A P N P M -CPR Framework for Hyperbolic Conservation Laws

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1 20th AIAA Computatoal Flud Dyamcs Coferece Jue 2011, Hoolulu, Hawa AIAA A P N P M -CPR Framewor for Hyperbolc Coservato Laws Z.J. Wag 1 ad Le Sh 2 Departmet of Aerospace Egeerg ad CFD Ceter, Iowa State Uversty, Ames, IA, Sog Fu 3 Departmet of Egeerg Mechacs, Tsghua Uversty, Bejg, Cha, Hax Zhag 4 Natoal Laboratory for Computatoal Flud Dyamcs, Bejg, Cha, ad Lapg Zhag 5 State Key Laboratory of Aerodyamcs, Mayag, Schua, Cha, The P N P M or recostructed dscotuous Galer method s a hybrd fte volume ad dscotuous Galer method, whch eghborg cells are used to recostruct a hgher order polyomal tha the soluto represetato the cell uder cosderato. The CPR method s a dscotuous odal formulato ufyg several well-ow methods a smple fte dfferece le maer. I ths paper, we preset several P N P M schemes uder the CPR framewor. May terestg schemes wth varous orders of accuracy ad effcecy are developed. Ther performace s llustrated wth several bechmar test cases. I. Itroducto OST producto CFD codes used the aerospace dustry s ether frst or secod order accurate. Ths meas Mthat the soluto error s proportoal to h or h 2, wth h beg the mesh sze. Numercal methods of 1 st ad 2 d order accuracy are called low order methods covetoally, whle 3 rd ad hgher order oes are defed as hghorder methods the aerospace commuty. For 3D smulatos, f the mesh resoluto s doubled both space ad tme, the computatoal cost creases by a factor of 16, but the soluto error oly decreases by a factor of 2 wth a 1 st order scheme ad a factor of 4 wth a 2 d order scheme. O the other had, the soluto error wth a 6 th order scheme decreases by a factor of 64 wth a 16 fold crease computatoal cost whe resoluto doubles. Obvously, f hgh accuracy s requred, low order methods are ot as effcet as hgh order methods wth grd refemet. Because of the potetal of hgh accuracy ad effcecy, hgh-order methods have receved cosderable research terest the global CFD commuty the last two decade. A varety of hgh order methods have bee developed. Refer to several boos 18,13,35 ad revews 9,34 for the state-of-the-art ad recet progresses the developmet of such methods. Most hgh order methods employ polyomals of degree 2 or hgher to approxmate the (uow) soluto. I two dmesos, at least sx degrees of freedom (DOFs) or soluto uows are requred to buld a degree 2 polyomal. Depedg o how may DOFs are avalable o a cell or elemet, multple cells may be eeded to buld the soluto polyomal. For example, at least 5 eghborg cells are requred to buld a degree 2 polyomal a fte volume method because each cell oly has oe DOF, the cell-averaged soluto. I a dscotuous Galer (DG) 2-6,10,25-26,32,39, resdual dstrbuto (RD) 1, spectral volume (SV) 21,33,38 /dfferece (SD) 19,22,24,30 or the 1 Wlso Professor of Egeerg, Departmet of Aerospace Egeerg ad CFD Ceter, 2271 Howe Hall, Assocate Fellow of AIAA. 2 PhD Studet, Departmet of Aerospace Egeerg ad CFD Ceter, 2271 Howe Hall. 3 Professor, Departmet of Egeerg Mechacs. Seor Member AIAA. 4 Academca of Chese Academy of Sceces, Natoal Laboratory for Computatoal Flud Dyamcs. 5 Professor, State Key Laboratory of Aerodyamcs. 1 Amerca Isttute of Aeroautcs ad Astroautcs Copyrght 2011 by Z.J. Wag. Publshed by the Amerca Isttute of Aeroautcs ad Astroautcs, Ic., wth permsso.

2 correcto procedure va recostructo (CPR) method 14-17,36-37, each cell has eough DOFs so that eghborg data s ot requred buldg the soluto polyomal. Such methods are compact because oly mmedate face eghbors play a role updatg the DOFs the curret cell. Compact methods are easy to mplemet o CPU ad GPU clusters, ad hghly scalable because the amout of data commucato s relatvely small. Although 2 d order fte volume schemes are ot strctly compact as eghbor s eghbors are used the soluto update, they ca be mplemeted a compact maer by messagg passg through oly mmedate eghbors o a parallel computer. Ths s (a) Recostructo stecl (a) Scheme stecl Fgure 1. Illustrato of the recostructo ad scheme stecls because the recostructo stecl s compact as show Fgure 1, whch shows both the recostructo stecl ad the scheme stecl. More recetly, hybrd methods amed P N P M 7-8, recostructed DG (RDG) 23, hybrd FV/DG 41, weghted tegral based schemes 40 have bee developed. The ey dea of these methods s to use multple DOFs o the curret cell ad ts eghbors to buld a soluto polyomal hgher tha that wth the DOFs o a sgle elemet. Ths soluto polyomal s the used to geerate hgh-order updates for the DOFs o the curret elemet. Ths hybrd approach thus offers a whole ew host of possbltes: How to recostruct the soluto polyomal ad at what degree? How to update the DOFs at the curret elemet? The umber of choces s qute large, ad fact, the FV ad DG/SV/SD/CPR methods ca be vewed as two extreme specal cases of the ew famly of possble methods. As metoed earler, may choces have already bee explored, ad some uque features have bee demostrated, e.g., More effcet per DOF tha ether the FV or DG method; Lower memory requremet for mplct schemes to acheve a gve order of accuracy tha the DG method. I the preset study, we test the hybrd approach the cotext of CPR, a fte dfferece-le odal formulato 14-17, We attempt to mata the smplcty the formulato, whle vestgatg the accuracy ad stablty of dfferet choces. I the ext secto, we brefly revew the CPR approach. I Secto III, the basc P N P M -CPR formulato wll be descrbed both 1D ad 2D. Numercal results are preseted Secto IV for the Euler equatos, ad coclusos are gve Secto V. II. Revew of the CPR Formulato For the sae of completeess, the CPR formulato s brefly revewed. The CPR formulato was orgally developed by Huyh uder the ame of flux recostructo, ad exteded to smplex ad hybrd elemets by Wag & Gao 35 uder lftg collocato pealty. The authors later decded to employ the ufed ame CPR for the method. I 14, CPR was further exteded to 3D hybrd meshes. The method s also descrbed two boo chapters 35. CPR ca be derved from a weghted resdual method by trasformg the tegral formulato to a dfferetal oe. Frst, a hyperbolc coservato law ca be wrtte as Q + F ( Q) = 0, (1) wth proper tal ad boudary codtos, where Q s the state vector, ad F = ( F, G) s the flux vector. Assume N that the computatoal doma Ω s dscretzed to N o-overlappg tragular elemets { V } = 1. Let W be a arbtrary weghtg fucto or test fucto. The weghted resdual formulato of (1) o elemet V ca be expressed as 2 Amerca Isttute of Aeroautcs ad Astroautcs

