How To Solve The Phemean Problem Of Polar And Polar Coordiates

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1 ISSN , Eglad, UK World Joural of Modellig ad Simulatio Vol. 8 (1) No. 3, pp Alterate treatmets of jacobia sigularities i polar coordiates withi fiite-differece schemes Alexys Bruo-Alfoso 1, Lube Cabezas-Gómez, Helio Aparecido Navarro 3 1 Departameto de Matemática, Faculdade de Ciêcias de Bauru, UNESP-Uiversidade Estadual Paulista, Bauru , Brazil Departameto de Egeharia Mecâica, Potifícia Uiversidade Católica de Mias Gerais, Coração Eucarístico, Belo Horizote, MG, , Brazil 3 Departameto de Egeharia Mecâica, Escola de Egeharia de São Carlos, Uiversidade de São Paulo, São Carlos , Brazil (Received Jauary 5 11, Accepted February 1) Abstract. Jacobia sigularities of differetial operators i curviliear coordiates occur whe the Jacobia determiat of the curviliear-to-cartesia mappig vaishes, thus leadig to ubouded coefficiets i partial differetial equatios. Withi a fiite-differece scheme, we treat the sigularity at the pole of polar coordiates by settig up complemetary equatios. Such equatios are obtaied by either itegral or smoothess coditios. They are assessed by applicatio to aalytically solvable steady-state heat-coductio problems. Keywords: fiite differece, polar coordiates, Jacobia determiat 1 Itroductio Partial differetial equatios (PDEs) serve as mathematical models for physical pheomea such as diffusio ad heat trasfer, wave propagatio ad statioary fields. Of course, besides the differetial equatio, oe should take ito accout boudary coditios ad, whe applicable, iitial coditios as well. Thus, depedig o the geometry of the spatial domai, the problem may be simplified whe writte i terms of curviliear coordiates. Moreover, the PDE usually cotais differetial operators like gradiet, divergece, Laplacia, ad curl. While these operator take simple forms i Cartesia coordiates (x,, their curviliear versios may be rather cumbersome. To fix ideas, we cosider a right-haded system of orthogoal coordiates (p, q), i.e., such that x p x q +y p y q = ad x p y q x q y p >. I two dimesios, the gradiet, divergece, Laplacia, ad curl, may be writte as [1] u = 1 h p p êp + 1 h q q êq, (1) U = 1 [ J p (h q U p ) + ] q (h p U q ) () u = 1 [ ( ) hq + ( )] hp, (3) J p h p p q h q q U = 1 [ J p (h q U q ) ] q (h p U p ), (4) The authors thak the Brazilia research fudig agecies CNPq, FAPESP ad FAPEMIG for fiacial support. A. B.-A. is grateful to G. V. B. de Souza for useful discussios. Correspodig author. Tel.: Fax: address: alexys@fc.uesp.br. Published by World Academic Press, World Academic Uio

2 164 A. Bruo-Alfoso & L. Cabezas-Gómez & H. Navarro: Alterate treatmets of jacobia sigularities respectively, where h p = x p + yp ad h q = x q + yq are the scale factors, J = h p h q is the Jacobia determiat, ê p = (x p, y p ) /h p ad ê q = (x q, y q ) /h q are the uit vectors of the curviliear coordiates, U p = U ê p ad U q = U ê q. I this way, oe may clearly see that differetial operators i Eqs. (1) (4) have sigularities at zeros of J. For short, these are called as Jacobia sigularities. Whe (p, q) are the usual polar coordiates (ρ, θ) satisfyig x = ρ cos(θ), y = ρ si(θ), the scale factors are h ρ = 1 ad h θ = ρ, the Jacobia determiat is J = ρ, ad the uit vectors are ê ρ = (cos(θ), si(θ))ad ê θ = ( si(θ), cos(θ)). Therefore, the oly Jacobia sigularity occurs at ρ =. I this sese, oe should carefully treat differetial equatios cotaiig the above metioed operators, wheever the spatial domai cotais the pole ρ =. I the available