Positivity-preserving and symmetry-preserving Lagrangian schemes for. compressible Euler equations in cylindrical coordinates.

Transcription

1 Postvty-reservng and symmetry-reservng Lagrangan shemes for omressble Euler equatons n ylndral oordnates Dan Lng, Juan Cheng and Ch-Wang Shu 3 Abstrat For a Lagrangan sheme solvng the omressble Euler equatons n ylndral oordnates, two mortant ssues are whether the sheme an mantan sheral symmetry (symmetry-reservng) and whether the sheme an mantan ostvty of densty and nternal energy (ostvty-reservng). Whle there were revous results n the lterature ether for symmetry-reservng n the ylndral oordnates or for ostvty-reservng n artesan oordnates, the desgn of a Lagrangan sheme n ylndral oordnates, whh s hgh order n one-dmenson and seond order n two-dmensons, and an mantan both sheral symmetry-reservaton and ostvty-reservaton smultaneously, s hallengng. In ths aer we desgn suh a Lagrangan sheme and rovde numeral results to demonstrate ts good behavor. Keywords: Lagrangan method; ylndral oordnates; symmetry-reservng; ostvtyreservng; omressble flows Graduate Shool, Chna Aademy of Engneerng Physs, Bejng 00088, Chna. E-mal: lngdan89@6.om. Researh s suorted n art by Sene Challenge Projet. No. JCKY06A50. Insttute of Aled Physs and Comutatonal Mathemats, Bejng 00088, Chna. E-mal: heng juan@am.a.n. Researh s suorted n art by NSFC grants and U63047 and Sene Challenge Projet. No. JCKY06A50. 3 Dvson of Aled Mathemats, Brown Unversty, Provdene, RI 09. E-mal: shu@dam.brown.edu. Researh s suorted n art by ARO grant W9NF and NSF grant DMS

2 Introduton There are two dfferent and tyal frameworks to desrbe the moton of flud flow, namely the Euleran framework and the Lagrangan framework. When we menton the latter framework, we refer to the knemat desrton whh onsders a tme deendent referene frame that follows the flud moton. Lagrangan methods are more sutable for roblems nvolvng nterfaes between materals or free surfaes and are wdely aled n many felds of mult-materal flow smulatons suh as n astrohyss or n nertal onfnement fuson (ICF). In these alatons, there often exst sheral-symmetr models suh as shere-shae asules. When suh models are smulated by Lagrangan methods n ylndral oordnates, t s a rtal and hallengng ssue to reserve the sheral symmetry n the ylndral oordnate system whh s dstnt from that symmetry. For examle, n the smulaton of an mloson roblem wth strong omressons, the reservaton of sheral symmetry s very mortant, sne the small devaton from sheral symmetry due to numeral errors may be amlfed by Raylegh-Taylor or other nstabltes whh may otentally rodue unredtably large errors. Earler strateges to desgn shemes n two-dmensonal ylndral oordnates to reserve sheral symmetry often sarfe momentum and energy onservaton, or at least momentuonservaton. In [3], a ell-entered Lagrangan sheme was develoed based on the ontrol volume dsretzaton. By dsretzng the soure tern the momentum equaton omatbly, the sheme was desgned to reserve one-dmensonal sheral symmetry n a two-dmensonal ylndral geometry usng an equal-angel-zoned grd wthout losng onservaton. Based on the frst order ontrol volume sheme of Mare n [], Cheng and Shu aled the methodology n [3] to obtan the sheral symmetry roerty. Ths modfed sheme an kee several good roertes, suh as symmetry, onservaton and the geometr onservaton law (GCL). In order to get hgher than frst-order symmetry-reservng shemes, Cheng and Shu n [6] resented a seond-order ell-entered Lagrangan sheme for solvng Euler equatons of omressble gas dynams n ylndral

3 oordnates. Ths sheme not only reserves symmetry but also reserves the onservaton for mass, momentum and total energy as well as the GCL. Another mortant ssue n omutatonal flud dynams s the ostvty-reservng roerty. As n a onservatve aroxmaton to Euler equatons where densty, momentum and total energy are solved dretly, the knet energy s omuted from mass and momentum and then subtrated from the total energy to rovde nternal energy. Therefore, at hgh Mah numbers, the nternal energy aears as a small dfferene of two large quanttes, and s rone to large erentage errors. It may easly beome negatve numerally whh may lead to nonlnear nstablty and a falure of the numeral sheme. To overome ths dffulty, many frst order ostvty-reservng Euleran shemes were develoed n earler years, for nstane, the lassal Godunov sheme [8], Lax-Fredrhs sheme [6, 0], the modfed HLLE sheme [8] and the HLLC sheme [] and so on. Some of them are desgned also u to seond order auray. Reently, Zhang and Shu roosed a general framework of hgh-order ostvty-reservng Euleran shemes suh as the Runge-Kutta dsontnuous Galerkn (RKDG) methods and the weghted essentally non-osllatory (WENO) fnte volume shemes n [0,, ]. Comared wth Euleran methods, ostvty-reservng Lagrangan shemes are less nvestgated. The oneerng work on ths ssue nludes the ostvty-reservng Godunovtye sheme based on the modfed HLL Remann solver [4], and the ostve and entro shemes [9]. In [5], Cheng and Shu onstruted hgh order ostvty-reservng Lagrangan shemes n one- and n two-dmensonal saes by develong an HLLC Remann solver and alyng the Zhang-Shu ostvty-reservng framework. More reently, ell-entered hgh order ostvty-reservng Lagrangan shemes for omressble flows n both onedmensonal and two-dmensonal saes were resented by Vlar et al n [8, 9] relyng on the two-state solver. We remark that these shemes are desgned n artesan oordnates and for roblems wthout soure terms. For equatons n non-artesan oordnates and wth soure terms, ostvty-reservng s more dffult to aheve. Ths s eseally the ase 3

4 when symmetry-reservng must also be taken nto onsderaton. In ths aer, we wll fous on desgnng hgh order ell-entered Lagrangan shemes whh an aheve ostvty-reservng and symmetry-reservng roertes smultaneously. Ths s not a straghtforward ombnaton of the symmetry-reservng tehnque n [3, 6] and the ostvty-reservng tehnque n [5, 8, 9], sne the desgn of one tehnque must ensure that the other roerty s not lost. In the one-dmensonal ase, for the ostvty-reservng roerty, we make an addtonal tme ste onstrant by ontrollng the hange rate of the ontrol volume to aheve ths goal wth any defnton of ostve aoust medane, manly followng [5, 8, 9, ]. For the extenson to two-dmensons, the desgn and analyss are smlar, however the ostvty-reservng lmter must be arefully aled n order not to affet the sheral symmetry reservaton when omuted on an equal-angle-zoned grd. For ths urose, our sheme s based on the work of Cheng and Shu n [6] and makes a areful balane between the orgnal symmetry-reservng framework and the new ostvty-reservng modfaton, n order to make sure one does not affet the erformane of the other. The fnal sheme thus has both symmetry-reservng and ostvty-reservng roertes, as well as the GCL and onservaton roertes. The remander of ths aer s organzed as follows: In Seton, we frst formulate the omressble Euler equatons n ylndral oordnates, desrbe the two-state Remann solver, and then desgn the frst order and hgh order ostvty-reservng Lagrangan shemes n ths one-dmensonal ase. In Seton 3, we show how to extend the ostvtyreservng tehnque to two-dmensonal ylndral oordnates wthout destroyng the o- rgnal sheral symmetry reservaton. In Seton 4, one- and two-dmensonal numeral examles are gven to verfy the erformane of our ostvty-reservng and symmetryreservng Lagrangan shemes. In Seton 5, we wll make some onludng remarks. 4

