A closed-form expression is derived for the numerical dispersion relation of the 2-D locally one-dimensional finite-difference time-domain (LOD-FDTD) method in lossy media. In contrast to the lossless formulation, we found that transverse-electric (TEz ) and transverse-magnetic (TMz ) waves in lossy media exhibit different numerical dispersion relations. Moreover, when the material relaxation-time constant is not well resolved by the integration time-step, the TMz case shows much worse accuracy than the TEz case. To remove this limitation, a split-field LOD-FDTD formulation for TMz waves is then considered, which exhibits the same dispersion relation as the LOD-FDTD method for TEz waves. The validity of the theoretical results is illustrated through numerical simulations.This work was supported in part by the Spanish Government (MINECO) and the European Commission (ERDF) under Research Projects TEC2014-55463-C3-2-P and TEC2014-55463-C3-3-P

Grande Sáez, Ana María

Pereda Fernández, José Antonio

IEEE Antennas and Wireless Propagation Letters

English

Producción CientíficaA closed-form expression is derived for the numerical dispersion relation of the 2-D locally one-dimensional finite-difference time-domain (LOD-FDTD) method in lossy media. In contrast to the lossless formulation, we found that transverse-electric (TEz) and transverse-magnetic (TM z ) waves in lossy media exhibit different numerical dispersion relations. Moreover, when the material relaxation-time constant is not well resolved by the integration time-step, the TM z case shows much worse accuracy than the TE z case. To remove this limitation, a split-field LOD-FDTD formulation for TM z waves is then considered, which exhibits the same dispersion relation as the LOD-FDTD method for TE z waves. The validity of the theoretical results is illustrated through numerical simulations.Ministerio de Economía, Industria y Competitividad  and the European Commission (ERDF) (Projects TEC2014-55463-C3-2-P and TEC2014-55463-C3-3-P

Pereda, José A.