literature, several approaches have bee developed to obtai umerical solutios of PDEs i curviliear coordiates where Jacobia sigularities occur. Some authors exclude the correspodig poits of the fiite-differece mesh [1, 4, 7]. I particular, Mohsei ad Coloius [4] have used a computatioal domai where r takes either positive or egative values, while θ π. I other approaches, a derivatio of a limitig PDE has bee proposed [6, 9]. Cotatiescu ad Lele [, 3] have developed a umerical method where goverig equatios for the flow at the polar axis are derived by series expasios. Moreover, Fukagata ad Kasagi [5] itroduced a sigle-valued represetatio of the velocity compoets at r =, based o the series expasio of Costatiescu ad Lele [, 3] ad cosiderig a eergy-coservative umerical scheme. However, accordig to Moriishi et al [8], those authors did ot cosider the mometum coservatio at the pole i order compute the velocity compoets at pole. I this sese, Moriishi et al [8] have itroduced a ovel treatmet of the pole, usig a discrete radial mometum equatio with a fully coservative covectio scheme at the pole. I other words, they obtaied the radial velocity at the pole based o the radial mometum equatio with a fully eergy-coservative covectio scheme. I this paper, a fiite-differece scheme i polar coordiates is set up to solve PDE problems. As usual, we cosider ρ ad θ π. Moreover, the pole ρ = belogs to the discretizatio mesh ad the correspodig equatio is derived. This is doe by followig two methods: (i) itegratio of both sides of the PDE over a ifiitesimal regio cotaiig the pole ad (ii) applyig differetiability coditios at the pole. To assess these approaches, we solve problems which model heat-trasfer processes. The cases where the ukow fuctio is either a scalar or a vector field are addressed. A cetral-differece discretizatio of secod order is used, icludig Dirichlet boudary coditios at the physical boudaries. Excellet agreemet betwee umerical ad aalytical solutios is obtaied i all cases. The extesio of the preset method to cylidrical ad spherical coordiates is straightforward. I this way, the preset ideas should fid further applicatio i modellig physical systems ad processes like lamiar ad turbulet isothermal fluid flow, covectio heat-trasfer pheomea, mechaical waves, electromagetic fields ad quatum states. Derivatio of complemetary coditios.1 A scalar fuctio i polar coordiates Let us cosider a physical problem modelled by a PDE i a domai D with boudary C i Cartesia coordiates. The origi O (, ) is assumed to be a ier poit of D, as show i Figure 1. Whe the boudary C is give by simple formulas i polar coordiates (ρ, θ), with ρ ad θ π, it may be advatageous to set up the mathematical problem i the variables ρ ad θ. To be cocrete, we deal with a statioary heat-trasfer process i a thi homogeeous circular plate where thermal eergy may be trasferred from/to the eviromet either across the plae circular faces or the thi cylidrical surface at the plate edge. More specifically, we assume that eergy iterchage across the plae faces is give by the source fuctio s(x,, bouded ad itegrable over D. Moreover, the two-dimesioal eergy-flux desity is deoted by the vector field U(x, = k u(x,, where k is the two-dimesioal thermal coductivity. The, because of the eergy coservatio, we have: U(x, = s(x,. (5) Cosequetly, the temperature distributio satisfies the Poisso equatio: WJMS for cotributio: submit@wjms.org.uk

