Improvement on spherical symmetry in two-dimensional cylindrical coordinates for a class of control volume Lagrangian schemes

Transcription

1 Improvement on spherial symmetry in two-imensional ylinrial oorinates for a lass of ontrol volume Lagrangian shemes Juan Cheng an Chi-Wang Shu 2 Abstrat In [4], Maire evelope a lass of ell-entere Lagrangian shemes for solving Euler equations of ompressible gas ynamis in ylinrial oorinates. These shemes use a noe-base isretiation of the numerial fluxes. The ontrol volume version has several istinguishe properties, inluing the onservation of mass, momentum an total energy an ompatibility with the geometri onservation law (GCL). However it also has a limitation in that it annot preserve spherial symmetry for one-imensional spherial flow. An alternative is also given to use the first orer area-weighte approah whih an ensure spherial symmetry, at the prie of sarifiing onservation of momentum. In this paper, we apply the methoology propose in our reent work [8] to the first orer ontrol volume sheme of Maire in [4] to obtain the spherial symmetry property. The moifie sheme an preserve one-imensional spherial symmetry in a two-imensional ylinrial geometry when ompute on an equal-angle-one initial gri, an meanwhile it maintains its original goo properties suh as onservation an GCL. Several two-imensional numerial examples in ylinrial oorinates are presente to emonstrate the goo performane of the sheme in terms of symmetry, non-osillation an robustness properties. Keywors: ontrol volume Lagrangian sheme; spherial symmetry preservation; onservative; ell-entere; ompressible flow; ylinrial oorinates Institute of Applie Physis an Computational Mathematis, Beijing 88, China. heng juan@iapm.a.n. Researh is supporte in part by NSFC grants an 934. Aitional support is provie by the National Basi Researh Program of China uner grant 25CB Division of Applie Mathematis, Brown University, Proviene, RI shu@am.brown.eu. Researh is supporte in part by ARO grant W9NF an NSF grant DMS-8986.

2 Introution The Lagrangian metho is one of the main numerial methos for simulating multiimensional flui flow, in whih the mesh moves with the loal flui veloity. It is wiely use in many fiels for multi-material flow simulations suh as astrophysis, inertial onfinement fusion (ICF) an omputational flui ynamis (CFD), ue to its istinguishe avantage in apturing material interfaes automatially an sharply. There are two kins of Lagrangian methos. One is built on a staggere isretiation in whih veloity (momentum) is store at verties, while ensity an internal energy are store at ell enters. The ensity / internal energy an veloity are solve on two ifferent ontrol volumes, see, e.g. [8,, 3]. This kin of Lagrangian shemes usually uses an artifiial visosity term, for example [8, 4, 5], to ensure the issipation of kineti energy into internal energy through shok waves. The other is base on the ell-entere isretiation in whih ensity, momentum an energy are all entere within ells an evolve on the same ontrol volume, e.g. [, 7, 5, 6, 7, 3]. This kin of shemes oes not require the aition of an expliit artifiial visosity for shok apturing. Numerial iffusion is impliitly ontaine in the Riemann solvers. It is a ritial issue for a Lagrangian sheme to keep ertain symmetry in a oorinate system ifferent from that symmetry. For example, in the simulation of implosions, sine the small eviation from spherial symmetry ue to numerial errors may be amplifie by Rayleigh-Taylor or other instability whih may lea to unexpete large errors, it is very important for the sheme to keep the spherial symmetry. In the past several eaes, many researh works have been performe onerning the spherial symmetry preservation in two-imensional ylinrial oorinates. The most wiely use metho that keeps spherial symmetry exatly on an equal-angle-one gri in ylinrial oorinates is the areaweighte metho [23, 2, 2, 22, 3, 4]. In this approah one uses a Cartesian form of the momentum equation in the ylinrial oorinate system, hene integration is performe on area rather than on the true volume in ylinrial oorinates. However, these areaweighte shemes have a flaw in that they may violate momentum onservation. Margolin 2

3 an Shashkov use a urvilinear gri to onstrut symmetry-preserving isretiations for Lagrangian gas ynamis [6]. In our reent work [8], we have evelope a new ell-entere ontrol volume Lagrangian sheme for solving Euler equations of ompressible gas ynamis in two-imensional ylinrial oorinates. Base on the strategy of loal oorinate transform an a areful treatment of the soure term in the momentum equation, the sheme is esigne to be able to preserve one-imensional spherial symmetry in a two-imensional ylinrial geometry when ompute on an equal-angle-one initial gri. A istinguishe feature of our sheme is that it an keep both the symmetry an onservation properties on the straight-line gri. However, our sheme in [8] oes not satisfy the geometri onservation law (GCL). In [4], Maire evelope a lass of high orer ell-entere Lagrangian shemes for solving Euler equations of ompressible gas ynamis in ylinrial oorinates. A noe-base isretiation of the numerial fluxes is given whih makes the finite volume sheme ompatible with the geometri onservation law. Both the ontrol volume an area-weighte isretiations of the momentum equations are presente in [4]. The ontrol volume sheme is onservative for mass, momentum an total energy, an satisfies a loal entropy inequality in its first-orer semi-isrete form. However, it oes not preserve spherial symmetry. On the other han, the first orer area-weighte sheme is onservative for mass an total energy an preserves spherial symmetry for one-imensional spherial flow on equal-angle polar gri, but it annot preserve the momentum onservation an oes not satisfy the entropy inequality. Numerial tests are given in [4] whih verify the robustness of the shemes. In this paper, we attempt to apply the strategy propose in [8] on Maire s first orer ontrol volume Lagrangian sheme [4] to improve its property in symmetry preservation while keeping its main original goo properties inluing GCL, onservation of mass, momentum an total energy. An outline of the rest of this paper is as follows. In Setion 2, we esribe the moifie sheme an isuss some ritial issues suh as GCL, onservation an spherial symmetry 3