5 One-dmensonal ase The Euler equatons for gas dynams n one-dmensonal ylndral oordnates an be gven by the followng ntegral forn the Lagrangan framework d ρrdr = 0, dt Ω(t) d ρurdr = (r) Γr (t) + (r) Γl (t) + dt Ω(t) d ρerdr = (ru) Γr (t) + (ru) Γl (t) dt Ω(t) Ω(t) dr, (.) where r > 0 denotes the radal dreton, ρ s densty, u s veloty, s ressure and E s sef total energy, Γ l (t) and Γ r (t) are the left and rght boundares of the ontrol volume Ω(t). The system (.) resents the onservaton of mass, momentum and total energy. The set of equatons s omleted by the addtonal equaton of state (EOS), whh has the followng general form = (ρ, e), wth the sef nternal energy e = E u. The thermodynam sound seed for the flud flow s defned as a = ρ s =. s ρ. Frst-order sheme Let I = [r, r + ] be the ell, r = r + r be the sze of the ell and = I ρrdr be the mass n the ell I, whh kees a onstant value durng the tme marhng aordng to the frst equaton n (.). Then we ntrodue the ell averaged value n the ell I as U = (ρ, u, E ), n whh the averaged values of densty, veloty and total energy are defned as follows ρ = V I ρrdr, u = I ρurdr, E = I ρerdr, (.) where V = I rdr denotes the volume of the ell obtaned by rotatng the ell I around the orgn of the oordnate (wthout the π fator). 5

6 Based on these notatons, we an rewrte the systen (.) n the followng ontrol volume formulaton ρ = V, d dt u = (r) r=r+/ + (r) r=r / + d dt E = (ru) r=r+/ + (ru) r=r /. I dr, (.3) Note that (.3) s satsfed by the exat soluton of the artal dfferental equatons (PDEs) (.) and s not a sheme yet. However, when the ont values at the rght-hand sde n (.3) are aroxmated usng the ell averages (.), t wll beome a sheme whh evolves these ell averages as well as moves the mesh. Moreover, we an get the fully dsrete fnte volume Lagrangan sheme by usng Euler forward tme dsretzaton whh s as follows, ρ n+ = V n+, u n+ = u n tn (r + r + ) + tn r P s, E n+ = E n tn (r + u r + + u ), (.4) where P s s the aroxmaton of the soure term, artularly n the frst order ase t an be taken as n. The nterell values + and u + are the ressure and veloty at the node r +, resetvely, obtaned from an exat or aroxmate Remann solver by gvng the left and rght states, whh are U n and U n + n the frst order ase. The sheme (.4) s not omlete wthout the tme ntegraton of the trajetory equaton, whh enables us to advane n the tme the grd oston, the ell sze and volume as r n+ = r n + t n u, r n+ = r n+ r n+, + V n+ = rn+ (r n+ + + r n+ ). (.5) Thus the numeral sheme deends on the hoe of the numeral flux, whh s generally obtaned by exatly or aroxmately solvng the Remann roblem at the ell nterfae r + wth the gven left and rght states resetvely. In ths aer, we wll utlze the two-state 6

7 Remann solver roosed n [3, 8]. Sef formulas of ths Remann solver are gven n Aendx A, whh wll be used below. More detals an be found n [3, 8].. Frst-order ostvty-reservng sheme For the desred roerty, we defne the set of admssble states by ρ G = U = u, ρ > 0, e = E u > 0 and a = ρ s > 0. (.6) E Lemma.. The set of admssble states G s a onvex set, referrng to [5, 8, ]. The sheme (.4) s alled ostvty-reservng f {U n G, =,..., N} mles {U n+ G, =,..., N}. By addng and subtratng tn r n n n the seond equaton of (.4), the sheme (.4) an be rewrtten as U n+ = where ρ n+ u n+ E n+ = H + W, = V n+ u n tn (r + r + ) + tn m r n n + tn ) E n tn (r + + u + r u r n (P s n ) (.7) H = W = V n+ u n tn (r + r + E n tn (r + r V n+ u + + u n + tn r n (P s n ) E n. ) + tn r n n ) u, (.8) It s obvous that U n+ s a onvex ombnaton of H and W. To ensure that U n+ reserve the ostvty roerty, we ould onsder the suffent ondton that both H and W are ostvty-reservng. an 7

8 Before that, we would also want to obey the lassal CFL ondton, whh reads as t n λ mn r n a n = t, (.9) where λ s the Courant number (we take λ = 0.5 n our omutaton), and a n s the sound seed determned by ell averages. Meanwhle, n order to ensure that the ells wll not degenerate after the tme ste t n, we would add another restrton on the tme ste as follows t n λ mn ( (u r n u + ) rn ), + = t + ( u ) +, (.0) + where + = max(, 0) and λ < / (we take λ = 0.4 n our omutaton). Ths ondton guarantees the sze of eah ell r > 0 and the oston r + > 0, then V > 0, whh an ensure the ostvty of densty based on the frst equalty n the sheme (.4). Motvated by [8, 9], we ut an addtonal onstrant on the tme ste as t n mn r + u + σv n r u = mn ρ n σ r + u r + u = t 3, (.) where 0 < σ <, then we an ensure that the volume hanges at most by a fator σ from t n to t n+ and hoe to aheve ostve nternal energy by determnng the fator σ. Now we fous on fndng the requred ondton to ensure the nternal energy e(h) = E(H) (u(h)) > 0 n the smlar way as that n [8]. Detaled dervaton s omtted and we an determne the fator σ n (.) as σ mn(, restrton t n mn r z + + r + z + ρ n en ) and obtan another tme ste n = t 4, (.) for the sheme (.4). In artular, for the aoust solver ase,.e. z + = z = ρ n + a n, ths ondton wrtes t n rn, whh an be reovered by the lass CFL ondton (.9) a n when λ = 0.5. As for the suffent ondton for W G. Smlarly, we only need to solve the quadrat nequalty e(w) = E(W) (u(w)) 0, whh s guaranteed f ( t n mn u n µ + m ) (u n µ ) + e n = t 5, (.3) 8

9 wth µ = r n (P s n ). In artular, when we hoose P s as n for the frst order aroxmaton of the soure term, the ondton (.3) always holds for any t n. Theorem.. For the frst order Lagrangan sheme (.4) wth the numeral fluxes defned n the two-state Remann solver and any general ostve waveseeds defnton, assumng U n G, U n+ ondton s ensured to be n the admssble set G under the followng tme ste onstrant t n mn( t, t, t 3, t 4, t 5 ), (.4) where t s the lassal CFL ondton defned n (.9), t s defned by (.0), t 3 reads as that n (.) wth σ mn(, ρ n en ), t n 4 s defned n (.) and t 5 s defned n (.3) wth µ = r n (P s n ). Now, let us make a summary about the frst-order ostvty-reservng sheme (.4) and gve the algorthm flowhart for t at eah tme ste: () Assumng U G at the n-th tme level, omute the numeral fluxes + and u + for all by (A.). () Comute the tme ste t n by (.4). (3) Udate the oston of eah ell vertex, and obtan the sze and volume of eah ell by (.5). (4) Comute the averaged values U at the (n + )-th tme level based on the sheme (.4)..3 Hgh-order sheme Now, we onsder a general hgh-order fnte volume Lagrangan sheme whh has the same general form as the frst order Lagrangan sheme (.4), where P s s the aroxmaton wth hgh order auray of the soure term r I dr. For nstane, f we erform a thrd order sheme, we an omute P s usng the Gauss-Lobatto quadrature rule P s =

10 For a hgh-order aurate saal dsretzaton, by the nformaton of the orresondng ell-average varables from the ell I and ts neghborng ells, we aly the tehnques of essentally non-osllatory (ENO) reonstruton and loal haraterst deomoston [0] to obtan reonstruton olynomals, through whh we an determne the values at the ell nterfaes {U }. Then the numeral fluxes ± + and u + for all an be omuted by the two-state Remann solver stated n (A.)-(A.). In ths seton, we onsder the ENO reonstruton by treatng the ell averages as ntegrals on the usual ontrol volumes. Ths s smly a standard reonstruton whh an be made arbtrarly hgh order aurate by nreasng the olynomal degrees and the stenls. We erform the ENO reonstruton on the onserved varables, namely densty, momentum and total energy (ρ, ρu, ρe), wth a set of olynomals {W (r) = (ρ (r), M (r), E (r)) } of degree k n the ell I for eah, whh mles ρ = ρ (r)rdr =, V I V M = I M (r)rdr = M (r)rdr V I E = I E (r)rdr = E (r)rdr V I m V = ρ u, m V = ρ E. (.5) If we use olynomals of degree k n the reonstruton roess, we obtan a (k + )-th order sheme..4 Hgh-order ostvty-reservng shemes In ths seton, we fous on how to desgn the hgh order sheme (.4) to be ostvtyreservng when alyng the ENO reonstruton (.5). Before that, let us onsder the K-ont Legendre Gauss-Lobatto quadrature rule n the nterval I, whh s exat for ntegrals of olynomals wth degree u to K 3, and we denote these quadrature onts n I as S = {r = r, r,..., r K, r K = r + }. (.6) 0