3 be advatageous to set up the mathematical problem i the variables ρ ad θ. v + v + To be cocrete, we deal with a statioary heat-trasfer process i a thi homogeeous circular plate where thermal eergy may be trasferred from/to the eviromet either across (,) = (, π ), the plae circular ρ faces or (7) the thi cylidrical surface at the plate edge. More specifically, we assume that eergy iterchage across w + w + the plae faces is give by the source fuctio s ( x,, bouded ad itegrable ( over,) = D. Moreover, (, π ). the World Joural of Modellig ad Simulatio, Vol. 8 (1) r No. ρ 3, rpp two-dimesioal eergy-flux desity is deoted by the vector field U ( x, = k u( x,, where k is Hece, complemetary coditios for a vector fuctio at ρ = are Eqs. (), (1) ad (7). Those the two-dimesioal thermal coductivity. The, because of the eergy coservatio, we have s(x, u(x, =. (6) r U r coditios are used i the fiite-differece scheme below. ( x, = s( x,. k (5) 3. Fiite-differeces ad complemetary coditios Cosequetly, the temperature distributio satisfies the Poisso equatio Furthermore, the polar versio of this equatio is: s( x, We cosider the computatioal domai to be a circular disk of radius R cetered at the pole. Hece, u( x, =. polar (6) k coordiates u + 1 take ρ + 1 values i the u ρ θ = s rages θ π ad ρ R. We uiformly divide these itervals ito N ad M parts, respectively. Furthermore, the polar versio of this equatio is k, The resultig mesh is give (7) by θ = ( 1) Δθ ad ρ m = mδρ, with Δ θ = π / N, Δρ = R /( M +1), = 1,,..., N ad m =,1,..., M + 1. u 1 1 u s + ad + the Jacobia =, Coveietly, N is take as a ρ ρ θ k sigularity at ρ = is apparet i the secod (7) eve umber. I this way, θ 1+ N / = π, i.e., π is amog the azimuthal ad third terms i the left-had side of this odes (see Figure ). The temperature distributio i a circular plate occupyig the regio R is supposed equatio. ad the Jacobia sigularity at ρ = is apparet i the secod to ad be third described terms by i the the fuctio left-had u side ( ρ, θof ) ad its ode values are deoted by um, = u( ρ m, θ ). this equatio. Fig. 1: Schematic represetatio of the physical domai D, its boudary C ad the Fig. pole : Computatioal (,). domai ad fiite-differece mesh i polar coordiates. Fig. 1. Schematic represetatio of the physical domai Fig.. Computatioal domai ad fiite-differece Because of the D, Jacobia its boudary sigularity C ad uder the ivestigatio, pole (,) the 3.1. polar-to-cartesia Scalar ukow mappig mesh imay polar lack coordiates iverse. I fact, all poits (,θ ) map ito the origi of the Cartesia coordiates, To assess i.e., the the complemetary pole. Therefore, equatios derived i subsectio.1, we calculate the temperature there is a certai value u such that distributio obeyig Eq. (7), with source fuctio s( ρ, θ ) = s [ 33( / ) ( ) ] 3 / 9 / 1/ ρ R ρ R cos(θ ) ad u (, θ ) = u, Because of the Jacobia sigularityboudary uder ivestigatio, 3 coditio u ( R, θthe ) = Tpolar-to-Cartesia (8) si ( θ π / 3). mappig may lack iverse. I fact, all poits (, θ) map ito the origi of for all θ π. Moreover, both sides of equatio (5) may be itegrated The the over odes Cartesia a small with disk coordiates, of radius N 1ε ad 1 i.e., m the Mpole. are ier Therefore, poits of the there mesh isi afigure. At those cetered at the pole. certai This value leads to u such that: poits, the fiite-differece versio of Eq. (7) is writte by usig cetral differeces as um 1, um, + um+ 1, um+ 1, um 1, um, 1 u m, + um, + 1 sm, u(, θ) = u + + =,,(8) (8) ( Δρ) ρ Δρ ρ ( Δθ ) k for all θ π. Moreover, both sides of Eq. (5) may be itegrated over a small disk of radius ε cetered at the pole. This leads to: π m (ε, θ) dθ = π ε s (ε), (9) k where s (ε) is the mea value of s(x, over the small disk. Sice this source