4 preservation about the sheme. In Setion 3, numerial examples are given to emonstrate the performane of the new moifie ell-entere Lagrangian sheme. In Setion 4 we will give onluing remarks. 2 The improvement on the ell-entere ontrol volume Lagrangian sheme of Maire in ylinrial oorinates 2. The ompressible Euler equations in a Lagrangian formulation in ylinrial oorinates The ompressible invisi flow is governe by the Euler equations whih have the following integral form in the Lagrangian formulation t ρv = Ω(t) t Ω(t) ρuv = Pns Γ(t) (2.) ρev = Pu ns t Ω(t) Γ(t) where ρ is the ensity, P is the pressure, u is the vetor of veloity, E is the speifi total energy, an n is the unit outwar normal to the bounary Γ(t). The geometri onservation law refers to the fat that the rate of hange of a Lagrangian volume shoul be ompute onsistently with the noe motion, whih an be formulate as V = u ns. (2.2) t Ω(t) In this paper, we seek to stuy the axisymmetri ompressible Euler system. Its speifi form in the ylinrial oorinates is as follows ρrr = t Ω(t) t Ω(t) rr = u nrl Γ(t) ρu t Ω(t) rr = Pn Γ(t) rl t Ω(t) ρu rrr = Pn Γ(t) rrl + Pr Ω(t) ρerr = Pu nrl t Ω(t) Γ(t) Γ(t) (2.3) where an r are the axial an raial iretions respetively. u = (u, u r ), where u, u r are the veloity omponents in the an r iretions respetively, an n = (n, n r ) is the unit outwar normal to the bounary Γ(t) in the -r oorinates. 4

5 The set of equations is omplete by the aition of an equation of state (EOS) with the following general form P = P(ρ, e) (2.4) where e = E 2 u 2 is the speifi internal energy. Espeially, if we onsier the ieal gas, then the equation of state has a simpler form P = (γ )ρe where γ is a onstant representing the ratio of speifi heat apaities of the flui. In the next subsetion, we will first summarie the ontrol volume sheme of Maire in [4]. 2.2 The ontrol volume sheme of Maire in ylinrial oorinates 2.2. Notations an assumptions We will mostly use the notations in [4]. The 2D spatial omain Ω is isretie into quarangular omputational ells. Eah quarangular ell is assigne a unique inex, an is enote by Ω (t). The bounary of the ell Ω is enote as Ω. Eah vertex of the mesh is assigne a unique inex p an we enote the ounterlokwise orere list of the verties of the ell Ω by p(). The ell Ω is surroune by four ells enote as Ω b, Ω r, Ω t, Ω l whih orrespon to the bottom, right, top an left positions respetively. A enotes the area of the ell Ω. V is the volume of the ell, that is, the volume of the irular ring obtaine by rotating this ell aroun the aimuthal -axis (without the 2π fator). Using these notations, the set of equations (2.3) an be rewritten in the following ontrol 5

6 volume formulation ( ) m = u nrl t ρ Ω m t u = Pn rl Ω m t ur = m t E = Pn r rl + Pr Ω Ω Ω Pu nrl (2.5) where m = Ω ρrr enotes the mass in the ell Ω, whih keeps onstant uring the time marhing aoring to the first equation in (2.3). ρ, u = (u, ur ) an E represent the ensity, veloity an total energy of the ell Ω whih are efine as follows ρ = ρrr, V Ω u = ρu rr, m Ω u r = ρu r rr, m Ω E = ρerr. m Ω The oorinates an veloity of the point p are enote as ( p, r p ) an u p = (u p, ur p ) respetively. l pp an l pp + enote the lengths of the eges [p, p] an [p, p + ], an n pp an n pp + are the orresponing unit outwar normals, where p, p + are the two neighboring points of the point p (see Figure 2.). In the ell-entere ontrol volume Lagrangian sheme presente in [4], the isrete graient operators over the ell Ω are onstrute by introuing two noal pressures at eah noe p of the ell Ω. These pressures are enote as πp an π p, see Figure 2.. They are relate to the two eges sharing the noe p. The half lengths an the unit outwar normals of the eges onnete to the point p are enote as follows l p = 2 l pp, l p = 2 l pp +, n p = n pp, n p = n pp +. (2.6) 6

7 u n p l p p p p r p p p l p (, u, p) r p n p p Figure 2.: Notations relate to the ell Ω. The pseuo raii r p an r p are efine as r p = 3 (2r p + r p ), r p = 3 (2r p + r p +). (2.7) Computation of noal veloity an pressure In the paper [4], the speifi way in etermining the noal veloity an pressure guarantees that the sheme satisfies the following suffiient onition for total energy onservation (p) (r pl pπ pn,r p where (p) is the set of the ells aroun the point p. + rpl pπ pn,r p ) =, (2.8) The speifi formulas to alulate the noal veloity u p an noal pressures π p an π p are as follows u p = M p (rp l p n p + r p l p n p )P + u, (p) P π p = p (u p u ) n p, P π p = p (u p u ) n p, (2.9) where P is the pressure of the ell Ω etermine by {ρ,u, E }. The 2 2 matries M p an M p are enote as M p = p r p l p (n p n p ) + p r p l p (n p n p ), M p = (p) M p (2.) 7