3 Q Q + F( Q) WdV = WdV + WF( Q) ds W F( Q) dv = 0. (2) V V V V Let Q be a approxmate soluto to the aalytcal soluto Q o V. O each elemet, the soluto belogs to the space of polyomals of degree or less,.e., Q P ( V ) (or P f there s o cofuso) wth o cotuty requremet across elemet terfaces. Let the dmeso of soluto Q, for the momet, s requred to satsfy (2) Q WdV + WF ( Q ) ds W F ( Q ) dv = 0. V V V P be K = (+1)(+2)/2. I addto, the umercal (3) Obvously the surface tegral s ot properly defed because the umercal soluto s dscotuous across elemet terfaces. Followg the dea used the Goduov method 11,31, the ormal flux term (3) s replaced wth a commo Rema flux, e.g., 20,27-28 F ( Q ) F( Q ) Fcom ( Q, Q+, ), (4) where Q + deotes the soluto outsde the curret elemet V. Istead of (3), the approxmate soluto s requred to satsfy Q WdV + WF com ds W F ( Q ) dv = 0. V V V Applyg tegrato by parts aga to the last term of the above LHS, we obta (5) Q WdV + W F ( Q ) dv + W F com F ( Q ) ds = 0. V V V (6) Here, the test space has the same dmeso as the soluto space, ad s chose a maer to guaratee the exstece ad uqueess of the umercal soluto. Note that the quatty F( Q ) volves o fluece from the data the eghborg cells. The fluece of these data s represeted by the above boudary tegral, whch s also called a pealty term, pealzg the ormal flux dffereces. The ext step s crtcal the elmato of the test fucto. The boudary tegral above s cast as a volume tegral va the troducto of a correcto feld o V, δ P ( V ), com Wδ dv = W[ F ] ds, (7) V V where [ F ] = F F ( Q ) s the ormal flux dfferece. The above equato s sometmes referred to as the lftg operator, whch has the ormal flux dffereces o the boudary as put ad a member of P ( V ) as output. Substtutg (7) to (6), we obta Q + F( Q ) + δ WdV = 0. V (8) If the flux vector s a lear fucto of the state varable, the F( Q ) P. I ths case, the terms sde the square bracet are all elemets of P. Because the test space s selected to esure a uque soluto, Eq. (8) s equvalet to 3 Amerca Isttute of Aeroautcs ad Astroautcs

4 Q + F( Q ) + δ = 0. (9) For olear coservato laws, F( Q ) s usually ot a elemet of P. As a result, (8) caot be reduced to (9). I ths case, the most obvously choce s to project F( Q ) to P. Deote F Π ( Q ) a projecto of F( Q ) to P. Oe choce s ( ) The (8) reduces to V Π F( Q ) WdV = F( Q ) WdV ( ) V. (10) Q + Π( F( Q )) + δ = 0. (11) Wth the troducto of the correcto feld δ, ad a projecto of F( Q ) for olear coservato laws, we have reduced the weghted resdual formulato to a dfferetal formulato, whch volves o explct tegrals. Note that for δ defed by (7), f W P, Eq. (11) s equvalet to the DG formulato, at least for lear coservato laws; f W belogs to aother space, the resultg δ s dfferet. We obta a formulato correspodg to a dfferet method such as the SV method. r (j vares from 1 to K), as show Next, let the DOFs be the solutos at a set of soluto pots (SPs) {, j} Fgure 2. The Eq. (11) holds true at the SPs,.e., Q, j ( F Q ) where Π j ( F( Q )) deotes the values of Π ( F( Q )) how the correcto feld δ ad the projecto Π ( F( Q )) + Π j ( ) + δ, j = 0, (12) at SP j. The effcecy of the CPR approach hges o are computed. Two approaches ca be used to compute ths dvergece as detaled 36. To compute δ, we defe +1 pots amed flux pots (FPs) alog each terface, where the ormal flux dffereces are computed, as show Fgure 2. We approxmate (for olear coservato laws) the ormal flux dfferece [ F ] wth a degree terpolato polyomal alog each terface, f,3 [ F ] f,2 [ F ] FP f f f, l l l [ F ] I [ F ] [ F ] L, (13) where f s a face (or edge 2D) dex, ad l s the FP dex, ad FP L l s the Lagrage terpolato polyomal based o the FPs a local terface coordate. For lear tragles wth straght edges, oce the soluto pots ad flux pots are chose, the correcto at the SPs ca be wrtte as δ 1 = α [ F ] S,, j j, f, l f, l f V f V l where α j, f, l are lftg costats depedet of the soluto, S f s the face area, V (14) s the volume of V. Note that the correcto for each soluto pot, amely δ, j, s a lear combato of all the ormal flux dffereces o f,1 [ F ] Fgure 2. Soluto pots (squares) ad flux pots (crcles) for = 2 4 Amerca Isttute of Aeroautcs ad Astroautcs

