Bild on Scrift: rur.pad UA Rur Zntrum für partill Diffrntialglicungn Bild mit Scrift: rur.pad UA Rur Zntrum für partill Diffrntialglicungn An ntropy stabl spactim discontinuous Galrkin mtod for t two-dimnsional comprssibl Navir-Stoks quations A. Hiltbrand and S. May Prprint 07-
An ntropy stabl spactim discontinuous Galrkin mtod for t two-dimnsional comprssibl Navir-Stoks quations Andras Hiltbrand and Sandra May In tis papr, w prsnt an ntropy stabl scm for solving t comprssibl Navir-Stoks quations in two spac dimnsions. Our scm uss ntropy variabls as dgrs of frdom. It is an xtnsion of an xisting spactim discontinuous Galrkin mtod for solving t comprssibl Eulr quations. T pysical diffusion trms ar incorporatd by mans of t symmtric SIPG or nonsymmtric NIPG intrior pnalty mtod, rsulting in t two vrsions ST-SDSC-SIPG and ST-SDSC-NIPG. T stramlin diffusion and sock-capturing trms from t original scm av bn kpt, but av bn adjustd appropriatly. Tis guarants tat t nw scm ssntially rducs to t original scm for t comprssibl Eulr quations in rgions wit undrrsolvd pysical diffusion. W sow ntropy stability for bot vrsions undr suitabl assumptions. W also prsnt numrical rsults confirming t accuracy and robustnss of our scms. ywords. Discontinuous Galrkin mtod, comprssibl Navir-Stoks quations, ntropy stability, ntropy variabls, intrior pnalty mtod, wall boundary conditions.. Introduction In tis contribution, w prsnt scms for solving t comprssibl Navir-Stoks quations in two spac dimnsions tat ar provn to b ntropy stabl. Our scms ar xtnsions of t mtod by Hiltbrand and Misra [0, 9] for solving yprbolic systms of consrvation laws. W us a vrsion spcific to t comprssibl Eulr quations. T scm by Hiltbrand and Misra as t following faturs: it uss ntropy variabls as dgrs of frdom instad of t classic consrvd variabls, uss a spactim ST discontinuous Galrkin DG approac on unstructurd grids, and involvs stramlin diffusion SD and sock-capturing SC trms. As a rsult, t scm can b sown to b ntropy stabl, is unconditionally stabl, is arbitrarily ig-ordr in smoot flow, and is robust in t prsnc of socks and discontinuitis. W xtnd t scm to solving t comprssibl Navir-Stoks quations by adding a suitabl tratmnt for t pysical diffusion trms rprsnting viscosity and at conduction. Tis work was supportd by ERC STG. N 30679, SPARCCLE Formrly: Sminar for Applid Matmatics, ETH Zuric, Rämistrass 0, 809 Zuric, Switzrland TU Dortmund, oglpotswg 87, 447 Dortmund, Grmany, sandra.may@mat.tu-dortmund.d
W considr bot t nonsymmtric NIPG and symmtric SIPG intrior pnalty formulation for tis purpos, rsulting in t vrsions ST-SDSC-NIPG and ST-SDSC-SIPG. In our xtnsion, w prsrv all t positiv qualitis of t original scm. In particular, w can sow ntropy stability of t rsulting numrical scm undr suitabl conditions for bot ST-SDSC-NIPG and ST-SDSC-SIPG. T qustion of wtr on still nds stramlin diffusion and sock-capturing trms wn approximating t comprssibl Navir-Stoks quations is quit controvrsial. For fficincy rasons, t pysical diffusion cannot b rsolvd vrywr in a typical computation. In rgions wr t pysical diffusion is sufficintly rsolvd,.g., in boundary layrs, t additional diffusion trms ar not ndd. Away from boundaris, t solution of t comprssibl Navir-Stoks quations can bav quit similar to t solution of t comprssibl Eulr quations wn t pysical diffusion is not sufficintly rsolvd. In ts rgions, w want to nsur tat our scm ssntially rducs to t original scm by Hiltbrand and Misra. Trfor, w do includ a suitabl xtnsion of t