11 Let ω α be the quadrature weghts suh that ω α > 0, α =,..., K, ω = ω K, and K ω α =. Assumng that we erform the ENO reonstruton as n (.5), we get a olynomal vetor {W (r) = (ρ (r), M (r), E (r)) } of degree k. We hoose K to be the smallest nteger satsfyng K 3 k +, then the K-ont Legendre Gauss-Lobatto quadrature rule s exat for the ntegrals nvolved n the reonstruton (.5), hene we have wth ω = K α= W V = ω α r α, K α= W ω α W α r α r = (ω W + r + ω W + ω K W r + + ) r, = ω K α= ω α W α r α = ω α= ( ) W V ω W + r r ω K W r + +, (.7) whh mles where m α = ρ α r α r n ρ = ρ (r)rdr = K ω α m α, V I V α= u = M (r)rdr = K ω α m α u α, I α= E = E (r)rdr = K ω α m α E α, I wth α= ρ α = ρ (r α ), u α = M (r α ) ρ (r α ), E α = E (r α ) ρ (r α ). (.8) Now, let us frst ntrodue the artfal numeral fluxes and u referrng to [8], whh are omuted from the left and rght states U + and U. In order to kee aordane wth + them, we also need to defne the artfal loal waveseeds relatve to ths term, z,u, n the ± same way as z. For examle, f we take the Dukowz defnton of z ± ± roosed n [7], z ± = ρ (a + Γ u u ), ± ± ± ± the artfal waveseeds z,u ± read as z,u ± = ρ (a + Γ u ± ± u ), ±

12 where the artfal veloty u and artfal ressure read as u = = z +,u u + + z,u u + + z +,u + z,u + z +,u + z,u z +,u + z,u + z +,u + z,u + z +,u z,u + z +,u + z,u + ( + (u + + ), u + ). If we take the aoust medane z ± = ρ a, there wll be no dfferene,.e. z,u ± ± ± = z. ± Then, by addng and subtratng tn r n the seond equaton n (.4) and tn r u n the thrd equaton n (.4) resetvely, the sheme (.4) beomes where and H = V n+ u n tn (r + r + U n+ = H + W, (.9) ) + tn r E n tn (r + u r + + u ) + tn r u, (.0) m V n+ W = u n + tn r (P s ). (.) E n tn r u We note that H an be exressed as the followng onvex ombnaton where m = K α= H = m F + ω ω α m α, and F = m ( m V n+ ˆF = u + tn tn E + V n+ (r ω, K α= (r ω F + ω Km K FK, (.) ω α m α u α, K r ) α= u r u ω α m α E α ),, (.3) )

13 V n+ ˆF K = u + E + tn ω K m K (r + + tn ω K m K (r + + u + r + ) r +. (.4) u ) We an see that H s a onvex ombnaton of three dfferent terms n (.). Consequently, f these terms belong to the onvex admssble set G, then so does H. Among these terms, F only onssts of the olynomal soluton at the tme level n at the nteror Gauss-Lobatto quadrature onts. To ensure ths quantty to be n G, a artular lmtaton wll be desgned n the later seton. We note that ˆF and ˆF K exatly mm the frst order sheme (.4) but wthout the soure term. Thus we would lke to aly the smlar analyss as that n the frst-order sheme and then obtan ˆF, ˆF K G under the followng ondton t n ω mn σ r n σ r u, n u u u = t 3, + t n ω mn ρ+ r n ρ r +, n = t 4, ( wth σ mn, ρ+ e + + ), σ mn z +,u + z + (, ρ + e + + z + ). + z,u + (.5) As for W, the densty s obvously ostve. We only need to fnd a suffent ondton to ensure e(w) = E(W) u (W) > 0, whh s formulated as ( ) t n µ mn r n(p s + ) r n(p µ + e n s ) (P s ) = t 5, (.6) wth µ = u + (P s )u n. Theorem.3. If the numeral fluxes are determned by the two-state Remann solver wth any general ostve waveseeds defnton, assume U n U α G, α =,..., K, then U n+ (.5) s also n the admssble set G f G and the olynomal solutons n the hgh order sheme (.4) wth the reonstruton n t n mn( t, t, t 3, t 4, t 5 ), (.7) 3

14 where t s the CFL ondton n (.9), t, t 3, t 4 and t 5 are defned n (.0), (.5)- (.6)..5 Postvty-reservng lmter In order to ensure the ostvty roerty for the hgh order sheme (.4), we have assumed that the values of reonstruton olynomals at Gauss-Lobatto quadrature onts U α G, α =,..., K. To ensure ths assumton, we need to make use of the artular lmter ntrodued n [5, 8, 9, ]. At the tme level n, assume the olynomal reonstruton n the ell I s the same as that we resented n the revous seton.4,.e W (r) wth degree k. Wth the assumton U n G, we modfy the reonstruton olynomal W (r) wth a onstant θ as follows W (r) = W + θ (W (r) W ), (.8) where θ [0, ] s to be determned, suh that U( W (r)) G for all r loated n the Gauss- Lobatto quadrature onts set S defned n (.6). In fat, we only need to requre the three onts W +, W, W + to be n G, where W = (ρ, M, E ) s defned n (.7). Referrng to [5, 8, ], the sef mlementaton an be taken as follows: Frst, let us enfore the admssblty of the densty. Choose a small number ε suh that ρ ε for all. In rate, we usually take ε = 0 3. For eah ell I, omute { ρ (r) = ρ + θ (ρ (r) ρ ), θ ρ = mn, ε } ρ ρ +, ρ ε ρ ρ, ρ ε ρ ρ. (.9) + Seond, enfore the ostvty of the nternal energy e = E u for eah ell. Defne ( ) Ŵ (r) = ( ρ (r), M (r), E (r)). For eah ell I, f mn e(ŵ + ), e(ŵ ), e(ŵ ) ε set + θ = ; otherwse, θ e(w ) = mn e(w ) e(ŵ + ), e(w ) e(w ) e(w ) e(ŵ ), e(w ) e(ŵ ). + Then we get the lmted olynomal W (r) = W + θ (Ŵ(r) W ). (.30) 4

15 It s easy to hek that the ell average of W (r) over I s not hanged and s stll W n, and W +, W, W + G. Moreover, the artular lmter does not destroy the hgh order auray n smooth regons and more detals and roof an be seen n [0]..6 Hgh-order tme dsretzaton To get a global hgh-order sheme, we generally make use of the lass thrd order SSP Runge-Kutta tye method to for tme dsretzaton, whose detals an be found n many referenes suh as [5, 8]. At eah ste, the ostvty-reservng lmter s erformed to modfy the olynomal. Note that the SSP Runge-Kutta shemes are onvex ombnatons of Euler forward tme steng, thus they are onservatve, stable and ostvty-reservng when the Euler forward tme steng s onservatve, stable and ostvty-reservng. 3 Two-dmensonal ase In ths seton, we fous on the omressble Euler systen the two-dmensonal ylndral oordnates. Its sef ntegral forn the Lagrangan framework an be desrbed as follows d dt d dt d dt d dt ρrdzdr = 0, Ω(t) ρu Ω(t) zrdzdr = n Γ (t) zrdl, ρu Ω(t) rrdzdr = n Γ (t) rrdl + dzdr, Ω(t) ρerdzdr = u nrdl, Ω(t) Γ (t) (3.) where z and r are the axal and radal dretons resetvely. u = (u z, u r ) where u z, u r are the veloty omonents n the z and r dretons resetvely, and n = (n z, n r ) s the unt outward normal to the boundary Γ (t) n the z-r oordnates. The geometr onservaton law (GCL) means that the rate of hange of a Lagrangan ontrol volume should be omuted onsstently wth the node moton, whh an be formulated as d dv = u nrdl. (3.) dt Ω(t) Γ (t) Ths roerty should also hold n the fully-dsretzed sheme. 5