Repositorio Documental de la Universidad de Valladolid

 This is a postprint version of the following published document:  José A. Pereda; Ana Grande. Numerical Dispersion Relation for the 2-D LOD-FDTD Method in Lossy Media. In: IEEE Antennas and Wireless Propagation Letters, 2017, Volume 16, pp. 2122-2125 DOI: https://doi.org/10.1109/LAWP.2017.2699692   © 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. 1Numerical Dispersion Relation for the 2DLOD­FDTD Method in Lossy MediaJosé A. Pereda and Ana GrandeAbstract—A closed­form expression is derived for the numer­ical dispersion relation of the 2D locally one­dimensional finite­difference time­domain (LOD­FDTD) method in lossy media. Incontrast to the lossless formulation, we found that transverse­electric (TE ) and transverse­magnetic (TM ) waves in lossymedia exhibit different numerical dispersion relations. Moreover,when the material relaxation­time constant is not well resolvedby the integration time­step the TM case shows much worseaccuracy than the TE case. To remove this limitation, a split­field LOD­FDTD formulation for TM waves is then consideredwhich exhibits the same dispersion relation as the LOD­FDTDmethod for TE waves. The validity of the theoretical results isillustrated through numerical simulations.Index Terms—Locally­one­dimensional finite­difference time­domain (LOD­FDTD) method, lossy media, numerical dispersion.I. INTRODUCTIONOver the past 15 years there has been a growing interestin developing efficient unconditionally­stable finite­differencetime­domain (FDTD) techniques such as the alternating­direction implicit (ADI)­ and the locally one­dimensional(LOD)­FDTD methods (see [1] and references within).Both ADI­ and LOD­FDTD techniques were early extendedto include lossy and dispersive materials [2]­[5]. As a result ofthe FD approximations, these methods suffer from numericaldispersion (phase errors) and numerical dissipation (amplitudeerrors). Both factors can be quantified by solving the numericaldispersion relation associated to the governing FD equations.This is a requisite for a mature understanding of the operationand accuracy limits of every FDTD algorithm.The numerical properties of the ADI­FDTD method in lossymedia have been extensively studied [6]­[11]. However, to thebest of our knowledge, the study of LOD­FDTD formulationsin lossy media has not been addressed yet.In this letter a closed­form expression is derived for thenumerical dispersion relation of the 2D LOD­FDTD methodin lossy media. Transverse­electric (TE ) and transverse­magnetic (TM ) waves are both considered. For the sake ofgenerality, weighted averages in time are used for discretizingthe conduction terms. In contrast to what occurs in the losslessformulation, we show that TE and TM waves in lossy mediaexhibit different numerical dispersion relations.Manuscript received March 17, 2017. This work was supported in part bythe Spanish Government (MINECO) and the European Commission (ERDF)under Research Projects TEC2014­55463­C3­2­P and TEC2014­55463­C3­3­P.J. A. Pereda is with the Dpto. de Ingeniería de Comunicaciones (DICOM),Universidad de Cantabria, Avda. Los Castros s/n, 39005 Santander, Cantabria,Spain (email: peredaj@unican.es).A. Grande is with the Dpto. de Electricidad y Electrónica, Universidad deValladolid, 47011 Valladolid, Spain (email: agrande@uva.es).For TE waves, we show that by properly selecting thevalues of the weighted­average parameters the accuracy of theformulation becomes independent of how well the materialrelaxation­time constant is resolved by the integration time­step . On the contrary, the same is not feasible for TMwaves. To remove this restriction a split­field LOD­FDTDformulation is proposed. The theoretical results are validatedby actual simulations.II. THE LOD­FDTD METHOD FOR TE WAVESMaxwell‘s curl equations for TE waves in isotropic andsource­free media with permittivity , permeability andconductivity can be expressed as(1a)(1b)(1c)Following the LOD technique [12], the Crank­Nicolsonscheme is first applied to (1). The resulting set of FDTDequations is then perturbed and factorized. Finally, after afurther splitting into two time sub­steps, we obtainSub­step 1(2a)(2b)(2c)Sub­step 2(3a)(3b)(3c)where and are central­difference operators defined asin [8], and . Weighted averages have beentaken for approximating the conduction terms in (2b) and (3a).We consider the weighted­average parameters subjected to theconstraints and2Note that the above LOD formulation involves parametersper electric field component, while parameters perelectric field component are needed in the ADI case [8]. Thereason for this is that and are not actually split inthe LOD case. Here, for comparison