Two-Potential Formalism for Numerical Solution of the Maxwell Equations

Transcription

1 Two-Potntial Foralis for Nurical Solution of th Maxwll Equations S. I. Trashkv,* A. N. Kudryavtsv** *Institut of Lasr Physics, Sibrian Branch, Russian Acady of Scincs (Novosibirsk) **Khristianovich Institut of Thortical and Applid Mchanics, Sibrian Branch, Russian Acady of Scincs (Novosibirsk) -ails: Abstract A nw forulation of th Maxwll quations basd on two vctor and two scalar potntials is proposd. Th us of ths potntials allows th lctroagntic fild quations to b writtn in th for of a hyprbolic syst. In contrast to th original Maxwll quations, this syst contains only volutionary quations and dos not includ quations having th charactr of diffrntial constraints. This fact aks th nw quations spcially convnint for nurical siulations of lctroagntic procsss; in particular, thy can b solvd by odrn powrful shock-capturing thods basd on approxiation of spatial drivativs by upwind diffrncs. Th lctroagntic fild both in vacuu and in an inhoognous atrial diu is considrd. Exapls of odling th propagation of lctroagntic wavs by ans of solving th forulatd syst of quations with th us of odrn high-ordr schs ar givn. Ky words: coputational lctrodynaics, two-potntial foralis, nurical solution of hyprbolic systs of quations, shock-capturing schs. 1. INTRODUCTION Th Maxwll quations for th basis of th classical lctrodynaics, which dscribs an xtrly wid rang of physical phnona obsrvd in natur and usd in various nginring dvics. Coputr odling of propagation of lctroagntic wavs and thir intraction with attr, basd on th nurical solution of th Maxwll quations, has bco on of th ost iportant tools of rsarch in any filds of scinc. It is sufficint to tll that thr ditions of th book of A. Taflov and S.G. Hagnss [1] daling with only on (though th ost popular on) nurical thod of solving ths quations wr publishd during th last dcad, and th last dition has or than a thousand pags. Particularly rapid dvlopnt of coputational lctrodynaics has bn rcntly obsrvd owing to its applications in vigorously progrssing iportant filds such as nonlinar optics, lasr physics, nanophotonics, and cration of nw atrials (photon crystals, soft attr, including liquid crystals and various sart dia). During any yars that passd sinc J.C. Maxwll drivd quations that ar known now undr his na, vrsatil forulations wr proposd: fro th original for in th coponnts of th lctric and agntic filds [2] to odrn approachs involving th diffrntial fors, th thory of spinors, and th Clifford algbra [3]. Bing quivalnt fro th physical viwpoint, diffrnt forulations ar oftn inquivalnt fro th viwpoint of convninc of thir us in practic, in particular, in nurical siulations of lctroagntic procsss. Th choic of an appropriat forulation can apprciably siplify th algorith usd for nurical odling and significantly incras th accuracy of th rsults obtaind. Thrfor, th sarch for th bst forulation will b undoubtdly continud in th futur. In this papr, w propos a nw forulation of lctroagntis quations, which involvs two vctor potntials and two scalar potntials. In our opinion, this forulation is lgant, posssss so dsirabl athatical proprtis, and offrs 1