5 all the faces of the cell. Coversely, a ormal flux dfferece at a flux pot o a face, say (f, l) results a correcto at all soluto pots j of a amout α j, f, l[ F ] f, l S f / V. For 1D coservato laws, Eq. (12) reduces to Q, j F( Q ) 1 + Π j ( L, j[ F ] L R, j[ F ] R ) 0, t + x h α + α = (15) where h s the legth of elemet, whch has two terfaces, the left oe ad rght oe, wth ut face areas ad ut face ormals of -1 ad 1 respectvely, so that [ F ] = [ F], [ F ] = [ F], α L, j ad α R, j are costat lftg coeffcets 1D. Due to symmetry, we have αl, j = α R, + 2 j. For the 1D case, detals ca be foud III. Hybrd P N P M -CPR Formulato A. 1D Formulato Let s start from the 1D coservato law to preset the basc dea. Cosder a 2 d order CPR formulato wth two soluto pots (SPs) wth each cell. I the most effcet CPR formulato, the two ed pots of the cell are used as the SPs (usually called the Lobatto pots) sce t s ot ecessary to recostruct the solutos there for flux computato. Obvously teror pots such as the Gauss pots ca also be used as the SPs as show Fgure 3. I order to dstgush schemes based o these two types of SPs, P N P M -CPR-L s used to deote schemes based o the Labatto pots, whle P N P M -CPR-G s used to deote schemes based o the Gauss pots. I a P N P M formulato, both the left ad rght cells are used recostructg a hgher order polyomal, deoted as U (vs. Q, the polyomal defed based o DOFs at the curret cell). We employ a terpolato based recostructo approach for smplcty sce the DOFs are odal values at a gve set of pots. If the SPs are Lobatto pots, multple values exst at cell terfaces. Not all of them ca be used the recostructo. As we always prefer local data, ths meas the terface solutos at the two eghborg cells are excluded the recostructo. Therefore the hghest degree of U s 3 for P 1 P M -CPR-L schemes. But for P 1 P M -CPR-G schemes, the hghest degree s 5 because solutos at all sx soluto pots ca be used the recostructo. The recostructo stecls are show Fgure 3 for both types of schemes. The recostructo polyomal U s the used to geerate hgh-order updates the followg fasho. Frst t s used to compute the teror dvergece F( U ) L L R (a) Lobatto pots as SPs (b) Gauss pots as SPs Fgure 3. Schematc of Labatto ad Gauss pots as the soluto pots term,.e.. Next U s also used to compute x the dfferece betwee the commo flux ad the teror flux, e.g., [ F ] L ad [ F ] R. The commo Rema flux s computed wth the recostructo polyomals at eghborg cells. For example, the commo flux at terface +1/2 s computed usg R F = Fˆ ( U ( x ), U ( x )). (16) + 1/2, com Rem + 1/ /2 The teror flux s also computed wth the recostructo polyomal so that the flux dfferece s Fally the P N P M -CPR scheme s F = F ˆ ( U ( x ), U ( x )) F ( U ( x )) R Rem + 1/ /2 + 1/2. (17) 5 Amerca Isttute of Aeroautcs ad Astroautcs