8 where p an p are the mass fluxes swept by the waves whih an be etermine in several ways, suh as the Dukowi approah or the aousti approah. We refer the reaer to the paper [4] for more etails. In this paper, we will only use the aousti approah, that is p = p = ρ a. (2.) where a is the loal isentropi spee of soun. After we get the noal veloity u p at point p, the point moves with the following loal kinemati equation t x p = u p, x p () = x p, (2.2) where x p = ( p, r p ) efines the position of point p at t > an x p enotes its initial position Spatial isretiation The semi-isrete finite volume sheme of the governing equations (2.5) is of the following form ( ) m (r t ρ pl pn p + rpl pn p) u p =, p p() m t u + p p() m t ur + p p() m t E + p p() (r p l p π p n, p (r p l p π p n,r p + r p l p π p n, p ) =, + r p l p π p n,r p ) A P =, (r pl pπ pn p + r pl pπ pn p) u p = (2.3) where n, p, n, p respetively. an n,r p, n,r p are the omponents of n p an n p along the an r iretions 8

9 2.2.4 Time isretiation The time isretiation for the equation of noal movement (2.2) is the Euler forwar metho given as follows n+ p r n+ p = n p + t n u,n p = r n p + t n u r,n p (2.4) where u,n p, u r,n p are the an r omponents of u p at the nth time step. As a first orer sheme, the time marhing for the semi-isrete sheme (2.5) an also be aomplishe by the Euler forwar metho. Thus the fully isretie sheme an be written as follows m ρ n+ ρ n u,n+ u r,n+ E n+ u,n u r,n E n p p() = t n p p() (r,n (r,n p p() (r,n (2.5) p l,n p n,n p p lp,n πp,n n,,n p p lp,n πp,n n,r,n p p p() (r,n p l,n p π,n p n,n p + r,n p l,n p n,n p ) un p + r,n p l,n p π,n p n,,n p ) p ) + r,n p l,n p π,n p n,r,n + r,n p l,n p π,n p n,n p ) un p + A n P n Here the variables with the supersripts n an n + represent the values of the orresponing variables at the nth an (n + )th time steps respetively. The sheme (2.5) is onsistent with the Euler equations (2.5) an has first orer auray in spae an time. In [4], the time step t n is ontrolle by both the CFL onition an the riterion on the variation of volume, that is, the CFL onition is satisfie as follows t e = C e min (l n /an ) where l n is the shortest ege length of the ell Ω, an a n is the soun spee within this ell. C e is the Courant number whih is set to be.5 unless otherwise state in the following tests. The riterion on the variation of volume is aomplishe as { } V n t v = C v V t (t n ) 9

10 where V t (t n ) = V n+ t n V n. The parameter C v =. is use in the numerial simulations. Finally, the next time step t n+ is given by t n+ = min( t e, t v, C m t n ) where C m = The improvement of the sheme on the symmetry property The ontrol volume sheme (2.5) is theoretially proven to have many goo properties suh as the onservation, GCL an entropy inequality, an is also verifie to have goo performane in pratial simulations. However, it has a limitation in not being able to preserve the spherial symmetry. In this paper, we attempt to improve the sheme in this aspet using the strategy of our reent work [8]. Somewhat ifferently from the approah of the expliit loal oorinate transform in [8], here we perform the improvement on the sheme (2.3) in -r oorinates iretly an use loal oorinate transform only in the symmetry-preserving proof but not in the atual implementation of the algorithm. In fat, the key ingreient to make the sheme satisfy the spherial symmetry property in a 2D ylinrial geometry is the treatment of the soure term. To be more speifi, we replae P in the soure term of the r-momentum equation in (2.3) by P a whih is efine as follows P a = 4 (π + π 2 + π 3 + π 4 ) (2.6) where π, π 2, π 3 an π 4 are the values of pressure relate to the two raial eges of the ell Ω (see Figure 2.2). We an easily see that A P a approximates Ω Pr with the same orer as A P. Thus, the ifferene between the moifie sheme an the original sheme lies only in the expression of the r-momentum equation. In summary, the moifie semi-isrete sheme

11 r 2, u2,p2, u,p 3, u,p 4 l t b r 2 Figure 2.2: Equi-angular polar gri for the ylinrial geometry. an be expresse as follows ( ) m (rp t l p n p + r p l p n p ) u p =, ρ m t u + p p() m t ur + p p() m t E + p p() p p() (r p l p π p n, p (r p l p π p n,r p + r p l p π p n, p ) =, + r p l p π p n,r p ) A P a =, (r p l p π p n p + r p l p π p n p ) u p =. (2.7) The fully isretie moifie sheme is of the following expression m ρ n+ ρ n u,n+ u r,n+ E n+ u,n u r,n E n p p() = t n p p() (r,n (r,n p p() (r,n (2.8) p l,n p n,n p p l,n p π,n p n,,n p p lp,n πp,n n,r,n p p p() (r,n p lp,n πp,n n,n p + r,n p l,n p n,n p ) un p + r,n p l,n p π,n p n,,n p ) p ) + r,n p l,n p π,n p n,r,n + r,n p l,n p π,n p n,n p ) un p The issue of onservation, GCL an entropy inequality A n Pa n Compare with the sheme (2.5) of Maire [4], we only use an alternative approah (2.6) to the pressure appearing in the soure term of the r-momentum equation to guarantee the sheme s spherial symmetry preservation property that will be shown in the following