2 iportant advantags for constructing high-accuracy nurical algoriths on its basis. In particular, this forulation allows wll-dvlopd (ainly, in coputational fluid dynaics) thods of solving hyprbolic systs of quations to b usd to th bst possibl xtnt in solving lctrodynaic probls. Th outlin of this papr is as follows. Th two-potntial foralis for lctroagntis quations in vacuu is dscribd in Sction 2. In Sction 3, it is gnralizd to th cas of an isotropic atrial diu. Th athatical proprtis of th drivd quations ar considrd in Sction 4. Exapls of using th nw approach for nurical siulation of lctrodynaic procsss ar furthr givn in Sction 5. Finally, so conclusions ar drawn in Sction TWO-POTENTIAL FORMALISM FOR THE MAXWELL EQUATIONS IN VACUUM Th Maxwll quations in an pty spac can b writtn as follows [4]: E B = J, (2.1) B + E = J, (2.2) E = ρ, (2.3) Β = ρ. (2.4) Hr, in addition to th lctric chargs ρ and currnts J, w also rtain th hypothtical agntic chargs ρ and agntic currnts J. Though agntic chargs and currnts hav nvr bn obsrvd in xprints, thy aris alost in all odrn fundantal thoris of th attr [5]; thrfor, it is rathr probabl that thy do xist in natur. In writing Eqs. (2.1)-(2.4), th asurnt units ar chosn in such a annr that th spd of light, as wll as th vacuu prittivity and prability ar qual to unity. It follows fro Eq. (2.4) that th agntic fild in th absnc of agntic chargs is solnoidal, and it can b xprssd via th agntic vctor potntial as B = A. (2.5) Substituting Eq. (2.5) into Eq. (2.2), w find that th curl of th vctor A / t + E vanishs in th absnc of agntic currnts; thrfor, this vctor can b xprssd via th gradint of th scalar lctric potntial ϕ : ϕ Α E =. (2.6) t Ths ar usual xprssions for filds in trs of th vctor and scalar potntials. On th othr hand, if thr ar no lctric chargs and currnts, th filds can b xprssd in a siilar annr via anothr pair of potntials, i.., lctric vctor potntial C and agntic scalar potntial ψ [6]: E = C, (2.7) ψ C B =. t (2.8) Lt us now assu that w hav nithr agntic nor lctric chargs and currnts. Thn th filds can b writtn both in th for of Eqs. (2.5), (2.6) and in th for of Eqs. (2.7), (2.8). Equating diffrnt xprssions for E and B, w obtain th syst of quations 2

3 A C + ϕ = 0, (2.9) C + A + ψ = 0. (2.10) To clos this syst, w hav to add gaug conditions for th potntials to Eqs. (2.9), (2.10). In particular, choosing th Lorntz gaug, w obtain ϕ + A = 0, (2.11) ψ + C = 0. (2.12) Equations (2.9) (2.12) giv th nw forulation of quations of lctrodynaics. In contrast to th standard forulation, which involvs th vctor agntic potntial and th scalar lctric potntial and yilds wav quations for th, th rsultant syst includs only first-ordr quations; thrfor, initial data for th drivativs of ths potntials with rspct to ti ar not ndd in forulating th initial-boundary valu probl for this syst. Morovr, all quations includd into syst (2.9) (2.12) ar volutionary quations, and th syst has no quations of th for of Eqs. (2.3) (2.4), which ar diffrntial constraints and ak th nurical solution of th original syst of th Maxwll quations substantially or difficult. Owing to thir construction, th drivd quations ar valid only if thr ar no sourcs (chargs and currnts) for th fild. Thy allow us to find th fr lctroagntic fild, which is th solution of th hoognous Maxwll quations with zro right-hand sids. In th gnral cas, whr both lctric and agntic chargs and currnts ar prsnt, in accordanc with th suprposition principl, th filds ar givn by th sus of Eqs. (2.6), (2.7) and (2.5), (2.8) [6]: ϕ Α E = C, ψ C B = A. t Nvrthlss, w can forally rtain th abov-proposd construction in th prsnc of sourcs as wll. For this purpos, w dtrin th vctor of vacuu polarization undr th action of lctric chargs P, such that P = ρ. (2.13) Thn th vctor fild E + P is solnoidal, and w can writ E = C P. Lt us now introduc th vctor of vacuu agntization inducd by lctric currnts, accordanc with th forula P = t M, in M J. (2.14) 3