6 Q, j F( U ) 1 + Π j + ( L, j[ F ] L R, j[ F ] R ) 0. t x h α + α = (18) Note that the lftg coeffcets ( αl, j, α R, j ) rema exactly the same. Obvously the hybrd scheme s uquely defed oce the recostructo polyomal U s determed. There are may choces o how U s recostructed. For the sae of accuracy ad stablty, the followg rules of thumb are establshed: The values of the recostructo polyomal at the soluto pots are detcal to the orgal solutos at the soluto pots,.e., where x, j deotes the jth soluto pot of cell. U ( x ) = Q ( x ), (19), j, j The recostructo stecl s symmetrc wth respect to cell. Ths s because upwdg s provded by the Rema flux so that a cetral recostructo stecl s preferred for the sae of accuracy ad stablty. Nearby data s always preferred tha far away data. A costraed least squares approach 7 s used whe the umber of DOFs o the recostructo stecl s larger tha the dmeso of the polyomal space, e.g., f 6 solutos are avalable to recostruct a degree 4 polyomal. Based o the above rules, the P N P M -CPR-L schemes are expected to be more effcet tha P N P M -CPR-G schemes sce the terface flux dfferece term s exactly the same as the CPR schemes. The oly dfferece s the flux dvergece term. I order to evaluate the performace of the P N P M -CPR formulato, we test the followg schemes: 1. P 1 P 3 CPR-L scheme I ths scheme, a uque degree 3 polyomal U s bult usg {Q -1, 1, Q,1, Q,2, Q +1, 2 }. 2. P 1 P 3 CPR-G-I scheme Ths scheme uses a complete stecl to buld a degree 3 polyomal by excludg two solutos further away from the curret cell. So the recostructo stecl s {Q -1, 2, Q,1, Q,2, Q +1, 1 }. 3. P 1 P 5 CPR-G-C scheme Ths scheme uses the complete stecl to buld the hghest polyomal - a degree 5 polyomal. So the recostructo stecl s {Q -1, 1, Q -1, 2, Q,1, Q,2, Q +1, 1, Q +1, 2 }. 4. P 1 P 3 CPR-G-C scheme Ths scheme uses the complete stecl to buld a degree 3 polyomal usg costraed least squares. The recostructo stecl s the same as that of P 1 P 5 CPR-G-C. 5. P 1 P 4 CPR-G-C scheme Ths scheme uses the complete stecl to buld a degree 4 polyomal usg costraed least squares. The recostructo stecl s the same as that of P 1 P 5 CPR-G-C. 6. P 2 P 6 CPR-L scheme I ths scheme, a uque degree 6 polyomal U s bult usg {Q -1, 1, Q -1, 2, Q,1, Q,2, Q, 3, Q +1, 2, Q +1, 3 }. 7. P 2 P 4 CPR-G-I scheme Ths scheme uses a complete stecl to buld a degree 4 polyomal by cludg two earby solutos from the eghborg cells. So the recostructo stecl s {Q -1, 3, Q,1, Q,2, Q,3, Q +1, 1 }. 8. P 2 P 6 CPR-G-I scheme Ths scheme uses a complete stecl to buld a degree 6 polyomal. So the recostructo stecl s {Q -1, 2, Q -1, 3, Q,1, Q,2, Q, 3, Q +1, 1, Q +1, 2 }. 6 Amerca Isttute of Aeroautcs ad Astroautcs

7 9. P 2 P 8 CPR-G-C scheme Ths scheme uses a complete stecl to buld the hghest polyomal - a degree 8 polyomal. All solutos from the eghborg cells are used the recostructo. 10. P 2 P 6 CPR-G-C scheme Ths scheme uses the complete stecl to buld a degree 6 polyomal usg costraed least squares. The above schemes are tested a grd refemet accuracy study usg the followg oe dmesoal lear wave equato: Q Q + = 0, x [0,1] (20) x wth the tal codto Q( x,0) = s(2 π x), ad perodc boudary codtos. The tme tegrato schemes used are the TVD Ruge Kutta schemes of 3rd or 4th order accuracy 12. The computato s carred out utl t = 1. The L 2 error s plotted Fgure 4. ad summarzed Table 1. Note that the hybrd P N P M -CPR formulato ca sgfcatly mprove the order of accuracy of the orgal CPR schemes. The hghest order of accuracy s 5 th wth the P 1 P M -CPR schemes ad 8 th wth the P 2 P M -CPR schemes. The umerc tests dcate that the costraed least squares recostructo s more stable tha the covetoal least squares method for the P M recostructo. Whe the umber of data tems the P M recostructo stecl s more tha the DOFs eeded for the hgher-order polyomal, P N P M -CPR schemes wth a complete stecl s more accurate tha those wth a complete stecl P1P3-CPR-G-C P1P3-CPR-G-I P1P3-CPR-L-C P1P4-CPR-G-C P1P5-CPR-G-C P2P4-CPR-G-I P2P6-CPR-G-C P2P6-CPR-G-I P2P6-CPR-L-C P2P8-CPR-G-C L L N N (a) P 1 P M CPR (b) P 2 P M CPR Fgure 4. L 2 error of the (a) P 1 P M CPR ad (b) P 2 P M CPR schemes for the 1D lear wave equato at t=1. B. 2D Formulato for a smplex 7 Amerca Isttute of Aeroautcs ad Astroautcs

8 The exteso of the P N P M -CPR schemes to a smplex s smlar to the 1D formulato ad qute straghtforward. Aga we have two choces to dstrbute the soluto pots, as show Fgure 5 the case of N = 1. If Lobatto pots are used the terpolato based recostructo, solutos at 6 uque locatos ca be used the recostructo, resultg a degree 2 polyomal. The scheme s amed a P 1 P 2 -CPR-L scheme. O the other had, f the soluto pots are the Gauss pots, oe of the SPs cocde wth each other. Thus all DOFs of the recostructo stecl ca be used for recostructg the hgher order polyomal U. Followg the rules of thumb descrbed earler, we ca desg the followg three schemes for N = 1: 1. P 1 P 3 CPR-G-C scheme I ths scheme, all DOFs the stecl are used to recostruct a degree 3 polyomal U usg a costraed least squares approach. 2. P 1 P 2 CPR-G-C scheme I ths scheme, all DOFs the stecl are used to recostruct a degree 2 polyomal U usg a costraed least squares approach. 3. P 1 P 2 CPR-G-I2 scheme I ths scheme, ot all DOFs the stecl are used the recostructo. Istead, oly the earest 6 DOFs from the three eghborg cells are selected ad a costraed least squares approach s employed to buld U. I2 stads for oly the earest 2 DOFs from each eghborg cell are used. Oce the hgher-order polyomal s obtaed, t s obvously used computg the teror dvergece term F( U ). There appear to be at least two optos how the correcto terms are computed multple dmesos. Opto CN (Correcto order N): The correcto term s computed exactly the same way as the CPR approach wth the same coeffcets α j, f, l, ad wth the same umber of flux pots alog each face. Therefore, o ew pots are added o the face. However, the flux dfferece s computed wth the recostructo polyomal U. For example, the commo flux s computed usg the recostructed solutos o both sdes of a face, ad the teror flux s also computed wth the recostructed hgher order polyomal at the curret cell. Opto CM (Correcto order M): The ormal flux dfferece s assumed to be of order M alog the face. As a result, extra flux pots are added o the faces to support a degree M polyomal. The ew correcto coeffcets α j, f, l are derved from the lftg operator gve Eq. 7. Obvously the recostructo polyomal s used to compute the flux dfferece at all the flux pots. (a) Labatto pots as SPs (b) Gauss pots as SPs Fgure 5. Dstrbuto of soluto pots for a smplex 2D I addto to the N = 1 schemes, the followg P 2 P M schemes are also tested: 4. P 2 P 3 CPR-G-C scheme I ths scheme, all DOFs the stecl are used to recostruct a degree 3 polyomal U usg a costraed least squares approach. 8 Amerca Isttute of Aeroautcs ad Astroautcs