purposes the parametersand used in the ADI formulation, areimplicitly assumed to be zero.Following the same approach as in [8], we find that thenumerical dispersion relation in the ­domain for (2) and (3)is(4)with(5a)(5b)and(6)where is the numerical wavenumber in the ­direction. Notethat (4) has the same form as in the ADI case [8, eq. (12)],but with different expressions for andBy doing in (4), we obtain the numericaldispersion relation in the frequency domain, which reads(7)where is the numerical complex permittivity.A number of FD schemes with different numerical proper­ties can be obtained depending on the specific values of theparameters . A convenient choice isand which leads to(8)As a consequence, the same dispersion relation as in the ADIcase (synchronized scheme) is obtained. Moreover, with thischoice, the accuracy does not depend on whether resolvesadequately , as in the Crank­Nicolson method [8].III. THE LOD­FDTD METHOD FOR TM WAVESMaxwell‘s curl equations for TM waves can be expressedas(9a)(9b)(9c)According to the LOD­FDTD method, (9) is discretized asfollowsSub­step 1(10a)(10b)(10c)Sub­step 2(11a)(11b)(11c)with . Note that now the electric field is split intime but the magnetic field is not.Following the same approach as in the TE case, thedispersion relation in the ­domain for (10) and (11) reads(12)where(13)and(14a)(14b)By doing in (12), we obtain the followingfrequency­domain dispersion relation(15)It is worth noting that (15) is exactly the same as the dispersionrelation previously obtained for the ADI­FDTD method in thecase of TM waves [9, eq. (4)]. Therefore, all the discussioncarried out in [9] applies to the present case and will notbe repeated here. A common choice for the weighted­averageparameters is However, we recall that,independently of the weighted­average parameter choice, toachieve good accuracy must be well resolved by . Toovercome this limitation a split­field formulation is proposedin the next section.30 10 20 30 40 50 60 70 80 900.10.20.30.40.50.6Propagation Angle, (Degrees)Relative Error, (%)  Phase (Theoretical)Phase (Simulation)Attenuation (Theoretical)Attenuation (Simulation)Fig. 1. Relative phase and attenuation errors against the propagation angle.Results calculated by the LOD­FDTD method for TE waves with = 1 8S/m, = 40 and = 20IV. A SPLIT­FIELD LOD­FDTD METHOD FOR TM WAVESConsider the following split­field version of (9) as governingequations:(16a)(16b)(16c)(16d)with Note that the above equations coincidewith the Berenger split­field approach when in the latter themagnetic conductivities are set equal to zero [4], [13].Applying the LOD­FDTD method to (16), the following setof difference equations is obtained:Sub­step 1(17a)(17b)(17c)(17d)0 10 20 30 40 50 60 70 80 9000.10.20.30.40.50.60.7Propagation Angle, (Degrees)Relative Error, (%)  Phase (Theoretical)Phase (Simulation)Attenuation (Theoretical)Attenuation (Simulation)Fig. 2. Relative phase and attenuation errors against the propagation angle.Results calculated by the LOD­FDTD method for TE waves with = 18S/m, = 40 and = 20Sub­step 2(18a)(18b)(18c)(18d)It is worth noting that in practice (17) and (18) reduce to a onestep formulation. The numerical dispersion relation and thenumerical permittivity for this formulation are found to be thesame as the ones given in (7) and (8), respectively. Therefore,as was discussed in section II, the condition is notrequired to achieve high accuracy.V. NUMERICAL RESULTSWith the aim of illustrating the validity of the above theo­retical results, the numerical phase and attenuation constantsin a homogeneous lossy material have been calculated. Theprocedure followed has been the same as the one described in[14] and previously used in [8] and [9]. All the results havebeen calculated at the frequency GHz. The permittivityand permeability of the lossy material are the same as in thefree space. Square cells with have beenused for discretizing the computational domain. The spatialresolution was defined as , being the exactwavelength in the lossy material (at 10 GHz). In the sameway, the temporal resolution was defined as , beingthe period of the wave.Figs. 1 and 2 display the relative errors for phase andattenuation against the propagation angle for TE waves withparameters and resolutions40 10 20 30 40 50 60 70 80 90-12-10-8-6-4-202468Propagation Angle, (Degrees)Relative Error, (%)  Phase (Theoretical)Phase (Simulation)Attenuation (Theoretical)Attenuation (Simulation)Fig. 3. Relative phase and attenuation errors against the propagation angle.Results calculated by the LOD­FDTD method for TM waves with = 18S/m, = 40 and = 20.and . The conductivity in Fig. 1 wasS/m, consequently and , where isthe maximum stable time step