fuctio is bouded, so it is s (ε), ad the right-had side of Eq. (9) teds to zero as ε +. Hece, i this limit we obtai: π (+, θ) dθ =. (1) I some cases, boudary coditios are such that temperature depeds o ρ aloe, i.e., u (ρ, θ) = u (ρ). Therefore, accordig to Eq. (1), the coditio at the pole reads: u ( + ) =. (11) I the fiite-differece calculatios preseted below, we may use discretized versios of Eq. (6) for ier mesh poits with ρ, ad Eqs. (8) ad (1) for mesh poits with ρ =. Thigs may become simpler whe the ukow fuctio is smooth, i.e., whe u(x, is kow to be differetiable. For istace, this will be the case whe u varies smoothly over a smooth boudary C ad the source tem is also smooth. Uder these coditios, U should be smooth alog ay straight lie passig alog the pole. By takig the a vector of the lie as ê ρ = (cos(θ), si(θ)), the lateral derivatives at the pole are: m WJMS for subscriptio: ifo@wjms.org.uk

4 166 A. Bruo-Alfoso & L. Cabezas-Gómez & H. Navarro: Alterate treatmets of jacobia sigularities u(ρ, θ) u(, θ) u( ρ, θ) u(, θ) u(ρ, θ ± π) u(, θ ± π) lim = lim = lim ρ ρ ρ + ρ ρ + ρ = (+, θ ± π), u(ρ, θ) u(, θ) lim = ρ + ρ (+, θ). Here we have take ito accout Eq. (8) ad idetity u( ρ, θ) = u(ρ, θ ± π). Hece, differetiability of u leads to: (+, θ) = (+, θ ± π), (1) for all values of θ. We also ote that, to determie the value of u, oe further coditio at the pole is eeded. Sice a arbitrary value of θ may be chose for the fiite-differece scheme, we take θ = ad obtai coditio: (+, ) = (+, π). (13) This equatio may be used for smooth solutios istead of Eq. (1). For the sake of completeess, we ote that for axially symmetric solutios, i.e., whe u (ρ, θ) = u (ρ), Eq. (1) leads to Eq. (11). Moreover, i other cases, by itegratig both sides of Eq. (1) from θ = to θ = π, oe obtais Eq. (1).. A vector fuctio i polar coordiates We ow cosider a vector fuctio U (x, = (X(x,, Y (x, ) satisfyig a PDE problem i a spatial domai D cotaiig the pole, as show i Fig. 1. For simplicity, we restrict ourselves to problems where the solutio is kow to vary smoothly. I other words, we assume both X(x, ad Y (x, to be differetiable scalar fuctios. Therefore, we may apply the ideas developed for u(x, i the previous sectio. However, we deal with their depedece o the polar coordiates, i.e., X (ρ, θ) ad Y (ρ, θ). O the oe had, from Eq. (8), there should be costats X ad Y such that: X(, θ) = X, Y (, θ) = Y, for all values of θ. O the other had, from Eq. (1), we get: X (+, θ) = X (+, θ + π), Y (+, θ) = Y (+, θ + π). (14) Now we deote the radial ad azimuthal compoets of the vector field U as v(ρ, θ) ad w(ρ, θ). Hece, v (ρ, θ) = U ê ρ = (X (ρ, θ), Y (ρ, θ)) (cos(θ), si(θ)) = X (ρ, θ) cos(θ) + Y (ρ, θ) si(θ), w (ρ, θ) = U ê θ = (X (ρ, θ), Y (ρ, θ)) ( si(θ), cos(θ)) = X (ρ, θ) si(θ) + Y (ρ, θ) cos(θ). Their values at the pole satisfy: v (, θ) = X (, θ) cos(θ) + Y (, θ) si(θ) = X cos(θ) + Y si(θ), (15) w (, θ) = X (, θ) si(θ) + Y (, θ) cos(θ) = X si(θ) + Y cos(θ). (16) Moreover, from these equatios we obtai the radial derivatives at the pole, amely: v ( +, θ ) = X ( +, θ ) cos(θ) + Y ( +, θ ) si(θ), (17) w ( +, θ ) = X ( +, θ ) si(θ) + Y ( +, θ ) cos(θ), (18) WJMS for cotributio: submit@wjms.org.uk