12 setion. Sine we o not make any hange on any other terms in the original sheme (2.5), the moifie sheme (2.8) an also satisfy the onservation for the mass, -momentum an total energy just as the original sheme. As to the onservation of the r-momentum, summing the r-momentum equation in (2.7) over all the ells, we have ( ) m u r = (r t pl pπ pn,r p + rpl pπ pn,r p ) A P a (2.9) p p() Swithing the summation over ells an the summation over noes at the right-han sie of Equation (2.9), it an be rewritten as ( ) m u r = t p (rp l p π p n,r p (p) + r p l p π p n,r p ) + A P a (2.2) Using the suffiient onition for total energy onservation (2.8), we have, ( ) m u r = A P a (2.2) t The equation (2.2) refers to the onservation of the r-momentum, whih, together with the onservation of the mass, -momentum an total energy, guarantees the moifie sheme (2.8) satisfies the Lax-Wenroff theorem. Sine the sheme (2.8) employs the same ompatible isretiation of the geometry onservation law (GCL) as that in the original sheme (2.5), it satisfies GCL naturally. As to the entropy inequality, sine we use the formulation of (2.6) to etermine the pressure in the soure term rather than P, we an not write a similar entropy inequality for the sheme (2.7) as that for the original sheme (2.3) The issue of symmetry preservation In this setion, we will prove the moifie sheme (2.8) an keep the spherial symmetry property ompute on an equal-angle-one initial gri. Theorem: The moifie sheme (2.8) an keep the one-imensional spherial symmetry property ompute on an equal-angle one initial gri. That is, if the solution has one- 2

13 imensional spherial symmetry at the initial time, then the omputational solution will keep this symmetry with the time marhing. Proof: Without loss of the generality, we only nee to prove the solution of the moifie sheme (2.8) an keep the spherial symmetry at the (n + )th time step, if the solution is known to be of spherial symmetry at the nth time step. Notie that, for the Lagrangian solution, symmetry preserving refers to the evolution of both the onserve variables an the gri. To failitate the proof, we first simplify the vertex inies of the ell Ω() as p =, 2, 3, 4 (shown in Figure 2.2) an rewrite the momentum equations in (2.7) along the ell s loal ξ-θ oorinates. The rewritten semi-isrete sheme is of the following form ( ) m = (r t plpn p + rpl pn p) u p, ρ p=,4 m t uξ = (rpl pπ pn,ξ p p=,4 m t uθ = (rp l p π p n,θ p p=,4 + r pl pπ pn,ξ p ) + A P a sin θ, + r p l p π p n,θ p ) + A P a osθ, m t E = (rp l p π p n p + r p l p π p n p ) u p (2.22) p=,4 where ξ is the raial iretion passing through the ell enter an the origin. For an equalangle-one gri, the ell shown in Figure 2.2 is an equal-sie trapeoi, it has the property that the angles between ξ an the two equal sies of the ell are the same. θ is the angular iretion whih is orthogonal to ξ, see Figure 2.2. u ξ an uθ are the omponent values of veloity in the loal ξ an θ iretions respetively, n,ξ p, n,ξ p an n,θ p, n,θ p are the omponents of n p an n p along ξ an θ iretions respetively. θ is set to be the angle between the loal ξ iretion an the oorinate. For the onveniene of notation, we aopt the onvention that variables without the supersript n + are those at the nth time step. Assume that at the nth time step the gri is a polar gri with equal angles (see Figure 2.2) an the ell averages of the onserve variables inluing ensity, momentum an total energy are symmetrial on this gri, namely 3

14 these variables in the ells with the same raial position are iential. Consier the ell Ω an its neighboring ells, in the loal ξ-θ oorinates of the ell Ω, we have ρ = ρ t = ρ b = ρ, ρ l = ρ, ρ r = ρ 2, P = P t = P b = P, P l = P, P r = P 2, ξ = ξ 4 = ξ l, ξ 2 = ξ 3 = ξ r, u = (u, ), u l = (u, ), u r = (u 2, ) (2.23) where {ξ k, k =, 4} are the istane of the four verties of the ell Ω from the origin respetively, an {u k, k =, 2} are the magnitue of the ell veloity in the relevant ells. Next we explain some notations onerning the gri geometry. For the ell Ω, the lengths of the ell eges are enote as {l 2, l 23, l 34, l 4 }. Sine the gri is symmetrial, we an efine them as l 2 = l 34 = l m, l 4 = l l, l 23 = l r. (2.24) For the onveniene of proof, in the following, we will projet all the variables relative to the etermination of the ell s an noal veloity to the ell s loal ξ-θ oorinates. For example, the outwar normal iretion of the ell s four eges in the loal ξ-θ oorinates are as follows n 2 = (n ξ 2, nθ θ 2 ) = ( sin 2 n 23 = (n ξ 23, n θ 23) = (, ), n 34 = (n ξ 34, nθ θ 34 ) = ( sin 2 n 4 = (n ξ 4, n θ 4) = (, )., os θ 2 ),, os θ 2 ), As to the symmetry of the gri, in the ase of a one-imensional spherial flow ompute on an equal angle polar gri, Maire in the paper [4] has alreay given the proof that the noal veloity is raial an inepenent of the angular ell position whih means the symmetry preservation of the gri. From the results of the vertex veloity shown in [4], we an eue that the veloities at eah vertex for the ell Ω in its loal ξ-θ oorinates are 4

15 of the following form u ξ = u ξ 4 = u + u (P P ) +, u θ = uθ 4 = u + u (P P ) + u ξ 2 = u ξ 3 = u + 2 u 2 (P 2 P ), + 2 u θ 2 = u θ 3 = u + 2 u 2 (P 2 P ) + 2 tan θ 2, tan θ 2 (2.25) where = ρ a, = ρ a an 2 = ρ 2 a 2. {a, a, a 2 } are the spees of soun in the ells {Ω l, Ω, Ω r } respetively. Next we will prove the symmetry preservation of the evolve variables suh as ensity, ell veloity an total energy. We will first write eah variable appearing at the right-han sie of (2.22) in etails. By the following simple manipulation, the noal pressures an be obtaine where u = (u, ), u = (u ξ, uθ ). Similarly π = P (u u ) n 4 = P + P (u u ) +, π = P (u u ) n 2 = P u sin θ 2, (2.26) π 2 = π 3 = π 4 = π = P u sin θ 2, π 4 = π = P + P (u u ) +, π 2 = π 3 = 2P + P 2 2 (u 2 u ) + 2. (2.27) 5