allowed by the conventionalFDTD method. In Fig. 2 the conductivity was S/m,so in this case and . It can be seenthat all the errors remain under even though is 10times greater than in Fig. 2. The theoretical results obtainedby directly solving (7) have been plotted by lines and thosecomputed by actual field simulations have been denoted bymarkers. Excellent agreement is observed between theory andsimulation. In fact, the difference between both sets of resultsis typically within 10Fig. 3 depicts the same case as in Fig. 2 but for TM waveswith Now very high phase and attenuationerrors can be seen for angles outside the diagonal ( ).Finally, in Fig. 4 the same example as in Fig. 2 is consideredagain. The difference is that in Fig. 4 the results have beenobtained by using the proposed split­field LOD­FDTD methodfor TM waves. It can be seen that these results replicate thoseobtained in Fig. 2 for the TE case.VI. CONCLUSIONThe numerical dispersion relation for the 2D LOD­FDTDmethod in lossy media has been derived in a closed­form. ForTE waves the numerical dispersion relation is, in general,different from that of the ADI case. However, if centralaverages are used to approximate the conduction terms thedispersion relation becomes the same as in the synchronized­scheme­ADI case [8]. A salient feature of this choice isthat the condition is not required in order toachieve high accuracy. For TM waves, both LOD and ADIformulations exhibit an identical numerical dispersion relation.Unfortunately, for any choice of the average parameters, thecondition is necessary to achieve high accuracy.To overcome this limitation we have proposed a split­fieldTM formulation that exhibits the same numerical dispersionrelation as for TE waves.0 10 20 30 40 50 60 70 80 9000.10.20.30.40.50.60.7Propagation Angle, (Degrees)Relative Error, (%)  Phase (Theoretical)Phase (Simulation)Attenuation (Theoretical)Attenuation (Simulation)Fig. 4. Relative phase and attenuation errors against the propagation angle.Results calculated by the split­field LOD­FDTD method for TM waves with= 18 S/m, = 40 and = 20REFERENCES[1] J. Shibayama, M. Muraki, R. Takahashi, J. Yamauchi, and H.Nakano,“Performance evaluation of several implicit FDTD methods foroptical waveguide analyses,” J. Lightw. Technol., vol. 24, no. 6, pp.2465–2472, Jun. 2006.[2] C. Yuan and Z. Chen, “On the modeling of conducting media withthe unconditionally stable ADI­FDTD method,” IEEE Trans. Microw.Theory Tech., vol. 51, pp. 1929–1938, Aug. 2003.[3] S. G. García, R. G. Rubio, A. R. Bretones, and R. G. Martín, “Extensionof the ADI­FDTD method to Debye media,” IEEE Trans. AntennasPropag., vol. 51, pp. 3183–3186, Nov. 2003.[4] V. E. do Nascimento, B.­H. V. Borges, and F. L. Texeira, “Split­field PML implementation for the unconditionally stable LOD­FDTDmethod,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 7, pp. 398–400, Jul. 2006.[5] J. Shibayama, R. Takahashi, J. Yamauchi, and H. Nakano, “Frequencydependent LOD­FDTD implementations for dispersive media,” Electron.Lett., vol. 42, no. 19, pp. 1084–1085, Sep. 2006.[6] S. Wang and J.H. Duyn, “A split­field iterative ADI method for simu­lating transverse­magnetic waves in lossy media,” IEEE Trans. Microw.Theory Tech., vol. 54, pp. 2169­2202, May 2006[7] W. Fu and E. L. Tan, “Stability and dispersion analysis for ADI­FDTDmethod in lossy media,” IEEE Trans. Antennas Propag., vol. 55, pp.1095­1102, Apr. 2007.[8] J. A. Pereda, A. Grande, O. González, and A. Vegas, “Analysis oftwo alternative ADI­FDTD formulations for transverse­electric wavesin lossy materials,” IEEE Trans. Antennas Propag., vol. 57, pp. 2047–2054, Jul. 2009.[9] J. A. Pereda, A. Grande, O. González, and A. Vegas, “The ADI­FDTDmethod for transverse­magnetic waves in conductive materials,” IEEETrans. Antennas Propag., vol. 58, pp. 2790­2793, Aug. 2010.[10] D.Y. Heh and E.L. Tan, “Lyapunov and matrix norm stability analysisof ADI­FDTD schemes for doubly lossy media,” IEEE Trans. AntennasPropag., vol. 59, pp. 979­986, Mar. 2011.[11] T. Tan and Q. H. Liu, “Unconditionally stable ADI/Crank­Nicolsonimplementation and lossy split error revisited,” IEEE Trans. AntennasPropag., vol. 61, pp. 5627­5636, Nov. 2013.[12] J. Shibayama, M. Muraki, J. Yamauchi, and H. Nakano, “Efficientimplicit FDTD algorithm based on locally one­dimensional scheme,”Electron. Lett., vol. 41, no. 19, pp. 1046–1047, Sep. 2005.[13] J. P. Berenger, “A perfectly matched layer for the absorption of elec­tromagnetic waves,” J. Comput. Physics, vol. 114, pp. 185­200, Oct.1994.[14] S. Ju, H. Kim, and H.­H. Kim, “A study of the numerical dispersionrelation for the 2­D ADI­FDTD method,” IEEE Microwave WirelessComp. Lett., vol. 13, pp. 405­407, Sept. 2003.