5 World Joural of Modellig ad Simulatio, Vol. 8 (1) No. 3, pp respectively. Therefore, by combiig Eqs. (14), (17) ad (18), we get: v ( +, θ + π ) = X ( +, θ ) cos(θ) + Y ( +, θ ) si(θ), w ( +, θ + π ) = X ( +, θ ) si(θ) + Y ( +, θ ) cos(θ). This meas that polar compoets of U satisfy: { v (+, θ) = v w (+, θ) = w (+, θ + π) (+, θ + π) (19) for all values of ϑ. Agai, to determie the values X ad Y, we oly eed to apply Eq. (19) to a sigle arbitrary value of θ. For simplicity, we take: θ = to obtai { v (+, ) = v (+, π) w (+, ) = w (+, π) () Hece, complemetary coditios for a vector fuctio at ρ = are Eqs. (15), (16) ad (). Those coditios are used i the fiite-differece scheme below. 3 Fiite-differeces ad complemetary coditios We cosider the computatioal domai to be a circular disk of radius R cetered at the pole. Hece, polar coordiates take values i the rages θ π ad ρ R. We uiformly divide these itervals ito N ad M parts, respectively. The resultig mesh is give by θ = ( 1) θ ad ρ m = m ρ, with θ = π/n, ρ = R/(M + 1), = 1,,..., N ad m =, 1,..., M + 1. Coveietly, N is take as a eve umber. I this way, θ 1+N/ = π, i.e., π is amog the azimuthal odes (see Fig. ). The temperature distributio i a circular plate occupyig the regio R is supposed to be described by the fuctio u(ρ, θ) ad its ode values are deoted by u m, = u(ρ m, θ ). 3.1 Scalar ukow To assess the complemetary equatios derived i subsectio.1, we calculate the temperature distributio obeyig Eq. (7), with source fuctio s(ρ, θ) = s [33 (ρ/r) 3/ 9 (ρ/r) 1/] cos(θ) ad boudary coditio u(r, θ) = T si 3 (θ π/3). The odes with N 1 ad 1 m M are ier poits of the mesh i Fig.. At those poits, the fiite-differece versio of Eq. (7) is writte by usig cetral differeces as: u m 1, u m, + u m+1, ( ρ) + u m+1, u m 1, ρ m ρ + u m, 1 u m, + u m,+1 ρ m ( θ) = s m, k, (1) with s m, = s(ρ m, θ ). It is worthy to otice that all poits with 1 m M ad either = 1 or = N lay at the boudary of the mesh i Fig., but they correspod to ier poits of the circular domai D. Hece, we may exted the validity of Eq. (1) by makig the replacemets u m, u m,n ad u m,n+1 u m,1. Moreover, for m = 1, we follow Eq. (8) ad make: u, = u, for all 1 N ad, for m = M, we take ito accout the discretized boudary coditio, i.e., u M+1, = T si 3 (θ π/3). () WJMS for subscriptio: ifo@wjms.org.uk

6 168 A. Bruo-Alfoso & L. Cabezas-Gómez & H. Navarro: Alterate treatmets of jacobia sigularities I this way, we have got a system of N M liear equatios for u ad the N M ukows u m, with 1 N ad 1 m M. The remaiig equatio follows from Eq. (1) ad has the form: N =1 (+, θ ) =. (3) Sice the radial derivative i this sum may be estimated i terms of three poits, amely: Eq. (3) gives a equatio for u, i.e., (+, θ ) = 4u 1, 3u u,, (4) ρ u = 1 3 N N (4 u 1, u, ). (5) =1 Cosequetly, Eqs. (1), () ad (5) form a system of 1 + N M equatios for u ad u m, with 1 N ad 1 m M. We also ote that for a smooth solutio we may use Eq. (13) istead of Eq. (1). Hece, from Eq. (4) we obtai: u = 3 (u 1,1 + u 1,1+N/ ) 1 6 (u,1 + u,1+n/ ), (6) which may be used istead of Eq. (5). For the sake of compariso, it is useful to bear i mid that the preset problem has a rather simple aalytical solutio, amely: u(ρ, θ) = 1 [ ] [ 3ρ ρ3 ( ρ 5/ ( ρ ) ] 7/ si(θ π/3) + T 4 R R 3 si(3θ) + 4σ cos (θ), R) R with σ = R s /(k T ) beig a dimesioless parameter which characterizes the itesity of the heatig source. Moreover, the two-dimesioal eergy-flux desity is give by: U (ρ, θ) = k u = êρ 1 ρ θ êθ. Therefore, its polar compoets v(ρ, θ) ad w(ρ, θ) satisfy: R v(ρ, θ) = 3 [ ] si(θ π/3) + ρ k T 4 R si(3θ) σ R w(ρ, θ) = 3 [ ] cos(θ π/3) + ρ k T 4 R cos(3θ) [ ( ρ ) 3/ ( ] ρ 5/ 5 7 cos (θ), R R) [ ( ρ 3/ ( ρ ) ] 5/ + 8σ si (θ). R) R Figs. 3 ad 4 display the cotour plot of u (x, ad the stream lies of U(x,, respectively. Vector U(x, is perpedicular to cotour lies ad poits toward the low-temperature side of the lie. 3. Vector ukow From the aalytical solutio preseted i the previous sectio, it is straightforward to show that the polar compoets of the eergy-flux desity satisfy the followig Dirichlet boudary coditios: R v(r, θ) = 3 [ si(θ π/3) + si(3θ)] + 4σ cos (θ). k T 4 (7) R w(r, θ) = 3 [ cos(θ π/3) + cos(3θ)]. k T 4 (8) WJMS for cotributio: submit@wjms.org.uk