16 For the simpliity of esription, we enote By the formula (2.6), we have P m = P u sin θ 2, P l = P + P (u u ) +, P r = 2P + P 2 2 (u 2 u ) + 2, u l = u + u (P P ) +, u r = u + 2 u 2 (P 2 P ) + 2. P a = P m = P u sin θ 2 (2.28) The other orresponing variables appearing at the right han sie of (2.22) an be esribe in the following etails l = l 4 = 2 l l, l = l 2 = l 3 = l 4 = 2 l m, l 2 = l 3 = 2 l r, (n,ξ, n,θ ) = (n,ξ 4, n,θ 4 ) = n 4 = (, ), (n,ξ 2, n,θ 2 ) = (n,ξ, n,θ ) = n 2 = ( sin θ 2 (n,ξ 3, n,θ 3 ) = (n,ξ 2, n,θ 2 ) = n 23 = (, ), (n,ξ 4, n,θ 4 ) = (n,ξ 3, n,θ ) = n 3 34 = ( sin θ 2 r + r 2 = (ξ l + ξ r ) sin(θ θ 2 ), r 3 + r 4 = (ξ l + ξ r ) sin(θ + θ 2 ), r 2 + r 3 = 2 sin θ os θ 2 ξ r,, os θ 2 ),, os θ 2 ), r + r 4 = 2 sin θ os θ 2 ξ l. (2.29) 6

17 Substituting ( ) into (2.22), we get /ρ m u ξ t u θ E = = = sin θ sin θ os θ(ξ 2 rl r u r ξ l l l u l ) sin θ [ (ξ 2 l + ξ r )l m P m sin θ + ξ l l l P l ξ r l r P r ] + A P m sin θ (ξ 2 l + ξ r )l m P m sin θ os θ + A P m osθ sin θ os θ(ξ 2 rl r u r P r ξ l l l u l P l ) sin θ os θ (ξ 2 rl r u r ξ l l l u l ) sin θ ( ξ r l r P r + ξ l l l P l + 2A P m ) A P m osθ + A P m osθ sin θ os θ(ξ 2 rl r u r P r ξ l l l u l P l ) os θ 2 (ξ rl r u r ξ l l l u l ) (ξ r l r P r ξ l l l P l ) + 2A P m os θ 2 (ξ rl r u r P r ξ l l l u l P l ) Sine the ell is an equal-sie trapeoi, we have. (2.3) m = ρ V = ρ r A = ρ ξ A sin θ (2.3) where r an ξ are the values of r an ξ at the ell enter respetively. Thus from (2.3) an (2.3), we have /ρ u ξ t u θ = ρ ξ A E os θ 2 (ξ rl r u r ξ l l l u l ) (ξ r l r P r ξ l l l P l ) + 2A P m os θ 2 (ξ rl r u r P r ξ l l l u l P l ) (2.32) Finally we obtain the moifie sheme (2.8) in the following etaile expression /ρ n+ /ρ os θ u ξ,n+ u θ,n+ = u ξ u θ + t (ξ 2 rl r u r ξ l l l u l ) (ξ r l r P r ξ l l l P l ) + 2A P m E n+ ρ ξ A (2.33) E os θ (ξ 2 rl r u r P r ξ l l l u l P l ) From the formula (2.33), we an see the ell veloity is raial an the magnitue of all the onserve variables is inepenent of the angular position of the ell at the (n + )th time step. The proof of the symmetry preservation property is thus omplete. 7

18 3 Numerial results in the two-imensional ylinrial oorinates In this setion, we perform numerial experiments in two-imensional ylinrial oorinates. Purely Lagrangian omputation, the ieal gas with γ = 5/3, the initially equal-angle polar gri an the moifie sheme (2.8) are use in the following tests unless otherwise state. Refletive bounary onitions are applie to the an r axes in all the tests. For the veloity of verties loate at the oorinate, we obtain it by imposing the bounary onition of ero normal veloity into the solver (2.9). 3. Auray test We test the auray of the moifie sheme (2.8) on a free expansion problem given in [2]. The initial omputational omain is [, ] [, π/2] efine in the polar oorinates. The gas is initially at rest with uniform ensity ρ = an pressure has the following istribution, p = ( 2 + r 2 ). The analytial solution of the problem is as follows, R(t) = + 2t 2, u ξ (, r, t) = 2t + 2t r 2, ρ(, r, t) = R 3, p(, r, t) = ( ) 2 + r 2 R 5 where R is the raius of the free outer bounary an u ξ represents the value of veloity in the raial iretion. We perform the test both on an initially equal-angle polar gri an a ranom polar gri, see Figure 3.. For the ranom polar gri, eah internal gri point is obtaine by an inepenent ranom perturbation on the angular iretion from a equal-angle polar gri R 2 8

19 r r Figure 3.: The initial gri of the free expansion problem with 2 2 ells. Left: equal-angle polar gri; Right: ranom polar gri. whih an be expresse as follows { (l ) θ, k =, K or l =, L θ k,l = (l ) θ + k,l θ, else k,l = k K os θ k,l, r k,l = k K sin θ k,l, k K, l L where ( k,l, r k,l ) is the -r oorinate of the gri points with the sequential inies (k, l), k =,..., K, l =,..., L in the raial an angular iretions respetively. K, L represent the number of gri points in the above mentione two iretions. θ = is the ranom number, is a parameter whih is hosen as.5 in this test. π..5 2(L ) k,l.5 Free bounary onition is applie on the outer bounary. Figure 3.2 shows the final gri. We an learly observe symmetry in the left figure. The errors of the sheme on these two kins of gri at t = are liste in Tables whih are measure on the interval [ K, 4 K] [, L] to remove the influene from the bounary. From both of the tables, we 5 5 an see the expete first orer auray for all the evolve onserve variables. 3.2 Non-osillatory tests Example (The Noh problem in a ylinrial oorinate system on the polar gri [9]). We test the Noh problem whih is a well known test problem wiely use to valiate Lagrangian sheme in the regime of strong shok waves. In this test ase, a ol gas with unit 9