4 Th fact that th right-hand sid of Eq. (2.14) is rally a solnoidal vctor fild follows fro th law of consrvation of th lctric charg. Thn Eq. (2.1) yilds Siilarly, introducing auxiliary vctors ψ C B = + M. M and P, such that P = ρ, (2.15) P = t M J, (2.16) w obtain xprssions for th lctric and agntic filds via th usual potntials: ϕ A E = + M B = A P., Equating, as prviously, diffrnt xprssions for E and B, w obtain th following rlations instad of Eqs. (2.9), (2.10): A C + ϕ = P + M, (2.17) C + A + ψ = P + M. (2.18) It is obvious, howvr, that, to dtrin th vctors P, P, M, and M for spcifid chargs and currnts, w hav first to solv syst (2.13) (2.16), which actually coincids with th original Maxwll quations. Thus, th gnralization of th nw approach to th cas with non-zro chargs and currnts is ssntially foral. Nvrthlss, Eqs. (2.17), (2.18) cannot b considrd as copltly uslss. Th point is that lctroagntic fild sourcs ar oftn prscribd in th for of known polarizations and agntizations. This ans that a particular solution of inhoognous Maxwll quations is known, which dos not ncssarily satisfy, howvr, th iposd initial and boundary conditions. In such a cas, th solution of th initial-boundary valu probl of intrst for us is a suprposition of this particular solution and a crtain solution of hoognous quations. This sought solution can b found fro syst (2.17), (2.18). W can asily s that th abov-drivd syst of first-ordr quations for th potntials is invariant with rspct to th Lorntz transforations. To writ this syst in an xplicitly rlativistic invariant for, w introduc th standard four-dinsional notations (,,, ) (, ) µ = = x, ν x = η x = ( t, x ) x x x x x t µ µν µ = =, µ x, µ x t, = =. As usually, th covariant and contravariant coponnts of th 4-vctors ar rlatd by ans of th tric tnsor of th Minkowski spac µ 4

5 ηµν W also introduc th 4-potntials µν = η = diag(1, 1, 1, 1). A µ = ( ϕ, A ), C µ ( ψ, ) = C, th antisytric tnsor of th lctroagntic fild F µν and th dual tnsor F µν : F µν 0 Ex Ey Ez Ex 0 Bz B y =, E y Bz 0 Bx Ez By Bx 0 0 Bx By Bz 1 Bx 0 Ez E y F µν µνστ = ε Fστ =. 2 By Ez 0 Ex Bz E y Ex 0 µνστ Hr ε is an absolutly antisytric tnsor of th fourth rank (Lvi-Civita sybol) that 0123 changs its sign in th cas of prutation of two arbitrary subscripts and such that ε = + 1. Th Maxwll quations can b writtn as a pair of quations whr th lctric and agntic 4-currnts ar µν ν µν ν µ F = J, F µ = J, = ( ρ, J ), J ν = ( ρ, J ). J ν 5 W introduc two antisytric tnsors of th lctric M µν and agntic M µν polarizations of vacuu, which ar coposd of th vctors P, M and P, M, rspctivly, in th sa annr as th lctroagntic fild tnsor is coposd fro th vctors E and B. Thn w obtain J ν µν = M, J = M. ν µν µ µν µν Divrgncs of th tnsor filds F M and F µν M µν ar qual to zro; thrfor, th filds can b xprssd via th 4-vctors of two potntials, which yilds th forulas µν µ ν ν µ µν F = A A + M, F µν µ ν ν µ µν = C C + M. Using th duality of th tnsors F µν and F µν, on can writ th rlations 1 µ C ν ν C µ + N µν = ε µνστ ( σ Aτ τ Aσ + M, στ ), (2.19) 2 µ ν ν µ µν 1 µνστ A A + N = ε ( σcτ τcσ + M, στ ). (2.20) 2 Rlations (2.19) and (2.20) ar quivalnt, both bing four-dinsional fors of Eqs. (2.17) and (2.18). Ths rlations can b supplntd by quations that xprss th Lorntz gaug condition for th potntials: A µ µ C µ µ = 0, = 0. µ