16 3. Prelmnares 3.. Bas notatons At frst, let us make some notatons referrng to [4, 6]. Ω s the quadrlateral omutatonal ell wth the unque ndex. The boundary of the ell Ω s Ω. Eah vertex of the grd has ts own unque ndex and the ounterlokwse ordered lst of the vertes of Ω s denoted by (). V and A denote the volume and the area of the ell Ω resetvely. There should be a remark that V s obtaned by rotatng ths ell around the azmuthal z-axs (wthout the π fator), whh an be formulated as V = rdrdz. Smlar to (.), here we also defne the ell averages of densty, veloty and total energy as follows: ρ = ρrdzdr, u z = ρu z rdzdr, V Ω Ω u r = (3.3) ρu r rdzdr, E = ρerdzdr, Ω Ω where = ρrdrdz s the mass n the ell Ω, whh kees a onstant durng the tme Ω marhng aordng to the frst equaton n (3.). Wth these assumtons, we an rewrte Ω the system (3.) n the followng form ρ =, V d dt uz = n z rdl, Ω d dt ur = n r rdl + Ω d dt E = Ω u nrdl. Ω dzdr, (3.4) We denote the oordnates and veloty of the vertex as (z, r ) and u = (u z, u r ) resetvely. l + and l stand for the length of the edges [, + ] and [, ], and n + and n are the orresondng unt outward norms, where and + are the neghborng vertes of the vertex, see Fgure 3.. 6

17 n l r r l n (, u, E ) Fgure 3.: Notatons of nodes and nodal varables In order to alulate the dsrete gradent oerator over the ell Ω, we need to denote the two nodal ressures at eah vertex as π and π, whh an be seen n Fgure 3.. These two ressures are related to the two edges sharng the vertex. Wth these, we also need to defne the half lengths and the unt outward normals of the edges onneted to the vertex n the followng way [6] l = l, l = l +, n = n, n = n +. (3.5) Besdes, the seudo-rad r and r are defned as r = 3 (r + r ), r = 3 (r + r +), (3.6) by whh the GCL n (3.) an be rewrtten as the followng form referrng to [6, ] d dt V (t n ) = () (r l n + r l n ) u. (3.7) We wll see that the formula s sgnfant for our analyss of ostvty reservaton n the later seton. Smlarly, we denote z = 3 (z + z ), z = 3 (z + z +), ξ = (z) + (r), ξ = (z ) + (r ). (3.8) 7

18 These notatons enable us to dsretze the rght-hand sde of (3.4) and get the followng sem-dsrete form where n = (n,z, n,r ρ = V, d dt uz = () d dt ur = () d dt E = () (r l π n,z (r l π n,r ) and n = (n,z, n,r ). + r l π n,z ), + rl π n,r ) + tn A P s, (r l π n + r l π n ) u, (3.9) 3.. Comutaton of the nodal ressure and veloty If we denote U = (ρ, u, E ), then the nodal ressure π and π an be determned n the followng way, π = z (u u ) n, π = z (u u ) n, (3.0) where and are the ressure values at the vertex whh are omuted from U and U resetvely. z and z are the aroxmatons of the aoust medane. For the sake of the symmetry-reservng roerty, we lmt the hoes of z and z to be the Godunov aoust solver for the two-dmensonal ase,.e. whh s dfferent from that n the one-dmensonal ase. z = ρ a, z = ρ a, (3.) The nodal veloty an be determned unquely by requrng the sheme to satsfy the onservaton of momentum and total energy [], [ ] u = M rl [ n + z (n n )u n ] + rl [ n + z (n n )u n ], (3.) () where the matrx M reads as M = () M, M = r l z (n n ) + r l z (n n ) (3.3) wth M beng the rojeton matrx along the two normals n and n. 8

19 3. Frst-order sheme Based on the notatons and assumtons n revous subsetons, we an rewrte (3.9) as the followng fully dsrete fnte volume Lagrangan sheme for the PDE system (3.) n the two-dmensonal ase U n+ = ρ n+ u z,n+ u r,n+ E n+ = V n+ u z,n u r,n tn tn E n tn () () (r l π n,z (r l π n,r + r l π n,z ) + rl π n,r ) + tn A P s (rl π n + rl π n ) u (). (3.4) One the nodal veloty u at the vertex has been determned by (3.), the ell vertex and the area and volume of the ell wll be udated as follows [], z n+ = z n + t n u z, r n+ = r n + t n u r, = A n+ V n+ () = 4 An+ (r,n+ () l,n+ n,r,n+ r n+. + r,n+ l,n+ n,r,n+ ), (3.5) 3.3 Frst-order ostvty- and symmetry-reservng sheme In ths seton, we wll dsuss how to obtan the sgnfant roertes of ostvty- and symmetry-reservng for the sheme (3.4). Before that, let us reall the general CFL ondton for the two-dmensonal ase, whh s formulated as t n λ mn l n a n where we agan take the Courant number λ = 0.5, l n = t, (3.6) s the length of the shortest edge of the ell Ω and a n s the sound seed omuted by the ell averages. At the same tme, to avod the degeneraton of the ells, the tme ste should also be restrted as follows ( ) t n λ A n mn (b + b + ), mn r n + () ( u r ), = t +, (3.7) 9

20 where b = () (r u z + + z +u r z u r + r +u z ), b = and λ <. In rate, we agan hoose t as 0.4. () (u r u z + u r + u z ), b + = max(b, 0) Now, we start wth the reservaton of ostvty. To aheve that, we ut an addtonal onstrant on the tme ste [9], whh an be formulated as follows t n mn () σv n (r l n + r l n ) u = mn ρ n σ (r l n + r l n ) u () = t 3, (3.8) wth the fator σ (0, ) to be determned. Smlarly, we defne the admssble set G as ρ G = U = u, ρ > 0, e = E u > 0 and a = ρ s > 0. (3.9) E Makng use of the sheme (3.4), we an know that the densty wll be ostve as long as the volume of the ell, V, s ostve, whh an be ensured under the restrton n (3.7) sne we udate the ell oston by (3.5). Therefore, we wll ay more attenton to the ostvty of the nternal energy. By addng and subtratng tn () (r l n,z +rl n,z )n and tn () (r l n,r +rl n,r )n n the seond and thrd equatons n (3.4) resetvely, we an artton the system (3.4) nto two arts, that s U n+ = ρ n+ u z,n+ u r,n+ E n+ = H + W, (3.0) 0

21 where and H = V n+ u z,n u r,n tn tn E n tn W = () () (r l π n,z (r l π n,r + rl π n,z ) + tn + rl π n,r ) + tn (rl π n + rl π n ) u () V n+ u z,n u r,n E n By ths way, we exress U n+ tn tn () () (r l n,z (r l n,r + rl n,z )n + r l n,r () () (r l n,z (r l n,r )n + tn + r l n,z + rl n,r )n )n, (3.). (3.) A P s as a onvex ombnaton onsstng of H and W. Thus, assumng U n G, f we are able to rove H, W G, then we an ensure that U n+ G. Here we follow a smlar way to get the suffent ondton for H G as that n onedmensonal ase and n [9], and by solvng e(h) > 0 we get σ mn(, ρn en n ) for (3.8) and the followng tme ste onstrant t n mn () () (r l z + r l z ) = t 4. (3.3) The suffent ondton for W G an be desrbed as e(w) > 0. Reallng that n [] (rl n,z + rl n,z ) = 0, () (r l n,r + r l n,r ) = A, (3.4) we an wrte e(w) as ( ) t e(w) = e n n (A P s A n ) tn (A P s A n )u r,n. (3.5)

22 Ths s a quadrat nequalty e(w) 0, whh s guaranteed by the followng ondton wth µ = A (P s n ). t n mn ( u r,n µ + m ) (u r,n ) µ + e n = t 5, (3.6) Theorem 3.. Consder the frst-order sheme (3.4) based on the two-state Remann solver n (3.0)-(3.3) for the aoust defnton of waveseeds z and z defned n (3.). Assume U n G, then U n+ s also n the set G under the followng tme ste onstrants t n mn ( t, t, t 3, t 4, t 5 ), (3.7) where t s the lass CFL ondton (3.6), t, t 3, t 4, t 5 are defned n (3.7), (3.8), ( ) (3.3), (3.6) resetvely wth σ mn, ρn en for t n 3 and µ = A (P s n ) for t 5. Now, let us onsder a one-dmensonal sheral symmetr roblem smulated on an equal-angled olar grd, then we know the ell average U = (ρ, u, E ) n the ell Ω are symmetr, whh means ρ, E and the omonent of u n the radal dreton are the same n all the ells wth the same radal oston, whle the omonent of u n the angular dreton s zero for all the ells. Aordng to [6], for the reservaton of symmetry, there s a seal requrement on the hoe of the aroxmaton P s for ressure n the soure term, whh should be determned as P s = ξ π + ξ π + ξ 3 π + 3 ξ 4π4 ξ + ξ + ξ +, (3.8) 3 ξ 4 where π, π, π 3 and π 4 are the values of ressure related to the two radal edges of the ell Ω. ξ, ξ, ξ 3 and ξ 4 are defned as (3.8).