7 3 / 5 / R w( ρ, θ ) 3 ρ ρ ρ cos( θ π / 3) cos(3θ ) 8σ si( θ ) 4 = + +. (38). k T R R R Fig. 3: Cotour Figures 3 ad 4 display the cotour plot of u ( x, ad the stream lies of U r plots of the steady-state temperature distributio, i uits of ( x,, respectively. Vector T U r, for a uiform circular plate of 3 radius R with the Dirichlet coditio ( x, is perpedicular World Joural to cotour of Modellig lies ad ad poits Simulatio, toward the Vol. low-temperature 8 (1) No. side 3, pp. of T ( R, θ ) = T si ( θ π / 3) ad heatig fuctio the lie. 169 (, ) 33( / ) 3 / 9( / ) 1/ s ρ θ = s ρ R ρ R cos(, with R s = k T. [ ] ) θ Fig. 3: Cotour plots of the steady-state temperature distributio, i uits Fig. of T 4:, Solid for a uiform orieted circular curves are plate stream of lies of the heat curret correspodig to temperature distributio ad 3 radius Fig. R with 3. the Dirichlet Cotour coditio plots T ( of R, θ ) the = T steady-state si ( θ π / 3) ad temperature distributio, heatig fuctio Fig. coditios 4. Solid i Fig. orieted 3. Dotted curves lies are are cotour stream lies lies Figure of3. the s( ρ, θ ) = s [ 33( / ) 9( / ) ] 3 / 1/ ρ Ri uits ρ Rof Tcos(, for θ ), with a uiform R s = k T. circular plate of radius R with the Dirichlet heat curret correspodig to temperature distributio 3.. Vector ukow coditio ad coditios i Fig. 3. Dotted lies are cotour lies From the aalytical solutio preseted i the previous sectio, it is straightforward to show that the polar T (R, θ) = T si 3 (θ π/3) ad heatig fuctio s(ρ, θ) = s [33 (ρ/r) 3/ 9 (ρ/r) 1/] cos(θ), with R s = k T i Fig. 3 The, i order to assess the complemetary equatios derived i subsectio., we may try to calculate the vector field U(ρ, θ). Of course, we should set up differetial equatios for v(ρ, θ) ad w(ρ, θ). O the oe had, sice U(ρ, θ) is a gradiet, its curl must vaish. Followig Eq. (4), this leads to: w + w ρ 1 ρ O the other had, by usig Eq. (), we rewrite Eq. (6) as: Fig. 4: Solid orieted curves are stream lies of the heat curret correspodig to temperature distributio ad v coditios i Fig. 3. Dotted lies are cotour lies i Figure 3. + v ρ + 1 w = s. (3) ρ θ 3.. Vector ukow From the aalytical solutio To setpreseted up the fiite-differece the previous sectio, scheme, it is straightforward we use theto same show mesh that the as polar i the previous sectio, deotig v m, = v(ρ m, θ ) ad w m, = w(ρ m, θ ). By usig the same discretizatio procedure, Eqs. (9) ad ((3) lead to: w m+1, w m 1, ρ v m+1, v m 1, ρ + w m, ρ m v θ =. (9) v m,+1 v m, 1 ρ m θ + v m, + w m,+1 w m, 1 ρ m ρ m θ =, = s m,. These later equatios are valid for 1 N ad 1 m M, provided replacemets v m, v m,n, w m, w m,n, v m,n+1 v m,1, ad w m,n+1 w m,1 are made whe eeded. Moreover, whe m = 1, we make use of Eqs. (15) ad (16) to write: The boudary Eqs. (7) ad (8) are writte as: v, = X cos(θ ) + Y si(θ ), (31) w, = X si(θ ) + Y cos(θ ). (3) R v M+1, k T = 3 4 [ si(θ π/3) + si(3θ )] + 4σ cos (θ ), R w M+1, k T = 3 4 [ cos(θ π/3) + cos(3θ )]. Furthermore, Eqs. (), (4), (31) ad (3) lead to the complemetary coditios: WJMS for subscriptio: ifo@wjms.org.uk