6 3. TWO-POTENTIAL FORMALISM FOR THE ELECTROMAGNETIC FIELD IN THE MEDIUM It is known that th Maxwll quations ( ) rains valid in a continuous diu as wll if th chargs and currnts in th right-hand sids of th quations ar undrstood as sus of fr and bound chargs and currnts. It is or convnint, howvr, to rtain only th dnsitis of fr chargs and currnts in th right-hand sids. For this purpos, th bound chargs and currnts ar xprssd via th diu polarization and agntization vctors P, P, M, M, as it was don in Sction 2 for fr chargs and currnts in vacuu. Introducing auxiliary vctors D = E + P, E = E M H = B M, B = B + P. w writ th Maxwll quations in th diu as D H = J, (3.1) B + E = J, (3.2) D = ρ, (3.3) Β = ρ, (3.4) whr th dnsitis of chargs and currnts in th right-hand sids ar undrstood as th dnsitis of fr chargs and currnts only. In what follows, th bar abov th vctors E and B is oittd bcaus thir diffrnc fro E and B is substantial only in hypothtical dia containing agntic onopols. In contrast to th Maxwll quations in vacuu, introduction of polarization and agntization vctors and transfr of th bound sourc filds fro th right-hand sid to th lft-hand sid of th quations ar not purly foral oprations. Th point is that th vctors D and H in ost of various dia can b rlatd to th vctors E and B by fairly sipl xprssions. In this papr, w confin ourslvs to considring linar isotropic dia without disprsion, which ar not ncssarily hoognous. For such dia, th rlations tak a particularly sipl for D = εe, B = µ H, (3.5) though th approach considrd hr can b rathr asily xtndd to or coplicatd (in particular, anisotropic and nonlinar) dia. In Eqs. (3.5), th rlativ prittivity ε and th th rlativ prability µ of th atrial ar known functions of spatial coordinats; in this for, ths rlations dscrib any dia of practical iportanc, in particular, photon crystals. Th vctor and scalar lctric potntials ar introducd with th hlp of th sa dfinitions as thos usd in Sction 2; th agntic potntials ar dscribd by th forulas D = C P, ψ C H = + M Substituting th xprssions of th fild via th lctric and agntic potntials into th atrial rlations (3.5), w obtain th syst 6.

7 A 1 1 ε C + ϕ = ε P + M, (3.6) C µ A + ψ = µ P + M. (3.7) Thus, Eqs. (3.6) and (3.7) can b considrd as xtnsion of Eqs. (2.17) and (2.18) to th cas of lctroagntic procsss in a atrial diu. Th polarization and agntization vctors in th right-hand sids of ths quations ar rlatd only to fr chargs and currnts. To clos th syst of quations, w hav to supplnt Eqs. (3.6) and (3.7) with gaug conditions for th potntials. Thy ar usually chosn in such a annr that th rsultant scondordr quations for th potntials ar as sipl as possibl. Th natural choic for a hoognous diu is [6] ϕ εµ + A = 0, (3.8) ψ εµ + C = 0, (3.9) which yilds sipl wav quations for th potntials. In th gnral cas of a diu with ε and µ dpndnt on spatial variabls, th choic of th bst gaug is not obvious. This probl was discussd in [7-9]. Th ost consistnt approach to obtaining th gaug conditions in a atrial diu ss to b writing th atrial rlations (3.5) in a rlativistically invariant for (for this purpos, it is ncssary to introduc a fourth rank tnsor dpndnt on th prittivity and th prability and 4-vlocity of th diu in ordr to dscrib th diu proprtis) and driving an quation for th 4-potntial in a oving diu. Th corrsponding calculations can b found in [8]. Th rsultant quation is apprciably siplifid if th gaug condition is chosn in th for ( εµ ) µ µ ν A + 1 U U A = 0, µ ν µ 1 u U µ =, u 1- u Hr U µ is th 4-vlocity vctor and u is th thr-dinsional vlocity of th diu. For a quiscnt diu, this condition rducs to Eq. (3.8). Basd on this fact, w tak Eqs. (3.8) and (3.9) as gaug conditions for th potntials. Togthr with Eqs. (3.6) and (3.7), thy for a closd syst of quations for th lctroagntic potntials in th diu. 4. SOME MATHEMATICAL PROPERTIES OF THE DERIVED EQUATIONS Introducing th quasi-vctor U = ( Dx, Dy, Dz, Bx, By, Bz ) and using th linar atrial rlations (3.5), th Maxwll quations (3.1), (3.2) without th righthand sids can b rwrittn in th atrix for as T U + A U x + Α U y + A U z = 0. (4.1) x y z 7