Numerical Dispersion Relation for the 2-D LOD-FDTD Method in Lossy Media

UCrea

1Numerical Dispersion Relation for the 2DLOD­FDTD Method in Lossy MediaJosé A. Pereda and Ana GrandeAbstract—A closed­form expression is derived for the numer­ical dispersion relation of the 2D locally one­dimensional finite­difference time­domain (LOD­FDTD) method in lossy media. Incontrast to the lossless formulation, we found that transverse­electric (TE ) and transverse­magnetic (TM ) waves in lossymedia exhibit different numerical dispersion relations. Moreover,when the material relaxation­time constant is not well resolvedby the integration time­step the TM case shows much worseaccuracy than the TE case. To remove this limitation, a split­field LOD­FDTD formulation for TM waves is then consideredwhich exhibits the same dispersion relation as the LOD­FDTDmethod for TE waves. The validity of the theoretical results isillustrated through numerical simulations.Index Terms—Locally­one­dimensional finite­difference time­domain (LOD­FDTD) method, lossy media, numerical dispersion.I. INTRODUCTIONOver the past 15 years there has been a growing interestin developing efficient unconditionally­stable finite­differencetime­domain (FDTD) techniques such as the alternating­direction implicit (ADI)­ and the locally one­dimensional(LOD)­FDTD methods (see [1] and references within).Both ADI­ and LOD­FDTD techniques were early extendedto include lossy and dispersive materials [2]­[5]. As a result ofthe FD approximations, these methods suffer from numericaldispersion (phase errors) and numerical dissipation (amplitudeerrors). Both factors can be quantified by solving the numericaldispersion relation associated to the governing FD equations.This is a requisite for a mature understanding of the operationand accuracy limits of every FDTD algorithm.The numerical properties of the ADI­FDTD method in lossymedia have been extensively studied [6]­[11]. However, to thebest of our knowledge, the study of LOD­FDTD formulationsin lossy media has not been addressed yet.In this letter a closed­form expression is derived for thenumerical dispersion relation of the 2D LOD­FDTD methodin lossy media. Transverse­electric (TE ) and transverse­magnetic (TM ) waves are both considered. For the sake ofgenerality, weighted averages in time are used for discretizingthe conduction terms. In contrast to what occurs in the losslessformulation, we show that TE and TM waves in lossy mediaexhibit different numerical dispersion relations.Manuscript received March 17, 2017. This work was supported in part bythe Spanish Government (MINECO) and the European Commission (ERDF)under Research Projects TEC2014­55463­C3­2­P and TEC2014­55463­C3­3­P.J. A. Pereda is with the Dpto. de Ingeniería de Comunicaciones (DICOM),Universidad de Cantabria, Avda. Los Castros s/n, 39005 Santander, Cantabria,Spain (email: peredaj@unican.es).A. Grande is with the Dpto. de Electricidad y Electrónica, Universidad deValladolid, 47011 Valladolid, Spain (email: agrande@uva.es).For TE waves, we show that by properly selecting thevalues of the weighted­average parameters the accuracy of theformulation becomes independent of how well the materialrelaxation­time constant is resolved by the integration time­step . On the contrary, the same is not feasible for TMwaves. To remove this restriction a split­field LOD­FDTDformulation is proposed. The theoretical results are validatedby actual simulations.II. THE LOD­FDTD METHOD FOR TE WAVESMaxwell‘s curl equations for TE waves in isotropic andsource­free media with permittivity , permeability andconductivity can be expressed as(1a)(1b)(1c)Following the LOD technique [12], the Crank­Nicolsonscheme is first applied to (1). The resulting set of FDTDequations is then perturbed and factorized. Finally, after afurther splitting into two time sub­steps, we obtainSub­step 1(2a)(2b)(2c)Sub­step 2(3a)(3b)(3c)where and are central­difference operators defined asin [8], and . Weighted averages have beentaken for approximating the conduction terms in (2b) and (3a).We consider the weighted­average parameters subjected to theconstraints and2Note that the above LOD formulation involves parametersper electric field component, while parameters perelectric field component are needed in the ADI case [8]. Thereason for this is that and are not actually split inthe LOD case. Here, for comparison purposes the parametersand used in the ADI formulation, areimplicitly assumed to be zero.Following the same approach as in [8], we find that thenumerical dispersion relation in the ­domain for (2) and (3)is(4)with(5a)(5b)and(6)where is the numerical wavenumber