9 5. P 2 P 3 CPR-G-I3 scheme I ths scheme, a complete stecl are used to recostruct a degree 3 polyomal U by excludg three solutos further away from the curret cell. I3 stads for oly the earest 3 DOFs from each eghborg cell are used. 6. P 2 P 3 CPR-L-C scheme I ths scheme, all DOFs the stecl are used to recostruct a degree 3 polyomal U usg a costraed least squares approach. 7. P 2 P 3 CPR-L-I2 scheme I ths scheme, a complete stecl are used to recostruct a degree 3 polyomal U by excludg oe soluto further away from the curret cell. I2 stads for oly the earest 2 DOFs from each eghborg cell are used. The above 2D schemes are tested wth a 2D lear wave equato: Q Q Q + + = 0, x y x [ 1,1], y [ 1,1], perodc boudary codtos, (21) uder the tal codto Q( x, y, 0) = s π ( x + y). The computato s carred out utl t = 1 o a set of rregular tragular mesh as show Fgure 6. The L 2 errors are plotted Fgure 7 ad summarzed Table 2. The results show that we get a extra order of accuracy wth the P 1 P M CPR ad P 2 P M CPR schemes. Due to the lmted umber of the avalable DOFs the P M recostructo stecl, the smulatos wth the P 1 P 3 CPR-G-C scheme ad the P 1 P 2 CPR-G-I2 scheme are ustable. I order to ehace the robustess of the P M recostructo, we eed to use more DOFs P M stecl tha eeded. For the same reaso, the P N P M -CPR schemes wth Gauss pots s preferred over the schemes wth Lobatto pots. Note that wth Lobatto pots, a CN correcto produces more stable ad accurate schemes tha a CM correcto; whle wth Gauss pots, a CM correcto s more stable ad accurate tha a CN correcto. We are ow performg aalyss to uderstad why ths s the case. Note also that complete stecls are more accurate tha the complete stecl. Ths s smlar to the 1D test Y X Fgure 6. Irregular 20x20x2 tragular mesh. 9 Amerca Isttute of Aeroautcs ad Astroautcs

10 (a) Gauss pots (b) Lobatto pots Fgure 7. L 2 error of the schemes wth (a) Gauss pots ad (b) Lobatto pots for the 2D lear wave equato at t=1. IV. Numercal Results for the Euler Equatos I ths sesso, umerc expermets are carred out for the Euler equatos for compressble flow. The twodmesoal Euler equatos ca be wrtte as Q F G + + = 0, (22) x y where the state varables Q ad vscd flux F ad G are: ρ ρu ρv u 2 ρ ρu + p ρuv Q =, F =, G = (23) ρv 2 ρuv ρv + p E u( E + p) v( E + p) I Eq. (23), p, u, v, ρ ad E are pressure, velocty compoet x ad y drectos, desty ad total eergy respectvely. The total eergy s related to the other varables accordg to p 1 2 E = + ρu. (24) γ 1 2 A. Accuracy Study wth Vortex Propagato Problem I ths case, we test the accuracy of the P N P M -CPR schemes for the two-dmesoal Euler equatos. The sotropc vortex propagato problem from Shu 29 s used. The tal codto has a mea flow of {ρ,u,v,p}={1,1,1,1} oto whch a sotropc vortex s added. The sotropc vortex cossts of perturbatos u, v γ ad temperature T, but o perturbato etropy S = p / ρ : ε 2 0.5(1 r ) ( δ u, δ u) = e ( y, x), (25) 2π 10 Amerca Isttute of Aeroautcs ad Astroautcs 2 ( γ 1) ε 2 1 r δt = e, 2 8γπ (26) δ S = 0, (27) Where r = x + y, ad the vortex stregth ε=5. The exact soluto of ths problem s just the passve covecto of the sotropc vortex wth the mea velocty. Fgure 8. shows the desty cotours at dfferet tme steps.