8 Th ost iportant proprty of quations involving th curls of th lctric and agntic filds is th fact that thy for a hyprbolic syst. By dfinition, syst (4.1) is hyprbolic [10] if, for an arbitrary unit vctor n = ( n, n, n ), th ignvalus of th atrix x y z nz / µ ny / µ nz / µ 0 nx / µ ny / µ nx / µ 0 A = nx A x + ny A y + nz A z = 0 nz / ε ny / ε nz / ε 0 nx / ε ny / ε nx / ε ar ral and it can b brought to a diagonal for by applying a siilarity transforation (i.., thr xists a coplt syst of ignvctors). Th ignvalus of A ar [11] λ 1,2 = c, λ 3,4 = 0, λ 5,6 = c, whr c = 1 / εµ. As a rul, it is th syst (3.1), (3.2) that is intgratd with rspct to ti in solving th Maxwll quations nurically, whil Eqs. (3.3) and (3.4) ar not considrd at all. Th lattr quations ar diffrntial constraints, actually rstricting th class of adissibl initial data. If th lctric and agntic filds ar solnoidal at th initial ti ont, thn ths filds rain solnoidal at all subsqunt ti ont in th cas of th xact solution of th diffrntial quations (3.1) and (3.2). In th nurical solution, howvr, this proprty can b violatd, and non-physical lctric and agntic chargs can appar insid th coputational doain. This is on of th ost svr probls in nurical siulation of lctroagntic procsss. Various tchniqus wr proposd to ovrco this probl; in particular, thy includ th corrction of th filds by solving Poisson s quation at ach ti stp [12], th us of odifid (so-calld prfctly hyprbolic) Maxwll quations [13], th us of xtndd ovrdtrind hyprbolic systs for corrcting th solution [14], tc. Th abov-ntiond probl is dirctly rlatd to th athatical proprtis of syst (3.1), (3.2), naly, to th xistnc of a zro ignvalu (of ultiplicity qual to 2). Indd, th ods corrsponding to this ignvalu ar non-physical: lctroagntic wavs always propagat in natur with th spd of light, which corrsponds to th ignvalus ± c. It can b asily donstratd that th subspac corrsponding to th zro ignvalu consists of vctor filds with th curl qual to zro and, thus, with th divrgnc that diffrs fro zro. Lt us now considr Eqs. (3.6)-(3.9), which dscrib th lctroagntic filds in th nw twopotntial foralis. Thy ar writtn in th atrix for as U + A U x + Α U y + A U z = 0, (4.2) x y z U. whr = ( Ax, Ay, Az, ψ, Cx, Cy, Cz, ϕ ) In this cas, th atrix A = nx A x + ny A y + nz A z is writtn as T 8

9 nz / ε ny / ε nx nz / ε 0 nx / ε n y ny / ε nx / ε 0 n z nx / εµ ny / εµ nz / εµ 0 A =. 0 nz / µ nz / µ nx nz / µ 0 nx / µ ny ny / µ nx / µ 0 n z nx / εµ ny / εµ nz / εµ Aftr calculations, w find that th atrix A = nx A x + ny A y + nz A z has two ignvalus, ach having ultiplicity qual to 4: λ 1,2,3,4 = c, λ 5,6,7,8 = c. Th lft-hand sid of Eqs. (4.2) can b transford to th following charactristic for (that dtrins th choic of th lft ignvctors, which is not uniqu, gnrally spaking): λ = c : λ = + c : ψ C 1 A ψ C 1 A n c + n c c 0, t t µ t n + n = ξ ξ µ ξ ϕ A ϕ A cn c c = 0, n ξ ξ ϕ A 1 C ϕ A 1 C n + c n + c c 0, t t ε t n + n = ξ ξ ε ξ ψ C ψ C + cn + c + c = 0, n ξ ξ Hr ξ is th coordinat along th dirction dtrind by th vctor n. Thus, in contrast to syst (4.1), th quations writtn with th us of th two-potntial foralis do not involv th zro ignvalu; all ods propagat with th spd of light, as it should b. Th Lorntz gaug conditions (3.8) and (3.9) for th potntials play hr an iportant rol. Indd, th quations for th potntials can b apprciably siplifid if th Coulob gaug is chosn: ϕ = ψ = 0, A = 0, C = 0. In this cas, howvr, th raining six quations for two vctor potntials hav xactly th sa structur as th Maxwll quations with th curls (3.1) and (3.2) with all abov-ntiond probls of aintnanc of th filds A, C solnoidal and non-physical ods propagating with zro vlocity. 5. EXAMPLES OF USING THE FORMALISM OF TWO POTENTIALS IN NUMERICAL SIMULATIONS Th nurical solution of Eqs. (3.6)-(3.9) can b prford with any thod aong th vast pool of tchniqus that hav bn dvlopd for hyprbolic systs of quations [15]. In this work, th spatial approxiation is prford by on of th odrn shock-capturing thods: th so-calld 9