20 r r Figure 3.2: The gri of the free expansion problem with 2 2 ells at t =. Left: equal-angle polar gri; Right: ranom polar gri. Table 3.: Errors of the sheme in 2D ylinrial oorinates for the free expansion problem using K L initially equal-angle polar gri ells. K = L Norm Density orer Momentum orer Energy orer 2 L.97E-2.3E-.64E-2 L.6E-.9E-.E- 4 L.53E E E-2.9 L.77E-2.3.E E L.28E E E-2.94 L.36E E E L.5E E E-3.96 L.9E E E-2.93 Table 3.2: Errors of the sheme in 2D ylinrial oorinates for the free expansion problem using K L initially ranom polar gri ells. K = L Norm Density orer Momentum orer Energy orer 2 L.94E-2.2E-.62E-2 L.6E-.9E-.E- 4 L.5E E E-2.92 L.84E E E L.27E E E-2.93 L.47E E E L.4E E E-3.95 L.25E E E

21 r r Figure 3.3: The results of the Noh problem with 2 2 ells at t =.6. Left: the original sheme (2.5); Right: the moifie sheme (2.8). ensity an ero internal energy is given with an initial inwar raial veloity of magnitue. The equal-angle polar gri is applie in the -irle omputational omain efine in the 4 polar oorinates by [, ] [, π/2]. The shok is generate in a perfet gas by bringing the ol gas to rest at the origin. The analytial post shok ensity is 64 an the shok spee is /3. The omparison of the final gri with 2 2 ells between the original sheme (2.5) an the moifie sheme (2.8) is given in Figure 3.3 whih emonstrates the improvement of the symmetry property for the moifie sheme. Figure 3.4 shows the results of the moifie sheme inluing the final gri with 2 2 ells an ensity as a funtion of raial raius for two ifferent angular onings (2 2, 2 4) at t =.6. From Figure 3.4, we observe the results are symmetrial an non-osillatory. The shok loation an the shok magnitue are loser to those of the analytial solution with the gri refinement in the angular iretion, whih reflets the onvergene tren of the numerial solution towar the analytial solution. Example 2 (The spherial Seov problem in a ylinrial oorinate system on the polar gri [2]). We perform our test on the spherial Seov blast wave problem in a ylinrial oorinate system as an example of a iverging shok wave. The initial omputational omain is a -irle region efine in the polar oorinates by [,.25] [, π/2]. The initial onition 4 onsists of unit ensity, ero veloity an ero speifi internal energy exept in the ells 2

22 r r r r.4 6 exat numerial 2*2 numerial 2* ensity raius Figure 3.4: The results of the Noh problem at t =.6. Left: initial gri with 2 2 ells; Mile: final gri with 2 2 ells; Right: ensity vs raial raius with 2 2 an 2 4 ells respetively. Soli line: exat solution; ashe line: omputational solution Figure 3.5: The results of the Seov problem with 2 2 ells at t =.. Left: the original sheme (2.5); Right: the moifie sheme (2.8). onnete to the origin where they share a total value of Refletive bounary onition is applie on the outer bounary. The analytial solution is a shok at raius unity at time unity with a peak ensity of 4. Figure 3.5 shows the omparison of the final gri with 2 2 ells between the original sheme (2.5) an the moifie sheme (2.8). We an see that the latter has a perfet symmetry property. The final gri, ensity as a funtion of the raial raius an surfae of ensity with 3 ells obtaine by the moifie sheme are isplaye in Figure 3.6. We also observe the expete symmetry in the plot of gri. The shok position an peak ensity oinie with those of the analytial solution very well an there is no spurious osillation, emonstrating the goo performane of the sheme in symmetry preserving, non-osillation an auray properties. Example 3 (The one-imensional spherial So Riemann problem). 22

23 r 4 exat numerial ensity 2 ensity raius Figure 3.6: The result of the Seov problem with 3 ells at t =.. Left: final gri; Mile: ensity vs raial raius. Soli line: exat solution; ashe line: omputational solution; Right: surfae of ensity..5.5 The moifie sheme (2.8) is teste on the So Riemann problem in the ylinrial oorinates. The initial omputational omain is a -irle region efine in the polar 4 oorinates by [, 2] [, π/2]. Its initial onition is as follows (ρ, u ξ, p) = (.,,.), ξ (ρ, u ξ, p) = (.25,,.), < ξ 2. Refletive bounary onition is applie on the outer bounary. The referene solution is the onverge result obtaine by using a one-imensional seon-orer Eulerian oe in the spherial oorinate with gri points. Figure 3.7 shows the numerial results of the gri an ensity as a funtion of the raial raius an the surfae of ensity performe by the moifie sheme with 4 equal-angle polar ells at t =.4. We observe the goo behavior of the sheme in symmetry an the goo agreement between the numerial result an the referene solution. Example 4 (Kier s isentropi ompression problem [, 4]). This problem is a self-similar isentropi problem whih is usually use to valiate the apability of a Lagrangian sheme in simulating a spherial isentropi ompression. At the initial time, the shell with a ring shape onstitutes the omputational region [ξ, ξ 2 ] [, π/2] in the polar oorinates, where ξ =.9 is the internal raius an ξ 2 = is the external 23