11 T=6 4 2 T=3 Y 0 T= RHO: X Fgure 8. Desty cotours for the vortex propagato problem at t=0, t=3 ad t=6. I the umercal smulato, the computatoal doma s tae to be [-10,10] [-10,10] wth perodc boudary codtos mposed o all of the outer boudares. As show Fgure 6, the rregular computatoal meshes are used. The computato s carred out utl t = 1. The L 2 error s plotted Fgure 9. ad summarzed Table 3. From the smulato results, we ote that the hghest order of accuracy we get s 3 rd at P 1 P x CPR schemes ad s 5 th at P 2 P x CPR schemes. We fd that for ths test case, smulatos of the Lobatto pots wth CN correcto ad the Gauss pots wth CM correcto are stll more stable ad accurate tha those of the other combatos. Icomplete stecl seems more accurate tha the complete stecl whch s the same as the prevous tests. (a)gauss pots (b) Lobatto pots Fgure 9. L 2 error of the schemes wth (a) Gauss pots ad (b) Lobatto pots for the 2D vortex propagato problem. B. Flow a Chael wth a Smooth Bump Ths teral aerodyamc problem s selected to test the order of accuracy of P N P M CPR scheme wth hghorder curved boudares. The chael has a heght of 0.8 ut ad a legth of 3 ut. The bump s defed as 2 25x e y = (28) The coarsest computatoal mesh whch has a total of 220 cells s show Fgure 10. The smooth bump s represeted wth quadratc segmets. Note that the mesh has mxed P1&P2 elemets ad the P2 elemets are used 11 Amerca Isttute of Aeroautcs ad Astroautcs

12 oly at the wall boudary. Characterstc boudary codtos are used at both the let ad outlet. The smulato s started from a uform free stream wth Mach umber 0.5 everywhere. Two approaches are used for the P M recostructo at the boudary cells. Oe s to use low order P N recostructo ad the other s to perform oe-sded P M recostructo. I order to crease the umber of DOFs the oe-sded P M recostructo stecl, the computatoal grd s geerated to mae sure every boudary cell has at most oe boudary face. The mplct lower-upper symmetrc Gauss-Sedel (LU-SGS) scheme s used for tme tegrato. The computed Mach cotour of P 2 P 3 -CPR-G-I3-C3 scheme s show Fgure 11. The L 2 orms of etropy error are plotted Fgure 12. ad summarzed Table 4. Because we eed to use oe-sded P M recostructo ear the boudares, oly the P N P M -CPR scheme wth gauss pots are used for ths test. From the smulato results, we ote that for the P 1 P 2 -CPR scheme wth P1 boudary, we get the smlar order of accuracy wth the P 1 P 1 -CPR. However, by usg the P2 recostructo o the teral cells the absolute etropy error s much smaller tha that of P 1 P 1 -CPR scheme. Sce the bggest etropy error s ear the wall boudary, the oe-sded P M recostructo o the boudary cells are eeded to acheve the optmal order of accuracy for P 1 P 2 ad P 2 P 3 CPR schemes. At the same tme, we got the smallest etropy error wth a complete P M stecl. I order to compare the performace of the P N P M -CPR scheme wth the classcal CPR scheme, the P 1 P 1 ad P 2 P 2 CPR results are show Fgure 11 ad Table 4 too. The results dcate that wth the same degree of freedom, P N P M -CPR scheme ca acheve much smaller error tha the orgal CPR schemes. Fgure 10. The coarsest tragular mesh wth P2 curved boudary for the smooth bump problem. Y X Fgure 11. Mach umber cotour for the smooth bump problem usg the P 2 P 3 -CPR-G-I3-C3 scheme. MACH Fgure 12. L 2 error for the smooth bump problem. 12 Amerca Isttute of Aeroautcs ad Astroautcs