original SD and SC trms in our nw scms. T artificial diffusion trms ar constructd suc tat ty liminat most oscillations around socks and vanis wit t corrct ordr of convrgnc in smoot flow. In particular w obsrv for smoot flow convrgnc ordrs of O k+ for polynomial dgrs of ordr k wit a potntially wors rat for t ST-SDSC-NIPG vrsion for vn polynomial dgr k. In t litratur, tr xists a varity of DG mtods for solving t comprssibl Navir- Stoks quations tat ar basd on discrtizing t consrvd variabls of t systm, s,.g., [4, 5, 6, 7, 8, 9, 0,,, 3, 5, 7, 8, 30] and t rfrncs citd trin. Svral of tm us t IP mtod for discrtizing t diffusion trm for xampl t work by Hartmann and Houston [7, 8]. Otrs us,.g., t local discontinuous Galrkin LDG approac or t Bassi-Rbay approac. A unifid comparison of typical discrtizations for t diffusion trms can b found in [] for t cas of an lliptic modl problm. To t bst of our knowldg t abov mtods do not allow for tortical stability rsults for t cas of t actual diffusion oprator of t comprssibl Navir-Stoks quations. Toug to a smallr xtnt, tr is also som work basd on using ntropy variabls as dgrs of frdom. In [], Hugs t al. xamin t proprtis of t pysical diffusion and at conduction trms for t comprssibl Navir-Stoks quations wn ntropy variabls ar usd as dgrs of frdom. In [34], Sakib t al. us a spactim finit lmnt approac for solving t rsulting quations. T autors us discontinuous lmnts in tim but continuous lmnts in spac. Bart [, 3] uss a spactim DG approac and discrtizs t diffusion trm using t SIPG approac. To t bst of our knowldg toug dos not xamin ntropy stability for t actual discrt formulation nor dos includ sock-capturing trms for simulations for t Navir-Stoks quations. Furtr, van dr gt and coworkrs [7, 6, 3] av workd on solving t comprssibl Navir-Stoks quations using bot consrvd variabls and ntropy variabls. T autors us a spactim DG approac in combination wit an IP discrtization of t diffusion trm but also do not provid an xplicit proof of ntropy stability. T work by Zakrzad and G. May [39] is on of t fw ons tat dos provid ntropy stability stimats for t fully discrt vrsions. T autors xamin ntropy stability for diffrnt discrtizations of t diffusion trm, in particular for a LDG discrtization, a BR-typ discrtization, and also a form of t SIPG discrtization. Howvr, t spcific vrsion of t mployd SIPG discrtization is diffrnt from t on considrd r. Furtr, t autors do not includ sock-capturing trms and do not considr boundary contributions. Finally, May [9, 8] compars on-dimnsional xtnsions of t scm by Hiltbrand and
Misra using t IP discrtization and t LDG discrtization for t diffusion trm and provids corrsponding ntropy stability rsults. In tis contribution, w xtnd t mtod basd on IP discrtization to two dimnsions. T proof of ntropy stability for t SIPG approac is mor callnging in two dimnsions as t 8 8 diffusion matrix writtn wit rspct to ntropy variabls only as rank 5. W also xamin t cas of adiabatic solid wall boundary conditions r wras all of t abov mntiond contributions assum compact support of t solution. Furtrmor, w provid improvd artificial diffusion trms compard to [9] as wll as xtnsiv numrical rsults in two dimnsions. Tis papr is structurd as follow: in sction, w sortly rviw t original scm of Hiltbrand and Misra for solving yprbolic systms of consrvation laws to kp tis work slf-containd. In sction 3, w discuss proprtis of t comprssibl Navir-Stoks quations in two dimnsions wn ntropy variabls ar usd as dgrs of frdom. In sction 4, w prsnt our