in the ­direction. Notethat (4) has the same form as in the ADI case [8, eq. (12)],but with different expressions for andBy doing in (4), we obtain the numericaldispersion relation in the frequency domain, which reads(7)where is the numerical complex permittivity.A number of FD schemes with different numerical proper­ties can be obtained depending on the specific values of theparameters . A convenient choice isand which leads to(8)As a consequence, the same dispersion relation as in the ADIcase (synchronized scheme) is obtained. Moreover, with thischoice, the accuracy does not depend on whether resolvesadequately , as in the Crank­Nicolson method [8].III. THE LOD­FDTD METHOD FOR TM WAVESMaxwell‘s curl equations for TM waves can be expressedas(9a)(9b)(9c)According to the LOD­FDTD method, (9) is discretized asfollowsSub­step 1(10a)(10b)(10c)Sub­step 2(11a)(11b)(11c)with . Note that now the electric field is split intime but the magnetic field is not.Following the same approach as in the TE case, thedispersion relation in the ­domain for (10) and (11) reads(12)where(13)and(14a)(14b)By doing in (12), we obtain the followingfrequency­domain dispersion relation(15)It is worth noting that (15) is exactly the same as the dispersionrelation previously obtained for the ADI­FDTD method in thecase of TM waves [9, eq. (4)]. Therefore, all the discussioncarried out in [9] applies to the present case and will notbe repeated here. A common choice for the weighted­averageparameters is However, we recall that,independently of the weighted­average parameter choice, toachieve good accuracy must be well resolved by . Toovercome this limitation a split­field formulation is proposedin the next section.30 10 20 30 40 50 60 70 80 900.10.20.30.40.50.6Propagation Angle, (Degrees)Relative Error, (%)  Phase (Theoretical)Phase (Simulation)Attenuation (Theoretical)Attenuation (Simulation)Fig. 1. Relative phase and attenuation errors against the propagation angle.Results calculated by the LOD­FDTD method for TE waves with = 1 8S/m, = 40 and = 20IV. A SPLIT­FIELD LOD­FDTD METHOD FOR TM WAVESConsider the following split­field version of (9) as governingequations:(16a)(16b)(16c)(16d)with Note that the above equations coincidewith the Berenger split­field approach when in the latter themagnetic conductivities are set equal to zero [4], [13].Applying the LOD­FDTD method to (16), the following setof difference equations is obtained:Sub­step 1(17a)(17b)(17c)(17d)0 10 20 30 40 50 60 70 80 9000.10.20.30.40.50.60.7Propagation Angle, (Degrees)Relative Error, (%)  Phase (Theoretical)Phase (Simulation)Attenuation (Theoretical)Attenuation (Simulation)Fig. 2. Relative phase and attenuation errors against the propagation angle.Results calculated by the LOD­FDTD method for TE waves with = 18S/m, = 40 and = 20Sub­step 2(18a)(18b)(18c)(18d)It is worth noting that in practice (17) and (18) reduce to a onestep formulation. The numerical dispersion relation and thenumerical permittivity for this formulation are found to be thesame as the ones given in (7) and (8), respectively. Therefore,as was discussed in section II, the condition is notrequired to achieve high accuracy.V. NUMERICAL RESULTSWith the aim of illustrating the validity of the above theo­retical results, the numerical phase and attenuation constantsin a homogeneous lossy material have been calculated. Theprocedure followed has been the same as the one described in[14] and previously used in [8] and [9]. All the results havebeen calculated at the frequency GHz. The permittivityand permeability of the lossy material are the same as in thefree space. Square cells with have beenused for discretizing the computational domain. The spatialresolution was defined as , being the exactwavelength in the lossy material (at 10 GHz). In the sameway, the temporal resolution was defined as , beingthe period of the wave.Figs. 1 and 2 display the relative errors for phase andattenuation against the propagation angle for TE waves withparameters and resolutions40 10 20 30 40 50 60 70 80 90-12-10-8-6-4-202468Propagation Angle, (Degrees)Relative Error, (%)  Phase (Theoretical)Phase (Simulation)Attenuation (Theoretical)Attenuation (Simulation)Fig. 3. Relative phase and attenuation errors against the propagation angle.Results calculated by the LOD­FDTD method for TM waves with = 18S/m, = 40 and = 20.and . The conductivity in Fig. 1 wasS/m, consequently and , where isthe maximum stable time step allowed by the conventionalFDTD method. In Fig. 2 the conductivity was S/m,so in this case and . It can be seenthat all the errors remain under even though is 10times greater than in Fig. 2. The