10 fifth-ordr Wightd Essntially Non-Oscillatory (WENO) sch [16]. This sch, which can b considrd as a rot dscndant of th faous Godunov sch [17], has bn wll approvd in solving probls of suprsonic arodynaics (s,.g., [18]). W will not discuss hr all spcific faturs of using th WENO sch for th nurical solution of th abov-dscribd quations of th nw foralis, bcaus w hop to dscrib th in a sparat papr. Intgration with rspct to ti was prford by th fourth-ordr Rung-Kutta-Gill thod [19], which rquirs a sallr aount of coputr ory for storag of auxiliary arrays than th standard fourth-ordr Rung-Kutta thod. As th first xapl, w considr propagation of an lctroagntic wav in a tallic wavguid with a squar cross sction. Th initial data ar dfind in th for of a transvrsly lctric TE 1,1 od of th wavguid [20] with a frquncy ω = π 6. Fig. 1. Th probl is solvd in a coputational doain 0 x 1, 0 y 1, 0 z 2 on a grid of clls. Sytry or antisytry conditions ar iposd on various coponnts of th potntials on th wavguid surfac to guarant satisfaction of th usual conditions for th coponnts of th lctric and agntic filds on an infinitly conducting surfac. Priodic boundary conditions ar applid in th longitudinal dirction at z = 0 and 1. Th stp of intgration with rspct to ti is dtrind fro th condition t = CFL(1/ x + 1 / y + 1 / z), whr th Courant-Fridrichs-Lwy nubr is CFL = 0.8. Figur 1 shows th coputd distribution of th longitudinal agntic fild B z on th wavguid surfac and in its nd cross sction at th ti t = 2 π / ω = Th lctric and agntic filds ar calculatd fro th vctor potntials in accordanc with Eqs. (2.5) and (2.7); a sixth-ordr cntral-diffrnc sch is usd for diffrntiation. 10

11 Fig. 2. Fig. 3. Th coputd distribution of th lctric fild coponnt E x along th lin x = y = 1 / 3 at th sa ti ont is copard in Fig. 2 with th xact solution. It is sn that ths solutions cannot b distinguishd in th scal of this figur. To gt a bttr ida about th accuracy nsurd by th WENO sch, w also solvd th sa probl using th Finit-Diffrnc Ti Doain (FDTD) thod, which is ost popular in coputational lctrodynaics. This thod is a scond-ordr finit-diffrnc sch on a staggrd spac-ti grid [1]. Figur 3 shows th ti volution of th rror (in th nor L 1 ) of dtrining B z for th two copard thods. As is sn fro Fig. 3, th WENO sch is or accurat than th FDTD thod by an ordr of agnitud. In th scond xapl, w considr propagation of an lctroagntic wav in a diu with a jup of atrial proprtis. W odl th noral incidnc (in th x dirction) of a planpolarizd wav onto th intrfac btwn two dia with diffrnt valus of dilctric prittivity. Th probl is solvd in th doain 0 z 2. W st ε = 4, and ε = 1 on th lft and on th right of th plan z = 1. Th agntic prability is µ = 1 vrywhr. Th doain is dividd into 400 clls. At th initial ti, th fild is qual to zro vrywhr insid th coputational doain. On th lft boundary, w ipos haronically varying (with a frquncy ω = 10 ) valus of th potntials corrsponding to a plan lctroagntic wav travling to th right. Th calculation is prford with CFL = 0.8 up to th ti t = 3 whn th wav rachs th right boundary of th coputational doain. At this ti, th solution in th intrval 0 < z < 0.5 includs only th incidnt wav, th solution in th intrval 0.5 < z < 1 is a suprposition of th incidnt and rflctd wavs, and th solution at z > 1 consists of th wav that passd to th scond diu. 11