xtnsions ST-SDSC-SIPG and ST-SDSC-NIPG for solving t comprssibl Navir-Stoks quations. Tis includs t discrtization of t pysical diffusion trm as wll as t suitabl xtnsion of t artificial diffusion trms. In sction 5, w sow ntropy stability of our scms undr suitabl assumptions. Finally, in sction 6 w prsnt numrical rsults in on and two spac dimnsions for picwis polynomial spacs of dgrs on, two, and tr. W conclud wit a summary in sction 7.. Rviw of t spactim DG mtod for yprbolic systms In tis sction, w rviw t spactim DG formulation for systms of yprbolic consrvation laws tat our nw mtod is basd on to kp tis work slf-containd. For mor dtaild information, w rfr to [0, 9]. Considr a systm of yprbolic consrvation laws on t opn domain R givn by U t + F U x + F U x = 0, x, t R +, wr U = u,..., u m T : R + R m, m N, is t vctor of consrvd variabls and F k : R m R m is t flux function in x k -dirction, k =,. W us t sort-and notation U t = t U and FU xk = xk FU. W assum t xistnc of a strictly convx ntropy function S : R m R and of ntropy flux functions Q k : R m R, k =,, suc tat t corrsponding ntropy inquality is satisfid. W not tat tis assumption is satisfid for t comprssibl Eulr quations. On can tn dfin ntropy variabls = S U U := SU u,..., SU u m T and apply a cang of variabls to gt U t + F x + F x = 0, x, t R +, wr F k = F k U for brvity. T mtod is basd on using ts ntropy variabls as dgrs of frdom instad of t usual consrvd variabls. Bfor dscribing t discrtization of quation, w will first st t prrquisits for t spactim ms. At t n t tim lvl t n, w dnot t tim stp as t n = t n+ t n and t updat tim intrval as = t n, t n+. For simplicity, w assum tat t spatial domain R is boundd and polydral and dividd into a triangulation T, i.., a non-ovrlapping st of triangls suc tat T =. Furtrmor, w tak t usual conditions of ms and 3
sap rgularity for grantd. For a gnric lmnt cll, w dnot = diam, N = { T : mas d > 0}, diamtr of, nigbours of. T ms widt of t triangulation is T = max. A gnric spactim lmnt is t prism. W also assum tat tr xists an arbitrarily larg constant C suc tat /C t n C for all tim lvls n. On a givn triangulation T wit ms widt T, t discrt solution w will us t suprscript for rfrring to discrt variabls is sougt in t spac { k = W L [0, T ] m : Wi } Pk in, 3 ac componnt i m wr P k is t spac of tr-dimnsional polynomials of ordr k on t prism. T discrtization of t consrvation law is givn by: find k suc tat B DG, Φ + B SD, Φ + B SC, Φ = 0 Φ k. 4 In t following, w will giv t dtails for ac of t tr quasilinar forms, wic ar all nonlinar in t first argumnt and linar in t scond... T DG quasilinar form T form B DG is givn by B DG, Φ = U Φ t + F k Φ x k dx dt n, k= + Un+,, n+,+ Φ n+, dx Un,, n,+ Φ n,+ dx n, n, + F,,,+; ν Φ, dσx dt, 5 n, wit N Φ n,±x = lim ε 0+ Φ x, t n ± ε =, ν = unit normal for dg pointing outwards from lmnt, Φ,±x, t = lim ε 0+ Φ x ± εν, t, x, 6 for all Φ k, and a b = m i= a ib i for a, b R m. W still nd to spcify t numrical fluxs tat w us. coos t upwind flux for t tmporal numrical flux U: To nabl tim marcing, w U n,, n,+ = U n,. 7 4