23 r rt lt l t 4 3 r rb b lb Fgure 3.: The loal ξ-θ oordnates for the ell Ω. z Theorem 3.. The frst-order sheme (3.4) wll kee both ostvty and symmetry smultaneously when t s used for a sheral-symmetr roblem on an equal-angle-zoned grd, f the tme ste t n satsfes the onstrants n Theorem 3. and P s s taken as (3.8). At the end of ths seton, we make a summary for our frst order ostvty- and symmetry-reservng Lagrangan sheme (3.4) for two-dmensonal ylndral oordnates and gve the followng algorthm flowhart. () Assumng U n G at the tme level n, omute the nodal ressure π, π and veloty u by (3.0)-(3.3) for all ells. () Comute the ressure n the soure term usng (3.8). (3) Comute the tme ste t n by (3.7). (4) Udate the oston of eah ell vertex and then omute the area A n+ and volume V n+ of eah ell by (3.5). (5) Comute the new averaged values U n+ by usng the sheme (3.4). 3.4 Hgh order sheme For hgh order auray, the values of U and U at the vertex n the sheme (3.4) wll not be the ell average U any more, but an be obtaned from the reonstruton olynomals. Consderng a hgh order sheme, wth both ostvty and symmetry reservaton, we need to reonstrut olynomal funtons n eah ell Ω based on the ell-average nformaton 3

24 of the ell Ω and ts neghbors. Then, the values of π, π and u an be determned by (3.0)-(3.3). Here P s s also determned as (3.8). Besdes, f we smulate a sherally symmetr roblem, we would hoe to kee the symmetry roerty wth the reonstruted olynomals, whh uts more restrtons on the reonstruton. We would frst need to transform the ell averages of the varables n the neghborng ells whh are nvolved n the reonstruton from the usual (z, r) oordnates to the loal olar oordnates (ξ, θ), where ξ stands for the radal dreton assng through the enter of the edge and the orgn, and θ refers to the angular dreton, orthogonal to ξ ounter-lokwsely. Also, we erform the ntegral on the area rather than over the usual ontrol volume to get the reonstruton olynomal [6], whh an avod the dffulty aused by dfferent values of r n dfferent ells. Ths aroah of reonstruton wll lmt the aroxmaton to be at most seond order aurate, regardless of reonstruton olynomal degrees. Ths s however not a restrton n the two-dmensonal ase as t s known that straght-edge quadrlateral based Lagrangan methods an be at most seond order aurate anyway []. In ths aer, we aly the same tehnque of reonstruton to get olynomals from the ell averages U n = (ρ, u, E ) as that n [6]. Hene we wll not gve more detals about t here. After we erform the reonstruton on the loal ξ-θ oordnates to obtan the olynomals, we transform thento the orgnal z-r oordnates for the alulaton of the sheme (3.4). Aordng to [6], after reonstrutng along eah edge, we get four lnear olynomals n the ell Ω, {U m, (z, r) = (ρ m, (z, r), u m, (z, r), E m, (z, r)), m =..., 4}, whh satsfy U m, (z, r)dzdr = A U. (3.9) Ω Here we defne the edge sequene m, m =,..., 4 of the ell Ω as those onnetng the vertes and, and 3, 3 and 4, 4 and resetvely. 4

25 3.5 Hgh order ostvty- and symmetry-reservng sheme Assume we have obtaned the reonstruton olynomals U m, (z, r) along eah edge, and by usng the relaton n (3.9) we an get U = 4A Ω 4 U m, (z, r)dzdr. (3.30) m= If we use a oordnate transformaton to onvert the ell Ω wth the general quadrlateral shae n the z-r oordnates to the square [, ] [, ] n the x-y oordnates (see Fgure 3.3), we an defne the set of Gauss-Lobatto quadrature onts for the ell Ω to be S = {(z α, r β ), α =,..., K, β =,..., K}, whh are the re-mages under the oordnate transformaton of the Gauss-Lobatto quadrature onts n the square [, ] [, ]. We requre the Gauss-Lobatto quadrature rule to be exat for olynomals of degree k +. Ths s beause a olynomal of degree k n the z-r oordnates beomes a olynomal of degree k n the x-y oordnates, sne the Jaoban of the oordnate transformaton s a blnear funton [5]. r 3 y (, ) (, ) 4 (, ) (, ) z x Fgure 3.3: The transformaton from the z-r oordnates to the x-y oordnates In fat, sne the reonstruton olynomals are lnear, we just need to aly the Smson quadrature rule, n whh the quadrature onts onsst of the ell vertes, the md-onts 5

26 of eah edge and the ell enter,.e. K = 3. ω = ω 3 =, ω 6 =. Based on these, we have 3 U = A Ω U m, (z, r)dzdr = A = A 3 3 α= β= U m, (g m, (x, y)) g m, (x, y) (x, y) dxdy g m, (x, y) ω α ω β (x, y) U m, (z α, r β ), (xα,yβ ) (3.3) where U m, (z α, r β ) s the value of U m, (z, r) at the orresondng Gauss-Lobatto quadrature onts. g m,(x,y) (x,y) s the Jaoban for the oordnate transformaton [5]. Then f we denote J α,β m, = g m,(x,y) and U m, α,β = U m,(z α, r β ), we an rewrte the ntegral (3.3) by the (x,y) (xα,y β ) summaton of the values at quadrature onts as follows U = 4 U m, (z, r)dzdr 4A Ω m= = 4 ω m U m, + (ω U + ω U 4 ), where ω = A ( ω 3 = A ( and ω ω J 3,, + ω ω J, 3, + 3 α= β= 3 3 α= β= m= ω α ω β J α,β, ω α ω β J α,β 3, ) ) (), ω = A (, ω 4 = A ( ω ω J,3, + ω ω J, 4, + 3 α= β= 3 3 α= β= ω = 4A ω J 3, 4,, ω = 4A ω J 3,,, ω = 4A ω J 3,3,, ω α ω β J α,β, ω α ω β J α,β 4, (3.3) ω = ω 4A J 3,3,, ω 3 = ω 4A J,3,, ω 3 = ω 4A J,3 3,, ω 4 = ω 4A J, 3,, ω 4 = ω 4A J, 4,. In fat, we do not need to know the values at the orresondng quadrature onts exet the nodes at the ell edges aordng to the Remarks 3.3 and 3.4 n [5], that s to say, we ) ),, an dretly exress t as U, = ( ) A ω U n 4ω U 4ω U, U, = ω U 3, = ω 3 ( ) A U n 4ω U 4ω 3U 3, ( ) A U n 4ω 3 U 3 4ω 4U 4, U 4, = ( ) A ω 4 U n 4ω 4 U 4 4ω U, (3.33) 6