24 r ensity referene numerial raius Figure 3.7: The results of the So problem at t =.4. Left: final gri with 4 ells; Mile: ensity versus raial raius. Soli line: exat solution; ashe line: omputational solution; Right: surfae of ensity in the whole irle region obtaine by a mirror image. raius. The initial ensity an pressure ρ, P are expresse as follows ( ξ 2 ρ (ξ) = 2 ξ 2 ξ2 2 ρ γ ξ2 + ξ2 ξ 2 ξ2 2 ξ2 ) ρ γ γ 2, P (ξ) = s(ρ (ξ)) γ where ρ = 6.3 4, ρ 2 = 2, s = 2.5 4, γ = 5/3. The pressure P (t) an P 2 (t) are impose ontinuously at the internal an external bounary respetively whih have the following representation P (t) = P 2γ a(t) γ, P2 (t) = P2 2γ a(t) γ, where P =., P 2 = an a(t) = ( t τ )2 in whih τ = is the fousing time of the shell an t [, τ) is the evolving time. Denoting ζ(ξ, t) to be the raius at time t of a point initially loate at raius ξ, its analytial solution is ζ(ξ, t) = a(t)ξ. The analytial solutions of three funamental variables for this problem in spherial geometry are as follows ρ(ζ(ξ, t), t) = ρ (ξ)ξa(t) 2 γ, u(ζ(ξ, t), t) = ξ t a(t), p(ζ(ξ, t), t) = P (ξ)ξa(t) 2γ γ. We test the moifie sheme (2.8) on the problem with 8 4, 6 8, 32 6 ells respetively. The final time is set to be t =.99τ. Figure 3.8 shows the initial an final 24

25 r r exat numerial raius time Figure 3.8: The results of the Kier problem with 4 2 gris. Left: initial gri; Mile: final gri at t =.99τ; Right: trajetory of external bounary ompare with the exat solution. Soli line: exat solution; ashe line: omputational solution. 3.5 exat numerial 8*4 numerial 6*8 3 numerial 32* exat numerial 8*4 numerial 6*8 numerial 32*6 5 exat numerial 8*4 numerial 6*8 numerial 32* ensity raius veloity raius pressure raius Figure 3.9: The results of the Kier problem at t =.99τ with three ifferent onings. Left: ensity vs raial raius; Mile: veloity vs raial raius; Right: pressure vs raial raius. Soli line: exat solution; ashe line: omputational solution. gris an the time evolution of the position of the external bounary with 4 2 gris. Figure 3.9 shows the results of ensity, veloity an pressure at the final time. From these figures, we an see the perfet symmetry in the gri. The trajetory of the external bounary oinies with the analytial solution quite well. The numerial solutions of ensity, veloity an pressure onverge to the analytial solutions asymptotially. Example 5 (Implosion problem of Laarus [2]). The implosion problem of Laarus is a self-similar problem. At the initial time, a sphere of unit raius with unit ensity an ero speifi internal energy is riven by the following inwar raial veloity u ξ (t) = αf ( ft) α (3.) 25

26 r referene t=.74 referene t=.8 3 angular ells t= angular ells t=.8 6 angular ells t=.74 6 angular ells t= ensity 5 ensity raius Figure 3.: The results of the Laarus problem. Left: final gri with 2 3 ells at t =.8; Mile: ensity vs raial raius at t =.74,.8 with 2 3 an 2 6 ells respetively. Soli line: referene solution; ashe line: omputational solution; Right: surfae of ensity at t =.8 with 2 3 ells r.3.4 where α = , f = εt δt 3, ε =.85, δ =.28. We test the problem on a gri of 2 3 ells in the initial omputational omain [, ] [, π/2] efine in the polar oorinates. The numerially onverge result ompute using a one-imensional seon-orer Lagrangian oe in the spherial oorinate with ells is use as a referene solution. We isplay the results of the moifie sheme (2.8) using 2 3 an 2 6 ells in Figure 3.. In the plot of gri, we notie the expete symmetry. In the plot of ensity, we observe the non-osillatory an aurate numerial solution an the onvergene teneny of the numerial results towar the referene solution. Example 6 (Coggeshall expansion problem [9]). This is a two-imensional aiabati ompression problem propose by Coggeshall. We attempt to apply it to test the performane of the moifie sheme (2.8) on a truly twoimensional problem. The omputational omain onsists of a quarter of a sphere of unit raius one with ells. The initial ensity is unity an the initial veloity at the gri verties is given as (u, u r ) = ( /4, r). The speifi internal energy of a ell is given as e = (3 /8) 2, where is the oorinate of the ell enter. Figure 3. shows the results of the gri an ensity plotte as a funtion of the raial raius along eah raial line at the time of.8 when the analytial ensity is expete to be flat with a value of From the figures, we an observe the numerial result agrees with the analytial solution exept 26

27 r r ensity raius Figure 3.: The results of the Coggeshall problem at t =.8. Left: initial gri; Mile: final gri; Right: ensity versus the raial raius. for the small region near the origin. Example 7 (Spherial Seov problem on the Cartesian gri) We test the spherial Seov blast wave in a ylinrial oorinate system on the initially retangular gri. The initial omputational omain is a square onsisting of 3 3 uniform ells. The initial ensity is unity an the initial veloity is ero. The speifi internal energy is ero exept in the ell onnete to the origin where it has a value of Figure 3.2 shows the results of the original sheme (2.5) an the moifie sheme (2.8). From the figures, we an observe the results from the moifie sheme are more satisfatory an more symmetrial even on the non-polar gri. Example 8 (Spherial Noh problem on the Cartesian gri [9]). At last, we test the spherial Noh problem on a Cartesian gri to verify the robustness of the sheme. This problem is a very severe test for a Lagrangian sheme omputing on a Cartesian gri, sine in this ase the gri near the axes is easy to be istorte whih has been aresse in [5]. The initial omain is [, ] [, ]. The initial state of the flui is uniform with (ρ, u ξ, u θ, e) = (,,, 5 ), where u ξ, u θ are the raial an angular veloities at the ell enter. Refletive bounary onitions are applie on the left an lower bounaries. Free bounary onition is use on the right an upper bounary. The analytial solution is the same as that in Example. Figure 3.3 shows the results of the original sheme (2.5) an the moifie sheme (2.8) with 5 5 initially uniform retangular ells at t =.6. 27

