FINITE VOLUME METHODS FOR SOLVING HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS ON CURVED MANIFOLDS

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1 BRAC Uiversi Joural, Vol. II, No., 005, pp FINITE VOLUME METHODS FOR SOLVING HYPERBOLIC PARTIAL DIFFERENTIAL EUATIONS ON CURVED MANIFOLDS Moshiour Rahama Deparme o Mahemaics & Naural Sciece BRAC Uiversi 66 Mohahali, Dhaa- mrahama@bracuiversi.ac.bd ABSTRACT The aural mahemaical area o ormulae coservaio laws o curve maiolds is ha o diereial geomer. Ricci developed his brach o mahemaics rom 887 o 896. Subseque wor i diereial geomer has made i a idespesible ool or solvig i mahemaical phsics. The idea rom diereial geomer is o ormulae hperbolic coservaio laws o scalar ield equaio o curved maiolds. The iie volume mehod is ormulaed such ha scalar variables are umericall coserved ad vecor variables have a geomeric source erm ha is aurall icorporaed io a modiied Riema solver. The orhoormalizaio allows oe o solve Caresia Riema problems ha are devoid o geomeric erms. The ew mehod is esed via applicaio o he liear wave equaio o a curved maiold. Ke words: iie volume mehods, curved maiolds, coservaio law, wave propagaio. I. INTRODUCTION I ma phsical applicaios, geomeric cosideraios require he use o umerical grids ha are o Caresia. I he soluio domai is a la maiold bu coais complicaed ieral or eeral boudaries, i is oe possible o iroduce a curviliear grid ha coorms o he boudaries. A udameall diere siuaio arises whe he soluio domai is a curved maiold, such as he surace o a sphere o radius r embedded ir. I his siuaio, he curvaure o he maiold modiies he uderlig damics. I his paper we are speciicall ieresed i he soluio o hperbolic parial diereial equaios o curved maiolds. For curviliear grids, a sadard umerical approach is o updae he hperbolic ssem i Caresia orm, or someimes reerred o as srog coservaio orm, i order o avoid he iroducio o source erms. However, his sraeg does o wor or geeral curved maiolds. Philosophicall speaig, here are wo approaches oe ca ae i order o solve PDEs o a curved maiold M R. The irs approach is o o direcl solve he equaios o he maiold, bu isead o solve he equaios i Caresia orm i R wih he help o a Lagrage muliplier o orce he soluio o remai o M R. The advaage is ha all o he geomer is hidde i a relaivel simple source erm; he disadvaage is ha oe mus solve o a higher-dimesioal domai. The aleraive o his sraeg is o solve direcl o he maiold. This removes he era space dimesio, bu iroduces geomeric source erms ad lu ucios ha eplicil var i space. I paricular, we prese i his wor a umerical scheme ha is o solve hperbolic parial diereial equaios o curved maiolds ad he basic mehods appl o equaios i Caresia coordiaes. For simplici, we ocus speciicall o maiolds ha ca be described b wo idepede coordiaes. II. CURVED MANIFOLDS Diereial geomer describes he geomeric srucure o a curved diereiable maiold, M. For eample, a maiold M ma represe he earl spherical surace o a plae, a curved

2 Moshiour Rahama spaceime i relaivi heor. A maiold is a se o pois ha loos locall Euclidea i ha his se ca be eirel covered b a collecio o local coordiae mappigs. Cosider a wo-dimesioal curved maiold ha is embedded i R. Le he coordiaes (, ) be he coordiaes o he maiold M. This coordiae ssem ca be relaed o he sadard Caresia coordiae ssem, (,, z), hrough he rasormaios = = z = z (, ) (, ) (, ) () A vecor, ( ( µ,µ ), i coravaria orm o he maiold M ca be rasormed o a vecor, z ( µ, µ, µ ), i Caresia space hrough he Jacobia J i he ollowig wa: µ µ µ µ = J = () z µ µ µ z z Thereore, he coordiae rasormaios direcl give us a aural basis i which o represe vecors o M. We will reer o such a basis as a coordiae basis. III. THE METRIC TENSOR The meric esor, g, is a smmeric esor ha provides a measure o legh o M. The meric relaes rue disaces as measured i R o he coordiae disaces measured i he coordiae ssem o he maiold. I paricular, he lie eleme ds = ( d) ( d) ( dz ) i R is relaed o d ad d hrough α β ds = g d d () The disace alog a curve C ( λ) parameerized b λ rom C(a) o C(b) is give b b α β d d L g a = (4) The surace area o Ω M ca be evaluaed b compuig he ollowig iegral: Surace Area o Ω = gd d (5) Ω (, ) where g is he square roo o he deermia o he meric esor. The compoes o he esor Γ are reerred o as he Chrisoel smbols or as coecio coeicies. The ivolve spaial derivaives o he meric esor g. I paricular, i a coordiae basis he ca be wrie as ollows h Γ = g g α g β g β α (6) h h h The Chrisoel smbols pla a impora role i wave propagaio o curved maiolds. IV. CONSERVATION LAWS ON CURVED MANIFOLDS Cosider he low o a subsace wih M sae M variables, q(, ) R. I he absece o a sources or sis, he ime rae o chage o he iegral o each sae variable over he volume V is ol depede o he lu o ha variable hrough he boudar, V. Mahemaicall, his is epressed wih he ollowig iegral coservaio law V (, ) dv ( q). d s = 0 q dv (7) where ( q) R M is he lu ucio, is he ouward poiig ui ormal vecor o V, ad s is he arclegh parameerizaio o V.The diereial orm o (7) is wrie q. ( q) = 0 (8) I order o solve (8), i is ecessar o epress he coservaio law i some basis. I la space, equaio (8) is oe wrie i a Caresia basis: q ( q) ( q) = 0 (9) 00

