribution Non-Commercial License (ttp://creativecommons.org/licenses/bync/3.0/) wic permits unrestricted, non-commercial use, distribution reproduction in any medium, provided te work is properly cited.