13 V. Coclusos A P N P M -CPR formulato has bee developed ad tested the preset study. For the sae of compactess, we lmt the recostructo stecl to clude the curret cell ad ts mmedate face-eghbors. The use of multple degrees of freedom a sgle cell ad ts eghbors opes a host of ew possbltes to buld schemes of varous orders ad dfferet characterstcs. The preset formulato herts the smplcty of the fte-dfferece-le CPR method, ad attempts to acheve very hgh order of accuracy wth relatvely few local DOFs. I the preset study, several schemes for 1D ad 2D coservato laws have bee developed, ad evaluated grd refemet accuracy studes. These prelmary tests dcate that the ew formulato offers potetal gas accuracy, effcecy ad flexblty. Possble future wor wll clude exteso to the Naver-Stoes equatos. Acowledgmets The preset research s partally supported by the Ar Force Offce of Scetfc Research uder grat FA , Natoal Scece Foudato of Cha uder grat , ad Iowa State Uversty. Refereces 1 T.J. Barth ad P.O. Fredercso, Hgh-order soluto of the Euler equatos o ustructured grds usg quadratc recostructo, AIAA Paper No , F. Bass ad S. Rebay, Hgh-order accurate dscotuous fte elemet soluto of the 2D Euler equatos, J. Comput. Phys. 138, 1997, pp F. Bass, S. Rebay, A hgh-order accurate dscotuous fte elemet method for the umercal soluto of the compressble Naver Stoes equatos. J Comp Phys, Vol. 131, No. 1, 1997, pp B. Cocbur ad C.-W. Shu, TVB Ruge-Kutta local projecto dscotuous Galer fte elemet method for coservato laws II: geeral framewor, Mathematcs of Computato 52, 1989, pp B. Cocbur ad C.-W. Shu, The Ruge-Kutta dscotuous Garler method for coservato laws V: multdmesoal systems, J. Comput. Phys., 141, 1998, pp B. Cocbur, C.-W. Shu, The local dscotuous Galer methods for tme-depedet covecto dffuso systems, SIAM J. Numer. Aal. Vol. 35, 1998, pp M. Dumbser, D. Balsara, E.F. Toro ad C.D. Muz, A ufed framewor for the costructo of oe-step fte-volume ad dscotuous Galer schemes, Joural of Computatoal Physcs, Vol. 227, 2008, pp Dumbser, M, PNPM schemes o ustructured meshes for tme-depedet partal dfferetal equatos. I eds. Z. J. Wag, Adaptve Hgh-order Methods Computatoal Flud Dyamcs. World Scetfc, Sgapore, 2011, pp J.A. Eaterars, Hgh-order accurate, low umercal dffuso methods for aerodyamcs, Progress Aerospace Sceces, Vol. 41, 2005, pp G. J. Gasser, F. Lorcher, C-D. Muz, ad J. S. Hesthave, Polymorphc odal elemets ad ther applcato dscotuous Galer methods, J. Comput. Phys., Vol. 228, 2009, pp S.K. Goduov, A fte-dfferece method for the umercal computato of dscotuous solutos of the equatos of flud dyamcs, Mat. Sb., Vol. 47, 1959, pp S. Gottleb, C-W. Shu, Total varato dmshg Ruge Kutta schemes. Math Comput, Vol. 67, 1998, pp Hesthave J. ad Warburto, T., Nodal dscotuous Galer Methods, Sprger, T. Haga, H. Gao ad Z. J. Wag, A Hgh-Order Ufyg Dscotuous Formulato for the Naver-Stoes Equatos o 3D Mxed Grds, Math. Model. Nat. Pheom., press. 15 H.T. Huyh, A flux recostructo approach to hgh-order schemes cludg dscotuous Galer methods, AIAA Paper H.T. Huyh, A Recostructo Approach to Hgh-Order Schemes Icludg Dscotuous Galer for Dffuso, AIAA Paper Huyh, H.T., Hgh-order methods by correcto procedures usg recostructos. I eds. Z. J. Wag, Adaptve Hgh-order Methods Computatoal Flud Dyamcs, World Scetfc, Sgapore, 2011, pp G.E. Karadas, ad S.J. Sherw, Spectral-hp elemet methods. Oxford Uversty Press, Oxford, D.A. Koprva ad J.H. Kolas, A coservatve staggered-grd Chebyshev multdoma method for compressble flows, J. Comput. Phys., Vol. 125, 1996, pp M.-S. Lou, A sequel to AUSM, Part II: AUSM+-up for all speeds, J. Comput. Phys., Vol. 214, 2006, pp Y. Lu, M. Vour, ad Z.J. Wag, Spectral (Fte) Volume Method for Coservato Laws o Ustructured Grds V: Exteso to Three-Dmesoal Systems, Joural of Computatoal Physcs Vol. 212, 2006, pp Y. Lu, M. Vour, ad Z.J. Wag, Dscotuous Spectral Dfferece Method for Coservato Laws o Ustructured Grds, Joural of Computatoal Physcs Vol. 216, 2006, pp Luo, H., Luo, L., Nourgalev, R., Mousseau, V.A., ad Dh N., A recostructed dscotuous Galer method for the compressble Naver Stoes equatos o arbtrary grds, Joural of Computatoal Physcs, Vol. 229, 2010, pp G. May ad A. Jameso, A spectral dfferece method for the Euler ad Naver-Stoes equatos, AIAA paper No , Amerca Isttute of Aeroautcs ad Astroautcs

14 25 J. Perare ad P.-O. Persso, The compact dscotuous Galer (CDG) method for ellptc problems, SIAM J. Sc. Comput., Vol. 30, No. 4, 2008, pp W.H. Reed ad T.R. Hll, Tragular mesh methods for the eutro trasport equato, Los Alamos Scetfc Laboratory Report, LA-UR , P.L. Roe, Approxmate Rema solvers, parameter vectors, ad dfferece schemes, J. Comput. Phys., Vol. 43, 1981, pp V.V. Rusaov, Calculato of teracto of o-steady shoc waves wth obstacles, J. Comput. Math. Phys. USSR 1, 1961, pp C.-W. Shu, Essetally o-oscllatory ad weghted essetally o-oscllatory schemes for hyperbolc coservato laws, Lecture Notes Mathematcs, Vol. 1697, 1998, pp K. Va de Abeele, C. Lacor, ad Z. J. Wag. O the stablty ad accuracy of the spectral dfferece method. J. Sc. Comput., Vol. 37, No. 2,2008, pp B. va Leer, Towards the ultmate coservatve dfferece scheme V. a secod order sequel to Goduov s method, J. Comput. Phys. Vol. 32, 1979, pp B. Va Leer ad S. Nomura, Dscotuous Galer for dffuso, AIAA Paper No , Z.J. Wag, Spectral (Fte) volume method for coservato laws o ustructured grds: basc formulato, J. Comput. Phys. Vol. 178, 2002, pp Z.J. Wag, Hgh-order methods for the Euler ad Naver-Stoes equatos o ustructured grds, Joural of Progress Aerospace Sceces, Vol. 43, 2007, pp. 1-47,. 35 Z.J. Wag, Adaptve Hgh-Order Methods Computatoal Flud Dyamcs, World Scetfc Publshg, Sgapore, May Z.J. Wag ad H. Gao, A ufyg lftg collocato pealty formulato cludg the dscotuous Galer, spectral volume/dfferece methods for coservato laws o mxed grds, Joural of Computatoal Physcs, Vol. 228, 2009, pp Z.J. Wag, H. Gao ad T. Haga, A Ufyg Dscotuous Formulato for Hybrd Meshes, Adaptve Hgh-Order Methods Computatoal Flud Dyamcs, Edted by Z.J. Wag, World Scetfc Publshg, Sgapore, Z.J. Wag, ad Y. Lu, Spectral (fte) volume method for coservato laws o ustructured grds II: exteso to twodmesoal scalar equato, J. Computatoal Physcs, Vol. 179, 2002, pp T. Warburto, A explct costructo of terpolato odes o the smplex, J. of Egeerg Mathematcs, Vol. 56, 2006, pp Xua L.J. ad Wu, J.Z., A weghted-tegral based scheme, Joural of Computatoal Physcs, Vol. 229, 2010, pp Zhag L., Lu W., He L., Deg X., Zhag H., A class of hybrd DG/FV methods for coservato laws II: Two-dmesoal cases, J. Computatoal Physcs, press. 14 Amerca Isttute of Aeroautcs ad Astroautcs