3 Fiie Volume Mehods or Solvig Hperbolic I ma ieresig applicaios, we eed o solve (8) o a smooh maiold M covered b a se o o-caresia basis vecors. I geeral, he coordiae basis represeaio o (8) aes he orm o a balace law: q ( q, ) ( q, ) = ψ c ( q, ) (0) where he lu has ow gaied a eplici depedece o he spaial coordiaes ad a geomericall iduced source erm has appeared. Le us cosider a coservaio law i which q is comprised o a scalar quai desi ρ (,) ad a vecor quai momeum, (,) ρ ( ) = µ q, Le he correspodig lu ucio be U ( q, ) = T We ca rewrie equaio (8) i esor orm: ρ ( g U ) = 0 g m m m µ ( g T ) = Γ T g µ : () () The ocus o his paper is o develop a accurae umerical mehod or he soluio o equaios () ad (). V. CARTESIAN FINITE VOLUME METHODS Le us is cosider Caresia coservaio laws o he orm (9). We cosruc a Caresia grid wih grid spacig ad ad le i = m i = where ( m, ) is he lower-le corer o he recagular compuaioal domai. I each grid cell ceered a ( i, ) ad a each ime, a iie volume mehod will produce a q, : approimaio o he average o ( ) i i i (, η, ) q ξ d ξ dη () I, he he ime averaged lu o q rom = o = across he cell ierace locaed a ad he cell ierace locaed a i ca be wrie as F F i, i, q q i, i,, τ dτ, τ dτ respecivel. Coservaio ow ells us ha mus be equal o i he grid cell ceered a ( i, ): (4) (5) i mius he lu o ou o F i i = i F, i F i F, i,, (6) All Caresia iie volume mehods ca be wrie i he orm (6). A ull umerical scheme is obaied b choosig a speciic sraeg or cosrucig he umerical lues F ad F. VI. THE SCALAR FIELD EUATION We appl he mehods or solvig hperbolic equaios o he scalar ield equaio o a curved maiold M. This equaio models he propagaio o acousic waves i a hi membrae whose shape is give b he maiold M. The scalar ield equaio ca be wrie as ϕ.( ϕ ) = 0. (7) 0

4 The pressure, ( ) ( ) ϕ ( ) (, ) p, u (, ) = ϕ (, ) Replacig (,) ϕ = ( ϕ ) p,, ad he luid veloci, u,, ca be obaied b aig appropriae emporal ad spaial gradies o he scalar ield: = (8) ϕ i equaio (7) b he above deiiios ad imposig ha (9) resuls i he ollowig ssem o balace laws or he pressure ad he compoes o he luid veloci: q = ψ c (0) g oe ca easil show ha he orhoormal equaios orm a sricl hperbolic ssem o coservaio laws. Cosider he propagaio o soud waves o a surace give b z = b(, ), where,, ad z are he sadard Caresia coordiaes. I his case, he rasormaio o surace coordiaes is quie simple sice ad parameerizes he surace:, = = ( ) (, ) ( ), b(, ) = = () z = z =. The surace ha deies M or a Gaussia dip wih he ucioal orm ( ) ( ) ( ) ) 0 0 b, = α ep () w The poi ( 0, 0 ) deie he locaio i he coordiae plae o he miimum value surace. Figures ad show his surace embedded i a hree-dimesioal Euclidea space wih diere values o he ampliudeα. The Euclidea space i which he surace is embedded has he Caresia g = dia,, meric[ ] ( ). The Jacobia mari ha rasorms a vecor o he surace o M io a Caresia vecor is Moshiour Rahama 0 J = 0 b b The meric ad he square roo o he deermia o he meric or his coordiae ssem are b b b g = b b b g = respecivel. b b Figure-: The surace ha deies M or geodesics iiialized as righ-goig curves alog he le boudar is proeced dow oo a coordiae plae. The depressio i he surace causes he geodesics o coverge. Figure-: The dip ges larger i ampliude, he geodesics coverge more ad eveuall wrap aroud he ceer o he surace. 0