8 This leaves us with ad 1 m M. + M N equatios for X, Y ad the ukows v m, ad w m,, with 1 N 4. Covergece of umerical results 17 A. Bruo-Alfoso & L. Cabezas-Gómez & H. Navarro: Alterate treatmets of jacobia sigularities The fiite-differece X = 3 (v schemes 1,1 v 1,1+N/ ) 1 6 (v described above lead to liear,1 v,1+n/ ), Y = 3 (w systems which may 1,1 w 1,1+N/ ) 1 6 (w be solved by usig a,1 w,1+n/ ). computatioal algebraic system such as Mathematica [11]. For simplicity, we take M = N, thus we are able Thisto leaves ivestigate us withthe + covergece M N equatios of results for X as, Ya ad fuctio the ukows of N. For v m, the ad case w m, of, scalar with 1ukow, N we ad defie 1 m M. ε N = max { um, u( ρm, θ) : m M, 1 N}, (51) 4 Covergece of umerical results as the The maximum fiite-differece error of the schemes fiite-differece described calculatio. above lead to liear systems which may be solved by usig a computatioal By usig algebraic N = system,, Ksuch, 6as, Mathematica fiite-differece [11]. For simplicity, results we for take the M temperature = N, thus we distributio are able to essetially ivestigatereproduce the covergece the exact of results solutio as displayed a fuctio i of Figure N. For3, thewhere case of σ scalar = 1. This ukow, is apparet we defie: Figure 5, where the maximum error ε N = ε Nmax is show { u m, as a u(ρ fuctio m, θ ) of : 1 / N m. Paels M, 1(a) ad (b) N} correspod, to Eq. (33) ad as the (34), maximum respectively. error of We theote fiite-differece that i both calculatio. cases the error By usig coverges N =, to zero,.. as., N 6, grows. the fiite-differece I fact, the error is essetially proportioal to 1/ N. maximum error ΕN a Ε N N N maximum error ΕN b Ε N 3.67 N Fig. 5: The maximum error of the fiite-differece scheme for scalar ukow as a fuctio of 1 / N, with N beig Fig. 5. The maximum error of the fiite-differece scheme for scalar ukow as a fuctio of, with beig the umber of agular the umber divisios. of agular Pael divisios. (a) (b) comes Pael from (a) the [(b)] use comes of Eq. from (5) the ad use (6) of as Eq. the(33) complemetary [(34)] as the coditio complemetary at the pole coditio at the pole. The aalysis of covergece for the case of vector ukow follows the same idea. Namely, we.1 defie r r.8 Ε ε = max U, U ( ρ, θ ) : N N m M, 1 N, (5) N maximum error ΕN.14.1 { m.6 m }.4 as the maximum error of the fiite-differece. calculatio. Figure 6 displays the maximum error ε N as a. fuctio of 1 / N. This error coverges. to.1 zero as. N grows,.3 beig.4 roughly.5.6 proportioal to 1/ N. 1 N Fig. 6. The maximum error of the fiite-differece scheme for vector ukow as a fuctio of 1/N, where / N N is the umber of agular divisios Fig. 6: The maximum error of the fiite-differece scheme for vector ukow as a fuctio of 1, where N is the umber of agular divisios. 5. Coclusios results for the temperature distributio essetially reproduce the exact solutio displayed i Fig. 3, where σ = 1. This is apparet i Fig. 5, where the maximum error ε N is show as a fuctio of 1/N. Paels (a) ad We (b) have correspod derived tocomplemetary Eq. (5) ad (6), coditios respectively. at We geometrical ote that i both sigularities cases the error of PDEs coverges i polar to zero coordiates as N both for grows. scalar Iad fact, vector the errorukows. is essetiallytheir proportioal fiite-differece to 1/N. versios were applied to solve heat-trasfer problems i The a circular aalysis ofdisk. covergece A very forgood the case agreemet of vector ukow betwee follows umerical the same ad idea. aalytical Namely, we solutios defie: was obtaied. As the umber of mesh poits { icreases, error decreases roughly as the iverse of the square of ε N = max Um, such umber, showig that the employed discretizatio } U(ρ m, θ ) : m M, 1 N, (33) scheme