27 whh s qute useful when mlementng the ostvty-reservng lmter. Therefore, the sheme (3.4) n the hgh-order ase an be rewrtten as 4 where U = 4ω m= ω + (ω + ω ), and () Obvously, U n+ U n+ = ω U + (ω ˆF + ω ˆF ), (3.34) () ω m U m, but the frst omonent reads as ˆF = ˆF = ωv n+ u,z u,r tn r ω m l π n,z ωv n+ tn r ω m l π n,r + tn ( ω 8 A )P s E tn ω r l π n u ωv n+ u,z u,r tn r ω m l π n,z tn r ω m l π n,r + tn ( ω 8 A )P s E tn ω r l π n u wth ω = 4 4 m= ω m, ω =, (3.35). (3.36) s a onvex ombnaton onsstng of U and ˆF, ˆF, hene we only need to ensure U and (), ˆF, ˆF are n the admssble set G. For eah ˆF, we hoe to aly exatly the same tehnques as those resented n the frst order ase to make sure ˆF G [9], smlarly for ˆF G, (). Let us defne all the orner nodes for eah ell as a set Q = {,,,, 3, 3, 4, 4} and assume the followng formula always holds where π,a q q Q r ql qπ,a q n q = 0, (3.37) stands for some artfal ressure and ts orresondng artfal veloty s u and πq,a, u are to be determned later. 7

28 By addng the artfal term (3.37) n (3.4), ˆF s hanged nto ˆF = ωv n+ u,z u,r tn (r ω m l π n,z tn (r ω m l π n,r rl π,a n,z ) r l π,a n,r ) + tn E tn ω (r l π n u r l π,a n u ) (. (3.38) ω 8 A )P s Lemma 3.3. For all, ˆF exatly mms the frst order sheme defned n (3.4), so does ˆF. The roof of ths lemma s gven n the Aendx B, n whh we also show how to defne and determne the artfal ressure π,a q and ts orresondng artfal veloty u n (3.37)-(3.38). Therefore, by usng the same analyss as that n the frst order ase, we have the followng onluson. Theorem 3.4. Consder the Lagrangan sheme (3.4) based on the two-state Remann solver defned n (3.0)-(3.3) wth the aoust defnton of z and z defned n (3.). Assume U n G and U m, α,β followng tme ste onstrant G for all m =, 4, α, β =, 3, then Un+ G under the t n mn ( t, t, t 3, t 4, t 5, t 6 ), (3.39) where t s the CFL ondton (3.6), t s defned n (3.7) and t 3 = mn, t 4 = mn, ω rql q z q q Q ω rql q z q, q Q, ρ ρ σ ω (u u ) rl n, σ ω (u u ) r l n, 8

29 ( ) ( wth σ mn, ρ e, σ mn, ρ e ), and where t 5 = mn ( ω u,r + ω, µ µ t 6 = mn ( ω u,r + ω, µ µ (u,r ) ) + e, ) (u,r ) + e µ = 8 A P s A π, µ = 8 A P s A π. 3.6 Postvty-reservng lmter To aheve the reservaton of symmetry, we need to erform the olynomal reonstruton and ostvty-reservng lmtaton along eah edge and n the loal ξ-θ oordnates for eah ell Ω as that we resented before, that s to say, we frst need to transform the olynomals n the z-r oordnates U m, (z, r) = (ρ m, (z, r), u m, (z, r), E m, (z, r)) nto the olynomals defned n the ξ-θ oordnates U m, (ξ, θ) = (ρ m, (ξ, θ), u m, (ξ, θ), E m, (ξ, θ)). Under the assumton U n G, we would lke to modfy the olynomal reonstruton U m, (ξ, θ) wth a onstant θ nto another olynomal Ũm,(ξ, θ) suh that the values of Ũ m, (ξ, θ) at ts orresondng Gauss-Lobatto quadrature onts an be set n G. The mlementaton s smlar to that for the hgh-order sheme n the one-dmensonal ase, whh an be desrbed as the followng modfaton on the reonstruton olynomal Ũ m, (ξ, θ) = U + θ (U m, (ξ, θ) U ), (3.40) where θ [0, ] s to be determned, suh that Ũm,(ξ, θ) G for all (ξ, θ) S. In fat, we do not need to modfy the values at all Gauss-Lobatto quadrature onts, we only need to modfy the values of U m, (ξ, θ) at the two nodes of ts orresondng edge and U m, defned n (3.33), whh reresents a lumed ontrbuton from all other Gauss-Lobatto quadrature onts. Frst, let us enfore the admssblty of the densty. Choose a small number ε suh that ρ ε for all. In rate, we usually take ε = 0 3. For the eah edge of eah ell Ω, 9

30 omute ρ m, (ξ, θ) = ρ + θ m,(ρ m, (ξ, θ) ρ ), { θm, = mn, ρ ε ρ ρm, ρ ε } ρ ρ m, ρ ε ρ ρ, m, (3.4) wth m =,, 3, 4 and m = + m where m = m (mod 4). Seond, enfore the ostvty of the nternal energy e for the ells. Defne Ûm,(ξ, θ) = ( ρ m, (ξ, θ), u m, (ξ, θ), E m, (ξ, θ)). For the eah edge of eah ell Ω, { } θm, e(u ) = mn, e(u ) e(û m), e(u ) e(u ) e(û m), e(u ). e(u ) e(û m,) Then we get the followng lmted olynomal relatve to eah edge Ũ m, (ξ, θ) = U + θ m,(ûm,(ξ, θ) U ). (3.4) After gettng the lmtaton fators θm, for m =,..., 4, we an get the modfed values at two node onts along eah edge of the ell Ω. Then we need to transform bak nto the z-r oordnates to udate the tme marhng. Performng the lmter n the loal ξ-θ oordnates an ensure that the values at the Gauss-Lobatto onts wth the same radal oston and dfferent angular oston are the same, thus the roerty of symmetry an be mantaned. Besdes, t s easy to hek that the ell average of Ũm,(ξ, θ) over Ω s not hanged and s stll U n, and Ũm,(ξ α, θ β ) G for all relevant α, β (nludng the lumed ones). Moreover, the artular lmter does not destroy the hgh order auray n smooth regons and an kee symmetry. 3.7 Hgh order tme dsretzaton To obtan a Lagrangan sheme wth unformly seond order auray both n tme and sae, the tme marh steng an be mlemented by a seond order strong stablty reservng (SSP) Runge-Kutta tye method, whh s detaled n [6]. 30

31 At eah ste, the reonstruton olynomals of eah ell an be obtaned based on the nformaton frotself and ts neghbors. Then the lmter oeraton s erformed to modfy the olynomal. 4 Numeral examles In ths seton, we hoose several hallengng numeral examles n one- and two-dmensonal ylndral oordnates to show the erformane of our frst order and hgh order ostvtyreservng and symmetry-reservng Lagrangan shemes. The examles are erformed on the deal gas wth γ = 5/3 unless otherwse stated. The Godunov aoust solver s used for all numeral tests,.e. z = ρa. All these examles enounter the roblem of negatve nternal energy f the usual hgh order Lagrangan sheme wthout the ostvty-reservng lmter s used. In order to use larger tme stes to mrove effeny as muh as ossble, we do not restrt the tme ste as strtly as resented n revous theorems n our atual ode. Instead, we just take the tme ste t n as the mnmum of the lassal CFL ondton and the ondton for avodng degeneraton of ells, defned by (.9)-(.0) and (3.6)-(3.7) relatve to the one- and two-dmensonal ases resetvely, to marh to the tme level n +. If the nternal energy obtaned s ostve, we wll ontnue to the next tme ste; otherwse, we wll ome bak to the revous tme level n, and take a smaller tme ste suh as tn, and roeed as before. The theorems n the revous setons ensure that we only need to ome bak a fnte number of tmes before we wll obtan a ostve nternal energy. In fat, n our followng numeral tests, the omng-bak ase haens only seldomly. 4. One-dmensonal tests Examle. Auray test. We frst test the auray of our shemes on a free-exanson roblem. The ntal 3

32 ondton s taken as ρ =, u = 0, = r 4, r [0, ]. Free boundary ondton s aled on the outer boundary. The errors and auray of the sheme at t = are lsted n Tables whh are measured on the nterval [r 0 N, r 9 0 N], where N s the total number of ells, to remove the nfluene from the boundary. erentage of the ells (averaged n sae and tme) n whh the ostvty-reservng lmter has been erformed s also lsted n Table 4.. We take the result of the thrd order ostvtyreservng Lagrangan sheme wth 0,000 ells as our referene soluton when omutng the errors. From these tables, we an learly see that the frst order and thrd order ostvtyreservng shemes wth the ostvty-reservng lmter have aheved the exeted order of auray n both L and L norms for all the evolved onserved varables. Table 4.: Errors of the frst order sheme n D ylndral oordnates usng N ntally unforells N Norm Densty order Momentum order Energy order 50 L 0.44E- 0.45E- 0.56E- L 0.5E- 0.85E- 0.8E- 00 L 0.3E E E L 0.78E E E L 0.E E E L 0.39E E E L 0.6E E E L 0.9E E E L 0.3E E E L 0.97E E E The 3