5 Fiie Volume Mehods or Solvig Hperbolic The ac ha he o-diagoal erms o his meric are ozero implies ha he coordiae lies o he surace are sewed. Figure- shows he sewig o coordiaig lies ha occurs i he regio ear he dip. The meric g derived above deies he geomer o he wo-dimesioal maiold o which we solve equaio (7). Oe wa o ge a eel or he geomer deied b g is o compue geodesics o he surace. Geodesics are curves ha represe he sraighes possible lies i M. I iggur- shows geodesics ha are iiialized alog he le boudar o be righ-goig. These geodesics are proeced dow o he coordiae plae or visualizaio purposes. The igure shows ha he geodesics ha avoid he dip begi o coverge ad heir pahs are alered. I he ampliude o he surace is icreased as i Figure-. he direcio o geodesic curves i M ca be chaged eirel. The geodesics show i Figure- ad resemble he pahs o phoos ravelig pas large clusers o galaies. Due o Eisei s geeral heor o relaivi, a cluser o galaies will warp he abric o spaceime similar o he surace show i he igures. Clusers wih more mass will have larger dips. Ligh rom disa obecs will arrive a he cluser ravelig i sraigh ad parallel pahs much lie he geodesics show i he igures. However, whe he ligh ecouers he curvaure o he dip produced b he cluser, ligh ras coverge as he do i he igures. Figure- shows he iersecio o wo ligh ras ha have passed b he dip. A observer a his iersecio would see he same obec comig rom wo diere direcios. This pheomea is called graviaioal lesig, ad is observed b asroomers. 7. CONCLUSIONS We have preseed i his paper a iie volume mehod or hperbolic parial diereial equaios o curved maiolds. The equaio is solved i a coordiae basis resulig rom he choice o coordiaes o he maiold. The claim is veriied b usig he algorihm o compue he soluio o he acousic equaios o a curved maiold. The Forra code ha is used o obai he soluio b usig he sadard clawpac soware pacage. REFERENCES [] Wul Rossma. Lecure oes o Diereial Geomer. [] J.M. Bardee ad L.T. Buchma. Numerical ess o evoluio ssems, gauge codiios, ad boudar codiios or d collidig graviaioal plae waves. Phs. Rev. D, 65, 00. [] J.Y-K. Cho ad L.M. Polvai. The emergece o es ad vorices i reel evolvig, shallow-waer urbulece o a sphere. Phsics o Fluids, 8:5 55, 995. [4] J.A. Fo. Numerical hdrodamics i geeral relaivi. Livig Rev. Rel., 000. [5] A. Hare ad J.M. Hma. Sel-adusig grid mehods or oe-dimesioal hperbolic coservaio laws. J. Comp. Phs., 50:5 69, 98. [6] R. Heies ad D.A. Radall. Numerical iegraio o he shallow waer equaios o a wised icosahedral grid. Par I: Basic desig ad resuls o ess. Mohl Weaher Review, :86 880, 995. [7] R. Heies ad D.A. Radall. Numerical iegraio iegraio o he shallow waer equaios o a wised icosahedral grid. Par II: A deailed descripio o he grid ad a aalsis o umerical accurac. Mohl Weaher Review, :88 887, 995. [8] C. Helzel. Numerical approimaio o coservaio laws wih si source erms or he modellig o deoaio waves. PhD hesis, Oo-vo-Guerice-Uiversi a Magdeburg, Magdeburg,Germa, 000. [9] J. Kevoria. Parial Diereial Equaios: Aalic Soluio Techiques. Spriger- Verlag, New Yor, secod ediio, 000. [0] J.O. Lagseh ad R.J. LeVeque. A wave propagaio mehod or hree-dimesioal hperbolic coservaio laws. J. Comp. Phs., 65:6 66, 000. [] R.J. LeVeque. Fiie Volume Mehods or Hperbolic Problems. Cambridge Uiversi Press, 00. [] J.M. Marý ad E. Muller. Numerical hdrodamics i special relaivi. Livig Rev. i Rel.,