For t spatial numrical flux F, w us a consistnt, consrvativ, and ntropy-stabl flux givn by F,,,+; ν = F k,,,,+ν k D,+, 8 k= wit D = D,,,+ ; ν. Hr, Fk, dnots an ntropy-consrvativ flux in x k - dirction. T xistnc of suc fluxs for any gnric consrvation law wit an ntropy framwork was sown by Tadmor [36]. Explicit xprssions of ntropy-consrvativ fluxs for t comprssibl Eulr quations av bn obtaind,.g., by Ismail and Ro []. T oprator D rprsnts a numrical diffusion oprator. For dtaild information also concrning t ntropy-consrvativ fluxs w rfr to [0, 9]... Stramlin diffusion and sock-capturing oprator If on only usd t B DG -form, i.., if on dfind t discrt solution as t solution of B DG, Φ = 0 Φ k, tn tis solution would typically xibit unpysical oscillations nar socks and contact discontinuitis. Trfor, a stramlin diffusion and a sock-capturing oprator ar addd, compar 4. Ts trms add artificial diffusion wr ndd in ordr to damp unpysical oscillations. T following form is usd for t stramlin diffusion oprator cf. [0, 3, 5, 4] B SD, Φ = n, wit intra-lmnt rsidual and scaling matrix U Φ t + Rs = U t + k= F k Φ DSD x k n,rs dx dt 9 F k xk, 0 k= D SD n, = C SD t n U. Hr, C SD dnots a positiv constant and is typically cosn to b 0. Furtr, U dnots t Jacobian DU and F k t Jacobian DFk. Not tat t intra-lmnt rsidual is wll dfind as t first drivativs ar takn of a polynomial function. T stramlin diffusion oprator adds numrical diffusion in t dirction of t stramlins. Howvr, on nds furtr numrical diffusion in ordr to rduc possibl oscillations at socks. For tis purpos, t following sock-capturing oprator similar to Bart [] is usd: B SC, Φ = n, wit Ũ = U Ṽn, for brvity and Dn, SC Φ t Ũ t + Ṽ n, = mas k= t n Φ x k Ũ x k dx dt, a x, t dx dt 5
bing t cll avrag. T scaling factor is D SC n, = wit ɛ := t n t Ũ t t n C SC Rs n, + k= t n x k Ũ x k dx dt + ɛ θ diam and θ / cosn as and Rs n, := U Rs Rs dx dt., b c Hr, C SC is a positiv constant, typically takn to b. W not tat in t original formulation of t sock-capturing trm [0, 9] bot an innr rsidual trm dfind by c and a boundary rsidual trm ntr t formula b. As t boundary rsidual trm as only littl influnc, w do not includ tis trm in our xtnsion to t comprssibl Navir-Stoks quations and trfor do not prsnt tis trm r..3. Entropy stability for nonlinar systms T dsign of t stramlin diffusion SD sock-capturing SC discontinuous Galrkin DG scm 4 is motivatd by t considration tat it as to b ntropy-stabl for a gnric nonlinar systm of consrvation laws, quippd wit an ntropy formulation. Tr olds t following torm. Torm. Partial rstatmnt of Torm 3. in [0]. Considr t systm of consrvation laws wit a uniformly convx ntropy function S and ntropy flux functions Q k k. For simplicity, assum tat t xact and approximat solutions av compact support insid t spatial domain. Lt t final tim b dnotd by t N. Tn, t stramlin diffusion sock-capturing discontinuous Galrkin scm 4 approximating is ntropy-stabl, i.., t approximat solutions satisfy SUN, x dx SU0, x dx. 3 On can also xtract t following proprty of t quasilinar form B DG from t proof of Torm. givn as proof of Torm 3. in [0]. Lmma.. Undr t conditions of Torm., tr olds B DG, SUN, x dx SU0, x dx. 4 3. T comprssibl Navir-Stoks quations T comprssibl Navir-Stoks quations in two spac dimnsions ar givn by U t + F U x + F U x = H U x + H U x, 5 6
wit and ρ U = ρu ρv, F U = E H U = 0 τ τ ρu ρu + p ρuv ue + p, F U =, H U = ρv ρuv ρv + p ve + p 0 τ τ,. τ u + τ v + κθ x τ u + τ v + κθ x Hr, ρ = ρx, t > 0 dnots t dnsity, u = ux, t t vlocity in x -dirction, v = vx, t t vlocity in x -dirction, p = px, t > 0 t prssur, and E = p γ + ρu + v t total nrgy wit γ > bing t adiabatic constant. Additionally, R > 0 is t gas constant, C v > 0 is t spcific at at constant volum, and θ = p Rρ > 0 rfrs to t tmpratur. T viscous strss tnsor τ is givn by T u u u τ = µ + + λ I, v v v wit suprscript T dnoting t transpos. W assum t viscosity paramtrs µ, λ and t conductivity κ > 0 to b