33 Table 4.: Errors of the thrd order sheme wth ostvty-reservng lmter n D ylndral oordnates usng N ntally unforells N Norm Densty order Momentum order Energy order lmted ells(%) 50 L 0.6E-5 0.8E-5 0.8E-5 L 0.7E E E L 0.34E E E-6.93 L 0.83E E E L 0.34E E E L 0.9E E E L 0.36E E E L 0.E E E L 0.40E E E L 0.5E E E Examle. Sedov blast wave n a ylndral oordnate [7]. The ntal omutatonal doman s [0,.5]. The ntal ondton s ρ =, u = 0, The sef nternal energy e s 0 exet n the ells onneted to the orgn where they share a total value of In the ratal smulaton, as we annot smulate vauum, e s usually set to be a small ostve value suh as 0 6. Here we take e to be a smaller ostve value, that s 0 4 whh s demonstrated to brng muh more hallenge to the sheme. The deal gas s used wth γ =.4. Refletve boundary ondton s aled on the outer boundary. The analytal soluton s a shok wth a eak densty of 6 at r = and at tme t =. The numeral results wth our frst and thrd order shemes usng 0 ells at t = are shown n Fgure 4. and onvergene lots an be seen n Fgure 4.. We an see the oston of the shok has been atured very aurately. Although the nternal energy and ressure are qute small, they an always be ket ostve. 33

34 6 5 exat st order 0 3rd order exat st order 0 3rd order 0 0. exat st order 0 3rd order densty 3 veloty 0. ressure radus radus radus Fgure 4.: The results of the Sedov roblem wth 0 ells at t =. 5 4 exat 3rd order 0 3rd order 40 3rd order exat 3rd order 0 3rd order 40 3rd order exat 3rd order 0 3rd order 40 3rd order 80 densty 3 veloty 0. ressure radus radus radus Fgure 4.: The onvergene results of the Sedov roblem on the meshes wth refnement at t =. Examle 3. The Noh roblen a ylndral oordnate system [5]. The Noh robles a lass test roblem whh s wdely used to valdate the erformane of Lagrangan shemes on strong dsontnutes. The ntal omutatonal doman s [0,]. The ntal densty s, the ntal ressure s 0, and the ntal veloty s dreted toward the orgn wth magntude. The analyt soluton s a shok generated by brngng the old gas to rest at the orgn. The densty behnd the shok s 6, and the shok seed s /3. But n ratal numeral smulaton, we an not take the ressure to be zero. In the lterature, the ressure s usually hosen as large as 0 5. However, n ths test, to verfy the erformane of the ostvty-reservng roerty n our sheme, we hoose the ntal ressure as small as 0 3, whh brngs sgnfant hallenge to the sheme. In fat, the thrd order Lagrangan sheme fals to omute t wthout the ostvty-reservng lmter, 34

35 even wth very small tme stes. Fgures show the results of our frst order and thrd order shemes on the dfferent grds at t = 0.6. We an observe that densty and ressure are ostve and the shok s atured very well, whh demonstrates the good erformane of our sheme when the ressure and nternal energy tend to zero. 6 4 exat st order 40 3rd order exat st order 40 3rd order exat st order 40 3rd order 40 densty 0 8 veloty ressure radus radus radus Fgure 4.3: The results of the Noh roblem wth 40 ells at t = exat 3rd order 0 3rd order 40 3rd order exat 3rd order 0 3rd order 40 3rd order 80 5 exat 3rd order 0 3rd order 40 3rd order 80 4 densty 0 8 veloty ressure radus radus radus Fgure 4.4: The onvergene results of the Noh roblem wth dfferent meshes at t = Two-dmensonal tests In ths subseton, we erform numeral exerments n two-dmensonal ylndral oordnates. Purely Lagrangan omutaton, the ntally equal-angled olar grd and the seond order sheme (3.4) wth (3.0)-(3.3) s used n the followng tests unless otherwse stated. Refletve boundary ondtons are aled to the z and r axes n all the tests. ξ = z + r s the radal oordnate. u ξ and u θ reresent the values of veloty n the radal and angular dretons n the ell s loal olar oordnates. 35

36 Examle. We test the auray of the sheme (3.4) on a free exanson roblem. The ntal omutatonal doman s [0, ] [0, π/] defned n the olar oordnates. At t = 0, we have ρ =, u ξ = 0, u θ = 0, = ξ 4. We erform the test on two dfferent tyes of grds as shown n Fgures The frst s an ntally equal-angled olar grd. The seond s an ntally non-unform smooth olar grd, for whh eah nternal grd vertex s obtaned by a smooth erturbaton from an equal-angled olar grd as follows z k,l = ξ k os θ l + ϵ sn(πξ k ) sn(4θ l ), r k,l = ξ k sn θ l + ϵ sn(πξ k ) sn(4θ l ), where ϵ = 0.0. ξ k = k, θ K l = l π, (z L k,l, r k,l ) s the z-r oordnates of the grd onts wth the sequental ndes (k, l), k =,..., K, l =,..., L n the radal and angular dretons resetvely. K, L reresent the number of ells n the above mentoned two dretons. Free boundary ondton s aled on the outer boundary. The grds at t = are gven n the rght fgures of Fgures In the fgures, we use the blak onts to reresent the ells where the ostvty-reservng lmter has been enated at t =. We an learly observe the symmetry-reservng roerty of the sheme on the equal-angled olar grd. The error and auray of the sheme on these two knds of grds at t = are lsted n Tables Due to the numeral sngularty, auray degeneray henomena may haen at the orgn and the free outer boundary, thus we remove several onts n these two areas to avod the nfluene of boundary ondtons n the onvergene results. Here we measure the error and auray on the nterval [ξ 0 K+, ξ 9 0 K] [θ, θ L ]. The tme-averaged erentage of the ells n whh the ostvty-reservng lmter has been erformed s also lsted n the tables. Here we take the result of the one-dmensonal thrd order ostvty-reservng Lagrangan sheme n the sheral oordnate wth 0,000 ells as our referene soluton. From the tables, we an see the exeted seond order auray n both L and L norms 36

37 for all the evolved onserved varables on both knds of grds. Fgure 4.5: The equal-angled olar grd of the free exanson roblem wth 0 0 ells. Left: ntal grd; Rght: grd at t =. The blak onts n the fgure reresent the ells where the ostvty-reservng lmter has been enated at t =. Table 4.3: Errors of the sheme n D ylndral oordnates for the free exanson roblem usng K L ntally equal-angled olar grd ells K = L Norm Densty order Momentum order Energy order lmted ells(%) 0 L 0.5E- 0.9E- 0.0E- L 0.89E- 0.9E- 0.68E L 0.6E E E-3. L 0.E E E L 0.8E E E-3.93 L 0.4E E E L 0.44E E E-4.07 L 0.0E E E

38 Fgure 4.6: The smooth non-equal-angled olar grd of the free exanson roblem wth 0 0 ells. Left: the ntal grd; Rght: the grd at t =. The blak onts n the fgure of the grd reresent the ells where the ostvty-reservng lmter has been enated at t =. Table 4.4: Errors of the sheme n D ylndral oordnates for the free exanson roblem usng K L ntally smooth non-equal-angled olar grd ells K = L Norm Densty order Momentum order Energy order lmted ells(%) 0 L 0.5E- 0.0E- 0.9E- L 0.E- 0.44E- 0.79E L 0.6E E E-3.4 L 0.3E E E L 0.8E E E-3.86 L 0.4E E E L 0.47E E E-4.83 L 0.E E E Examle (The sheral Sedov roblen a ylndral oordnate system on the olar grd [7]). The sheral Sedov blast wave roblen a ylndral oordnate systes a ommonly used examle of a dvergng shok wave. The ntal omutatonal doman s a -rle regon 4 defned n the olar oordnates by [0,.5] [0, π/]. The ntal ondton s, ρ =, u ξ = 0, u θ = 0. The sef nternal energy s 0 4 exet n the ells onneted to the orgn where they share a total value of Refletve boundary ondton s aled on the outer boundary. The analytal soluton s a shok wth a eak densty of 4 at radus unty at tme unty. The 38