12 Fig. 4. Fig. 5. Th distributions of th only non-zro coponnt of th lctric fild intnsity E x and th corrsponding coponnts of th vctor of lctric induction D x at th final ti instant ar copard in Figs. 4 and 5 with xact analytical xprssions for th. Obviously, th nurical and analytical solutions ar in good agrnt. As it could b xpctd, th tangntial coponnt of th lctric fild intnsity is continuous on th intrfac btwn two dia (Fig. 4), whras th tangntial coponnt of th lctric induction vctor has a jup (Fig. 5). Sall diffrncs btwn th nurical and analytical solutions ar obsrvd only on th intrfac and nar th lading front of th transittd wav around z = 2. Ths diffrncs ar causd by using a cntral-diffrnc forula for calculating th filds fro th potntials. At th abov-indicatd points, th svn-point stncil of this forula intrscts grid points whr th diffrntiatd function has a kink, which lads to a dcras in accuracy. Obviously, th rsults can b iprovd by using on-sidd diffrnc forulas that do not intrsct th kink points of th diffrntiatd function for calculating th filds at such points. Ths considrations ar illustratd in Figs. 6 and 7, which show th distributions of th coponnt C of th lctric vctor potntial and th coponnt A of th agntic vctor y potntial, rspctivly, at t = 3. Th knik in th distribution of C y on th intrfac of two dia, which corrsponds to a jup of th tangntial coponnt D x of th lctric induction vctor, is clarly visibl. At th sa ti, th distribution of A looks sooth, as it should b bcaus thr is no jup of th tangntial coponnt of th agntic fild B y on th intrfac of two dia with an idntical valu of µ. Figurs 6 and 7 illustrat on or advantag of forulating th lctroagntis quations with th us of potntials: thy ar soothr than th filds thslvs. It should b notd that th coputation of propagation of lctroagntic wavs in a diu with discontinuitis of dilctric prittivity by FDTD-typ thods basd on approxiation of spatial drivativs by cntral diffrncs without involving spcial tchniqus (.g., rplacnt of th jup of atrial proprtis by a zon of thir rapid, but still continuous variation) will invitably lad to rgnc of noticabl non-physical oscillations of th nurical solution. x x 12