have a secod order accuracy. Preset as theapproach maximumis error promisig of the fiite-differece for applicatio calculatio. i the solutio Figure 6of displays other the physical maximum problems error ε N of asiterest, a like lamiar fuctio ad ofturbulet 1/N. This isothermal error coverges fluid to zero flow, as Ncovectio grows, beig heat-trasfer roughly proportioal pheomea, to 1/Nmechaical. waves, electromagetic fields ad quatum states i curviliear coordiates. The preset approach is quite WJMS for cotributio: submit@wjms.org.uk geeral for smooth solutios ad does ot avoid the pole. This idea may be exteded to spherical ad other curviliear coordiates. Of course, further umerical tests i time-depedet ad o-liear problems, like those ivolvig 1 N

9 World Joural of Modellig ad Simulatio, Vol. 8 (1) No. 3, pp Coclusios We have derived complemetary coditios at geometrical sigularities of PDEs i polar coordiates both for scalar ad vector ukows. Their fiite-differece versios were applied to solve heat-trasfer problems i a circular disk. A very good agreemet betwee umerical ad aalytical solutios was obtaied. As the umber of mesh poits icreases, error decreases roughly as the iverse of the square of such umber, showig that the employed discretizatio scheme have a secod order accuracy. Preset approach is promisig for applicatio i the solutio of other physical problems of iterest, like lamiar ad turbulet isothermal fluid flow, covectio heat-trasfer pheomea, mechaical waves, electromagetic fields ad quatum states i curviliear coordiates. The preset approach is quite geeral for smooth solutios ad does ot avoid the pole. This idea may be exteded to spherical ad other curviliear coordiates. Of course, further umerical tests i time-depedet ad o-liear problems, like those ivolvig Navier-Stokes equatios, should be performed i order to establish the overall applicability of the proposed methodology. Refereces [1] V. Arpaci. Coductio Heat Trasfer. Addiso-Wesley Publishig Compay, New York, [] G. Costatiescu, S. Lele. A ew method for accurate treatmet of flow equatios i cylidrical coordiates usig series expasios. i: CTR Aual Research Briefs, Ceter for Turbulece Research, NASA Ames ad Staford Uiversity Press, Staford, CA, 1, [3] G. Costatiescu, S. Lele. A highly accurate techique for the treatmet of flow equatios at the polar axis i cylidrical coordiates usig series expasios. Joural of Computatioal Physics, 3, 183: [4] G. Davies. A ote o a mesh for use with polar coordiates. Numerical Heat Trasfer, 1979, : [5] K. Fukagata, N. Kasagi. Highly eergy-coservative fiite differece method for the cylidrical coordiate system. Joural of Computatioal Physics,, 181: [6] M. Griffi, E. Joes, et al. A computatioal fluid dyamic techique valid at the ceterlie for o-axisymmetric problems i cylidrical coordiates. Joural of Computatioal Physics, 1979, 3: [7] K. Mohsei, T. Coloius. Numerical treatmet of polar coordiate sigularities. Joural of Computatioal Physics,, 157: [8] Y. Moriishi, O. Vasilyev, T. Ogi. Fully coservative fiite differece scheme i cylidrical coordiates for icompressible flow simulatios. Joural of Computatioal Physics, 4, 197: [9] N. Ozisik. Boudary Value Problems of Heat Coductio. Dover Publicatios Icorporated, New York, [1] K. Riley, M. Hobso. Essetial Mathematical Methods for Physical Scieces. Cambridge Uiversity Press, Cambridge, 11. [11] S. Wolfram. The Mathematica Book,, Cambridge Uiversity Press, WJMS for subscriptio: ifo@wjms.org.uk

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11 Alterate treatmets of jacobia sigularities 173