theoretical results obtainedby directly solving (7) have been plotted by lines and thosecomputed by actual field simulations have been denoted bymarkers. Excellent agreement is observed between theory andsimulation. In fact, the difference between both sets of resultsis typically within 10Fig. 3 depicts the same case as in Fig. 2 but for TM waveswith Now very high phase and attenuationerrors can be seen for angles outside the diagonal ( ).Finally, in Fig. 4 the same example as in Fig. 2 is consideredagain. The difference is that in Fig. 4 the results have beenobtained by using the proposed split­field LOD­FDTD methodfor TM waves. It can be seen that these results replicate thoseobtained in Fig. 2 for the TE case.VI. CONCLUSIONThe numerical dispersion relation for the 2D LOD­FDTDmethod in lossy media has been derived in a closed­form. ForTE waves the numerical dispersion relation is, in general,different from that of the ADI case. However, if centralaverages are used to approximate the conduction terms thedispersion relation becomes the same as in the synchronized­scheme­ADI case [8]. A salient feature of this choice isthat the condition is not required in order toachieve high accuracy. For TM waves, both LOD and ADIformulations exhibit an identical numerical dispersion relation.Unfortunately, for any choice of the average parameters, thecondition is necessary to achieve high accuracy.To overcome this limitation we have proposed a split­fieldTM formulation that exhibits the same numerical dispersionrelation as for TE waves.0 10 20 30 40 50 60 70 80 9000.10.20.30.40.50.60.7Propagation Angle, (Degrees)Relative Error, (%)  Phase (Theoretical)Phase (Simulation)Attenuation (Theoretical)Attenuation (Simulation)Fig. 4. Relative phase and attenuation errors against the propagation angle.Results calculated by the split­field LOD­FDTD method for TM waves with= 18 S/m, = 40 and = 20REFERENCES[1] J. Shibayama, M. Muraki, R. Takahashi, J. Yamauchi, and H.Nakano,“Performance evaluation of several implicit FDTD methods foroptical waveguide analyses,” J. Lightw. Technol., vol. 24, no. 6, pp.2465–2472, Jun. 2006.[2] C. Yuan and Z. Chen, “On the modeling of conducting media withthe unconditionally stable ADI­FDTD method,” IEEE Trans. Microw.Theory Tech., vol. 51, pp. 1929–1938, Aug. 2003.[3] S. G. García, R. G. Rubio, A. R. Bretones, and R. G. Martín, “Extensionof the ADI­FDTD method to Debye media,” IEEE Trans. AntennasPropag., vol. 51, pp. 3183–3186, Nov. 2003.[4] V. E. do Nascimento, B.­H. V. Borges, and F. L. Texeira, “Split­field PML implementation for the unconditionally stable LOD­FDTDmethod,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 7, pp. 398–400, Jul. 2006.[5] J. Shibayama, R. Takahashi, J. Yamauchi, and H. Nakano, “Frequencydependent LOD­FDTD implementations for dispersive media,” Electron.Lett., vol. 42, no. 19, pp. 1084–1085, Sep. 2006.[6] S. Wang and J.H. Duyn, “A split­field iterative ADI method for simu­lating transverse­magnetic waves in lossy media,” IEEE Trans. Microw.Theory Tech., vol. 54, pp. 2169­2202, May 2006[7] W. Fu and E. L. Tan, “Stability and dispersion analysis for ADI­FDTDmethod in lossy media,” IEEE Trans. Antennas Propag., vol. 55, pp.1095­1102, Apr. 2007.[8] J. A. Pereda, A. Grande, O. González, and A. Vegas, “Analysis oftwo alternative ADI­FDTD formulations for transverse­electric wavesin lossy materials,” IEEE Trans. Antennas Propag., vol. 57, pp. 2047–2054, Jul. 2009.[9] J. A. Pereda, A. Grande, O. González, and A. Vegas, “The ADI­FDTDmethod for transverse­magnetic waves in conductive materials,” IEEETrans. Antennas Propag., vol. 58, pp. 2790­2793, Aug. 2010.[10] D.Y. Heh and E.L. Tan, “Lyapunov and matrix norm stability analysisof ADI­FDTD schemes for doubly lossy media,” IEEE Trans. AntennasPropag., vol. 59, pp. 979­986, Mar. 2011.[11] T. Tan and Q. H. Liu, “Unconditionally stable ADI/Crank­Nicolsonimplementation and lossy split error revisited,” IEEE Trans. AntennasPropag., vol. 61, pp. 5627­5636, Nov. 2013.[12] J. Shibayama, M. Muraki, J. Yamauchi, and H. Nakano, “Efficientimplicit FDTD algorithm based on locally one­dimensional scheme,”Electron. Lett., vol. 41, no. 19, pp. 1046–1047, Sep. 2005.[13] J. P. Berenger, “A perfectly matched layer for the absorption of elec­tromagnetic waves,” J. Comput. Physics, vol. 114, pp. 185­200, Oct.1994.[14] S. Ju, H. Kim, and H.­H. Kim, “A study of the numerical dispersionrelation for the 2­D ADI­FDTD method,” IEEE Microwave WirelessComp. Lett., vol. 13, pp. 405­407, Sept. 2003.

Numerical dispersion relation for the 2D LOD-FDTD method in lossy media

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