13 Fig. 6. Fig CONCLUSIONS Thus, a nw forulation of lctroagntic fild quations basd on th us of two vctor potntials and two scalar potntials is proposd. This forulation allows th Maxwll quations both in vacuu and in a atrial diu to b writtn in th for of a hyprbolic syst possssing a nubr of dsirabl proprtis. In particular, it consists only of volutionary quations and has no rlations having th charactr of diffrntial constraints and lading to significant probls in th nurical solution of th Maxwll quations in th standard forulation. Morovr, all ignvalus of th Jacobi atrix of th drivd syst of quations corrsponds to physical ods propagating with th spd of light; thr ar no non-physical ods corrsponding to th zro ignvalu and obtaind in th frquntly usd approach whr only th quations containing curls of th vctor filds ar solvd, whras th quations with divrgncs ar ignord. All ths facts allow powrful odrn shock-capturing thods basd on approxiation of spatial drivativs by upwind diffrncs to b usd to solv th nw syst nurically. Exapls of nurical siulations of propagation of lctroagntic wavs by solving th quations in th nw forulation ar givn. On of th odrn schs is usd: a fifth-ordr WENO sch. It is donstratd that such a nurical approach allows th solution to b obtaind with high accuracy, including probls that involv jups of atrial proprtis of th diu. Th authors wish to xprss thir cordial gratitud to S.K. Godunov for discussions of various aspcts of th thory of hyprbolic quations and spcific faturs of th nurical solution of ovrdtrind systs, which stiulatd this work to a larg xtnt. REFERENCES 1. A. Taflov, S. C. Hagnss. Coputational Elctrodynaics. Th Finit-Diffrnc Ti-Doain Mthod, 3 rd d. Artch Hous, Boston / Nw York, J. C. Maxwll. A Tratis on Elctricity and Magntis. Vol Posnr Morial Collction. Carngi Mllon Univrsity, W. E. Baylis. Elctrodynaics: a Modrn Gotric Approach, 2 nd d. Birkhäusr, Boston, V. I. Strashv, L. M. Toil chik. Elctrodynaics with a Magntic Charg. Nauka i Tkhnika, Minsk, 1975 (in Russian). 13

14 5. J. Polchinski. Monopols, duality, and string thoris. Int. J. Mod. Phys. A., Vol. 19, No. S1, pp (2004). 6. Yu. V. Novozhilov, Yu. A. Yappa. Elctrodynaics. Nauka, Moscow, B. R. Chawla, S. S. Rao, H. Unz. Potntial quations for anisotropic inhoognous dia. Procdings of IEEE, Vol. 55, No. 3, pp (1967). 8. V. A. Ugarov. Spcial Thory of Rlativity. 2-nd d. Nauka, Moscow, 1977 (in Russian). 9. W. S. Wiglhofr, N. K. Gorgiva. Vctor potntials and scalarization for nonhoognous isotropic dius. Elctroagntics, Vol. 23, pp (2003). 10. S. K. Godunov. Equations of Mathatical Physics. 2-nd d. Nauka, Moscow, 1979 (in Russian). 11. J. S. Shang, R. M. Fithn. A coparativ study of charactristic-basd algoriths for th Maxwll quations. J. Coput. Phys, Vol. 125, No. 2, pp (1996). 12. C. K. Birdsall, A. B. Langdon. Plasa Physics via Coputr Siulation. McGraw Hill, Nw York, C.-D. Munz, R. Ons, R. Schnidr, E. Sonnndrückr, U. Voß. Divrgnc corrction tchniqus for Maxwll solvrs basd on a hyprbolic odl. J. Coput. Phys, Vol. 161, No. 2, pp (2000). 14. D. P. Babii, S. K. Godunov, V. T. Zhukov, O. B. Fodoritova. Diffrnc approxiations of ovrdtrind hyprbolic quations of th classical athatical physics. Zh. Vych. Mat. Mat. Fiz, Vol. 47, No. 3, pp (2007). 15. A. G. Kulikovskii, N. V. Pogorlov, A. Yu. Snov. Mathatical Aspcts of Nurical Solution of Hyprbolic Systs. Chapan & Hall / CRC, G.-S. Jiang, C.-W. Shu. Efficint iplntation of wightd ENO schs. J. Coput. Phys, Vol. 126, No. 1, pp (1996). 17. S. K. Godunov. Diffrnc thod of th nurical calculation of discontinuous solutions of hydrodynaic quations. Mat. Sb., Vol. 47(89), No. 3, pp (1959). 18. A. N. Kudryavtsv, T. V. Poplavskaya, D. V. Khotyanovsky. Application of high-ordr schs for nurical siulation of unstady suprsonic flows. Mat. Modl, Vol. 19, No. 7, pp (2007). 19. R. J. Thopson. Iproving round-off in Rung-Kutta coputations with Gill s thod. Co. of th ACM, Vol. 13. No. 12, pp (1970). 20. M.B. Vinogradova, O. V. Rudnko, A. P. Sukhorukov. Thory of Wavs, 2nd d. Nauka, Moscow, 1990 (in Russian). 14