15 Table 1. Grd refemet Accuracy study wth the 1D lear wave equato. Scheme No. of Cells L2 error L E E P 1 P 3 -CPR-L-C E E E E E P 1 P 3 -CPR-G-I E E E E E P 1 P 5 -CPR-G-C E E E E E P 1 P 3 -CPR-G-C E E E E E P 1 P 4 -CPR-G-C E E E E E P 2 P 6 -CPR-L-C E E E E E P 2 P 4 -CPR-G-I E E E E E P 2 P 6 -CPR-G-I E E E E E P 2 P 8 -CPR-G-C E E order 15 Amerca Isttute of Aeroautcs ad Astroautcs

16 Table 1. Cotued Scheme No. of Cells L2 error L E E P 2 P 6 -CPR-G-C E E E order Table 2. Grd refemet Accuracy study wth 2D lear wave equato. Scheme No. of Cells L2 error L2 10x10x2 6.28E-03-20x20x2 5.63E P 1 P 2 -CPR-G-C-C2 40x40x2 6.24E x80x2 7.15E x160x2 8.66E x10x2 1.26E-03-20x20x2 9.78E P 2 P 3 -CPR-G-C-C2 40x40x2 7.02E x80x2 4.75E x160x2 3.17E x10x2 3.06E-04-20x20x2 1.79E P 2 P 3 -CPR-G-C-C3 40x40x2 1.09E x80x2 6.66E x160x2 4.16E x10x2 2.26E-04-20x20x2 1.22E P 2 P 3 -CPR-G-I3-C3 40x40x2 6.92E x80x2 3.85E x160x2 2.28E x10x2 1.83E-03-20x20x2 1.25E P 2 P 3 -CPR-L-C-C2 40x40x2 8.59E x80x2 5.70E x160x2 3.83E x10x2 1.45E-03-20x20x2 9.51E P 2 P 3 -CPR-L-I2-C2 40x40x2 6.12E x80x2 3.89E x160x2 2.59E x10x2 2.23E-03-20x20x2 1.25E P 2 P 3 -CPR-L-I2-C3 40x40x2 7.36E x80x2 4.46E x160x2 2.91E order 16 Amerca Isttute of Aeroautcs ad Astroautcs

17 Table 3. Grd refemet Accuracy study wth the vortex propagato problem. Scheme No. of Cells L2 error L2 20x20x2 2.76E-03-40x40x2 3.30E P 1 P 2 -CPR-G-C-C2 80x80x2 3.34E x160x2 3.60E x320x2 4.19E x20x2 1.41E-03-40x40x2 1.98E P 1 P 2 -CPR-G-I2-C2 80x80x2 2.33E x160x2 2.85E x320x2 3.60E x20x2 1.00E-03-40x40x2 1.20E P 2 P 3 -CPR-G-C-C2 80x80x2 1.07E x160x2 7.56E x320x2 5.19E x20x2 4.65E-04-40x40x2 3.58E P 2 P 3 -CPR-G-C-C3 80x80x2 2.28E x160x2 1.30E x320x2 7.30E x20x2 3.01E-04-40x40x2 2.05E P 2 P 3 -CPR-G-I3-C3 80x80x2 1.28E x160x2 6.84E P 2 P 4 -CPR-G-C-C4 P 2 P 3 -CPR-L-C-C2 P 2 P 3 -CPR-L-C-C3 order 320x320x2 3.63E x20x2 3.84E-04-40x40x2 1.80E x80x2 5.94E x160x2 1.86E x320x2 6.00E x20x2 1.41E-03-40x40x2 1.42E x80x2 1.11E x160x2 7.96E x320x2 5.64E x20x2 1.83E-03-40x40x2 1.36E x80x2 9.46E x160x2 6.33E x320x2 4.22E Amerca Isttute of Aeroautcs ad Astroautcs

18 Table 3. Cotued Scheme No. of Cells L2 error L2 20x20x2 1.30E-03-40x40x2 1.12E P 2 P 3 -CPR-L-I2-C2 80x80x2 7.87E x160x2 5.59E x320x2 4.01E x20x2 1.84E-03-40x40x2 1.33E P 2 P 3 -CPR-L-I2-C3 80x80x2 9.54E x160x2 6.63E x320x2 4.61E order Table 4. Grd refemet Accuracy study wth the smooth bump problem. Scheme No. of Cells L2 error L2 P 1 P 1 -CPR-G-C-C1 P1 Boudary P 1 P 2 -CPR-G-C-C2 P1 Boudary P 1 P 2 -CPR-G-C-C2 oe-sded P2 Boudary P 1 P 2 -CPR-G-I2-C2 oe-sded P2 Boudary P 2 P 2 -CPR-G-C-C3 P2 Boudary P 2 P 3 -CPR-G-C-C3 oe-sded P3 Boudary P 2 P 3 -CPR-G-I3-C3 oe-sded P3 Boudary order E E E E E E E E E E E E E E E E E E E E E Amerca Isttute of Aeroautcs ad Astroautcs