39 fnal grd and the surfae of densty and ressure obtaned by the seond order sheme wth ells are dslayed n Fgures 4.7. We observe the exeted symmetry n the lots of grd, densty and ressure. The densty and ressure as a funton of ξ at all the ell enters on 0 0, and grds are shown n Fgure 4.8 resetvely. In the fgures, we observe that the values of densty and ressure wth the same ξ onde wth eah other omletely, whh learly demonstrates the symmetry-reservng roerty of the sheme. The shok oston, eak densty and ressure agree wth the analytal solutons better wth the refnement of grd, whh verfes the good erformane of the sheme n symmetryreservng, ostvty-reservng, non-osllaton and auray roertes. The enters of all the ells where the ostvty-reservng lmter has been effetve wth the tme marhng are shown n the rght fgure of Fgure 4.8, whh shows that the lmter always ats along the front of the shok wave n ths test. Fgure 4.7: The results of the Sedov roblem on an equal-angled olar grd wth ells at t =. Left: the grd; Mddle: densty ontour; Rght: ressure ontour. 39

40 Fgure 4.8: The results of the Sedov roblem on an equal-angled olar grd at t =. Left: ρ vs ξ at all the ell enters on 0 0, and grds resetvely. Mddle: vs ξ at all the ell enters on 0 0, and grds resetvely. Sold lne: exat soluton; Symbols: seond order sheme. Rght: Blak onts: the ell enters where the ostvty-reservng lmter has been enated wth the tme marhng on the grd, Red lnes: grds at t = 0 and t =. Examle 3 (Imloson roblem of Lazarus []). In ths self-smlar mloson roblem, ntally a shere of unt radus has the followng ondton, ρ =, u ξ (t) = αf ( ft) α, u θ(t) = 0, e = 0 4, where α = , f = εt δt 3, ε = 0.85, δ = 0.8. We test the roblem on an equal-angled olar grd of ells n the ntal omutatonal doman [0, ] [0, π/] defned n the olar oordnates. Free boundary ondton s aled on the outer boundary. The numerally onverged result omuted usng a onedmensonal thrd-order Lagrangan ode n the sheral oordnate wth 0,000 ells s used as a referene soluton. We dslay the results of the seond order sheme n Fgures In the lots of the grd, densty ontour and ressure ontour, we note the exeted symmetry. In the lot of densty and ressure as a funton of ξ at all the ell enters, we observe the non-osllatory, symmetry-reservng and ostvty-reservng roertes of the sheme. The enters of all the ells where the ostvty-reservng lmter has been erformed wth the tme marhng are shown n the rght fgure of Fgure 4.0. In the fgure, we ould see that the lmter ats near the shok front and the outer boundary n ths test. 40

41 Fgure 4.9: The results of the Lazarus roblem on an equal-angled olar grd wth ells at t = 0.8. Left: grd; Mddle: densty ontour; Rght: ressure ontour. Fgure 4.0: The results of the Lazarus roblem on an equal-angled olar grd wth ells at t = 0.74, 0.8. Left: ρ vs ξ at all the ell enters; Mddle: vs ξ at all the ell enters. Sold lne: referene soluton; Symbols: seond order sheme. Rght: Blak onts: the ell enters where the ostvty-reservng lmter has been enated wth the tme marhng, Red lnes: the grds at t = 0 and t = Examle 4 (The Noh roblen a ylndral oordnate system on the olar grd [5]). In ths test ase, the erfet gas has the followng ntal ondton, ρ =, u ξ =, u θ = 0, e = 0 3. The equal-angled olar grd s aled n the -rle omutatonal doman defned n the 4 olar oordnates by [0, ] [0, π/]. Free boundary ondton s aled on the outer boundary. The shok s generated by brngng the old gas to rest at the orgn. The analytal ost shok densty s 64 and the shok seed s /3. Fgure 4. shows the results of the seond 4

42 order sheme nludng the fnal grd, densty ontour and ressure ontour wth 00 0 ells at t = 0.6. The densty and ressure as a funton of ξ at all the ell enters on 5 5, 50 0 and 00 0 grds are gven n Fgure 4. resetvely, where we observe the results are onvergent, symmetr, ostvty-reservng and non-osllatory near the shok. Fgure 4.: The results of the Noh roblem on an equal-angled olar grd wth 00 0 ells at t = 0.6. Left: the grd; Mddle: densty ontour; Rght: ressure ontour. The blak onts n the fgure of the grd reresent the ells where the ostvty-reservng lmter has been enated. Fgure 4.: The results of the Noh roblem on equal-angled olar grds wth 5 5, 50 0 and 00 0 ells resetvely at t = 0.6. Left: ρ vs ξ at all the ell enters. Rght: vs ξ at all the ell enters. Sold lne: exat soluton; Symbols: numeral soluton. Examle 5 (Sheral Sedov roblem on the Cartesan grd). The sheral symmetry roblem smulated on the ntally retangular grd s demonstrated to be muh more hallengng for a Lagrangan sheme due to the shok dreton 4

43 beng not algned wth the grd lne. In ths examle, we test the sheral Sedov blast wave roblen a ylndral oordnate system on the ntally retangular grd. The ntal omutatonal doman s a.5.5 square onsstng of unforells. Its ntal ondton s ρ =, u z = 0, u r = 0. The sef nternal energy e s 0 4 exet n the ell onneted to the orgn where t has a value of Refletve boundary ondton s aled on the rght and to boundares. Fgures show the results of our seond order sheme (3.4) wth (3.0)-(3.3). From the fgures, we an observe the results of our seond order sheme are ostvty-reservng and roughly symmetr even on ths non-olar grd. Fgure 4.3: The results of the Sedov roblem on a Cartesan grd wth ells at t =. Left: grd; Mddle: densty ontour ; Rght: ressure ontour. The blak onts n the fgure of grd reresent the ells where the ostvty-reservng lmter has been enated. 43

44 Fgure 4.4: The results of the Sedov roblem on a Cartesan grd wth ells at t =. Left: ρ vs ξ at all the ell enters; Rght: vs ξ at all the ell enters. Sold lne: exat soluton; Symbols: seond order sheme. Examle 6 (Sheral Noh roblem on the Cartesan grd [5]). In ths examle, we test the sheral Noh mloson roblem on a Cartesan grd to verfy the robustness of the sheme. The ntal doman s [0, ] [0, ]. The ntal state of the flud s (ρ, u ξ, u θ, e) = (,, 0, 0 3 ). Free boundary ondton s aled on the rght and to boundares. The analytal soluton s the same as that n Examle 4. Fgures show the results of the seond order sheme wth ntally unform retangular ells at t = 0.6. From these fgures, we an see that there s no grd dstorton along the axes, the sheral symmetry of the shok front s reserved well, the shok oston s orret and the ostvty of densty and ressure s mantaned very well, whh demonstrate the robustness of the sheme on the Cartesan grd. 44

45 Fgure 4.5: The results of the Noh roblem on a Cartesan grd wth ells at t = 0.6. Left: the grd; Mddle: densty ontour; Rght: ressure ontour. The blak onts n the fgure of the grd reresent the ells where the ostvty-reservng lmter has been enated at t = 0.6. Fgure 4.6: The results of the Noh roblem on a Cartesan grd wth ells at t = 0.6. Left: ρ vs ξ at all the ell enters. Rght: vs ξ at all the ell enters. Sold lne: exat soluton; Symbols: numeral soluton. 5 Conluson In ths aer, we fous on the methodology to desgn ostvty-reservng and symmetry - reservng Lagrangan shemes n one- and two- dmensonal ylndral oordnates for solvng omressble Euler equatons wth general equatons of state. Frstly, we develo the frst order ostvty-reservng Lagrangan sheme and hgh order ostvty-reservng Lagrangan sheme by usng ostvty-reservng lmter for Euler equatons n one-dmensonal ylndral oordnates, whh are erformed based on the two-state Remann solver [3, 8]. Then 45