28 r r exat numerial 3.5 exat numerial ensity 2.5 ensity raius.5.5 raius Figure 3.2: The results of the Seov problem with 3 3 gris at t =.. Left: the original sheme (2.5); Right: the moifie sheme (2.8). Top: final gri. Bottom: ensity vs raial raius.soli line: exat solution; symbols: omputational solution. From these figures, we an see that there is no gri istortion along the axes, the spherial symmetry is preserve better an the shok position is orret for the moifie sheme, whih emonstrate the robustness of the moifie sheme in this problem on the Cartesian gri. 4 Conluing remarks In this paper we apply the methoology propose in our previous work [8] on Maire s first orer ontrol volume Lagrangian sheme [4]. The purpose of this work is to improve the sheme s property in symmetry preservation while maintaining its original goo properties inluing geometri onservation law (GCL) an onservation of mass, momentum an total energy. The moifie sheme is proven to have one-imensional spherial symmetry in the two-imensional ylinrial geometry for equal-angle-one initial gris. Several twoimensional examples in the ylinrial oorinates have been presente whih emonstrate 28

29 r r r r Figure 3.3: Gri an ensity ontour for the Noh problem with 5 5 Cartesian ells at t =.6. Top: the original sheme (2.5); Bottom: the moifie sheme (2.8). Left: whole gri; Right: oom on the region with shok. the goo performane of the moifie sheme in symmetry, non-osillation an robustness. The improvement of the moifie sheme in auray onstitutes our future work. Referenes [] D.J. Benson, Momentum avetion on a staggere mesh, Journal of Computational Physis,, 992, [2] D.J. Benson, Computational methos in Lagrangian an Eulerian hyrooes, Computer Methos in Applie Mehanis an Engineering, 99, 992, [3] E.J. Caramana, D.E. Burton, M.J. Shashkov an P.P. Whalen, The onstrution of ompatible hyroynamis algorithms utiliing onservation of total energy, Journal of Computational Physis, 46, 998,

30 [4] E.J. Caramana, M.J. Shashkov an P.P. Whalen, Formulations of Artifiial Visosity for Multiimensional Shok Wave omputations, Journal of Computational Physis, 44, 998, [5] J.C. Campbell an M.J. Shashkov, A tensor artifiial visosity using a mimeti finite ifferene algorithm, Journal of Computational Physis, 72, 2, [6] J. Cheng an C.-W. Shu, A high orer ENO onservative Lagrangian type sheme for the ompressible Euler equations, Journal of Computational Physis, 227, 27, [7] J. Cheng an C.-W. Shu, A thir orer onservative Lagrangian type sheme on urvilinear meshes for the ompressible Euler equation, Communiations in Computational Physis, 4, 28, [8] J. Cheng an C.-W. Shu, A ell-entere Lagrangian sheme with the preservation of symmetry an onservation properties for ompressible flui flows in two-imensional ylinrial geometry, Journal of Computational Physis, 2, oi:.6/j.jp [9] S. V. Coggeshall an J. Meyer-ter-Vehn, Group invariant solutions an optimal systems for multiimensional hyroynamis, Journal of Mathematial Physis, 33, 992, [] B. Després an C. Maeran, Lagrangian gas ynamis in two imensions an Lagrangian systems, Arhive for Rational Mehanis an Analysis, 78, 25, [] R.E. Kier, Laser-riven ompression of hollow shells: power requirements an stability limitations, Nulear Fusion,, 976, 3-4. [2] R. Laarus, Self-similar solutions for onverging shoks an ollapsing avities, SIAM Journal on Numerial Analysis, 8, 98,

31 [3] W. Liu, J. Cheng, C.-W. Shu, High orer onservative Lagrangian shemes with Lax- Wenroff type time isretiation for the ompressible Euler equations, Journal of Computational Physis, 228, 29, [4] P.-H. Maire, A high-orer ell-entere Lagrangian sheme for ompressible flui flows in two-imensional ylinrial geometry, Journal of Computational Physis, 228, 29, [5] P.-H. Maire, R. Abgrall, J. Breil an J. Ovaia, A ell-entere Lagrangian sheme for ompressible flow problems, SIAM Journal of Sientifi Computing, 29, 27, [6] L.G. Margolin an M.J. Shashkov, Using a urvilinear gri to onstrut symmetrypreserving isretiations for Lagrangian gas ynamis, Journal of Computational Physis, 49, 999, [7] C.D. Mun, On Gounov-type shemes for Lagrangian gas ynamis, SIAM Journal on Numerial Analysis, 3, 994, [8] J. von Neumann an R.D. Rihtmyer, A metho for the alulation of hyroynamis shoks, Journal of Applie Physis, 2, 95, [9] W.F. Noh, Errors for alulations of strong shoks using an artifiial visosity an an artifiial heat flux, Journal of Computational Physis, 72, 987, [2] W.D. Shul, Two-imensional Lagrangian hyroynami ifferene equations, Methos of Computational Physis, 3, 964, -45. [2] L.I. Seov, Similarity an Dimensional Methos in Mehanis, Aaemi Press, New York, 959. [22] A. Solov ev an M. Shashkov, Differene sheme for the Dirihlet partile metho in ylinrial metho in ylinrial oorinates, onserving symmetry of gas-ynamial flow, Differential Equations, 24, 988,

32 [23] M.L. Wilkins, Calulation of elasti plasti flow, Methos of Computational Physis, 3, 964,