constant. W us λ = 3 µ. W furtr assum t rlation btwn µ and κ/r to b givn by t Prandtl numbr Pr = 4γ/9γ 5 via κ R = γc vµ RPr = γ γ Pr µ. In ordr to writ t comprssibl Navir-Stoks quations in t form [ F U t + ] [ ] [ ] U D U D F = U Ux, 6 U D U D U on nds to dfin suitabl matrics D ij U, i, j =,. W do not giv t spcifics r w rfr t intrstd radr to [5]. W mpasiz tat t rsulting matrix D = D ij i,j=, wic is formulatd wit rspct to t consrvd variabls is not symmtric. Trfor, w rwrit 6 using ntropy variabls as dgrs of frdom. For t transformation to ntropy variabls, w us t pysical ntropy and t corrsponding ntropy flux in t following way S = ρs γ, Q = ρus γ, U x Q = ρvs, s = logp γ logρ. 7 γ Tis rsults in t ntropy variabls writtn in trms of primitiv variabls and s for simplicity γ s = γ ρu + v, p ρu p, ρv p, ρ p T. 8 7
Tn, w can rformulat t comprssibl Navir-Stoks quations 5 in ntropy variabls as follows [ F U t + ] [ ] [ ] A A F = x, 9 A A wit [ ] A A A = A A 0 0 0 0 0 0 0 0 0 4 3 v 4 4 0 3 v v 4 0 0 3 v 4 3 v 3v 4 = µ 0 0 v4 v 3 v 4 0 v 4 0 v v 4 4 0 v4 3 3 v v 4 v 3 v 4 4 3 v v 3 + χv 4 0 v 3 v 4 3 v v 4 3 v v 3 0 0 0 0 0 0 0 0 0 0 v4 v 3 v 4 0 v4 0 v v 4 0 3 v 4 0 3 v v 4 0 0 4 3 v 4 4 3 v 3v 4 0 3 v 3v 4 v v 4 3 v 4 v 3 0 v v 4 3 v 3v 4 4 3 v 3 v + χv 4 0 and χ = γ γ Pr. T matrix A as t following proprty []. Lmma 3.. T matrix A R 8 8 givn in 0 is symmtric positiv smi-dfinit. In t following lmma, w xamin A furtr. Lmma 3.. Lt t matrix R R 5 8 b givn by x 0 0 0 0 0 0 0 0 0 0 0 0 0 R = 0 0 0 0 0 0 0 0 0 0 0 0 0 0. 0 0 0 0 0 0 0 Dfin Tr olds:  = RAR T. i T matrix  R5 5 is givn by a a3 a3 a 4 a 7 a 8 a 33 a34 a37 a38  = a 4 a43 a 44 a 47 a 48 a 7 a73 a 74 a 77 a 78 a 8 a83 a 84 a 87 a 88 and tr olds A = R T ÂR. ii T matrix  is symmtric positiv dfinit for µ, κ > 0. 8
iii Lt EA dnot t st of ignvalus of t matrix A. Tn, EA = E {0} wit t dimnsion of t ignspac corrsponding to t ignvalu 0 bing 3. Proof. i Follows by dirct computation, xploiting tat columns 3 and 6 and rows 3 and 6, rspctivly, av t sam ntris. ii Follows by dirct computation,.g., by vrifying tat all lading principal minors ar positiv. iii Dfin 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 R xt = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Not tat R xt is an ortogonal matrix and tat  R xt AR T xt = 0 0 0.. Tis dirctly implis t claim. In sction 5, w will prov ntropy stability for t ST-SDSC-SIPG mtod for t comprssibl Navir-Stoks quations. To do so, w will nd t quotint of t largst and smallst ignvalu of  to b uniformly boundd. W can driv tis proprty from t following assumption wic rquirs uniform bounddnss of t computd solution. Assumption 3. Ass. for ST-SDSC-SIPG. W assum tat tr ar uniform lowr bounds ρ 0 > 0, p 0 > 0 suc tat ρ ρ 0 and p p 0. W furtr assum tat tr ar uniform uppr bounds ρ M, u M, v M, p M > 0 suc tat ρ ρ M, u u M, v v M, and p p M. Lmma 3.3. Undr Assumption 3., tr xist bounds λ and Λ suc tat 0 < λ λ... λ 5 Λ, wr λ i ar t ignvalus of  wit dnoting t discrt solution. Proof. Undr t Assumption 3., t ntropy variabls, 3, 4 ar uniformly boundd from abov. In addition, 4 = ρ ρ p 0 p M > 0. Tus, all t ntris in A as wll as in  ar boundd. Tis dirctly lads to an uppr bound on t largst ignvalu C i,j a i,j = tra A = trâ  = i λ i. 3 9
Lt us dnot t uppr bound of t ignvalus by Λ and assum tat t ignvalus ar sortd 0 λ... λ 5. Tn w av A lngty but dirct calculation yilds λ λ λ Λ... λ 5 Λ = dt Â Λ 4. 4 dt  = 8 3 κ R µ3 p 7. 5 Tis is boundd from blow by Assumption 3. and trfor tis stabliss a lowr bound on t ignvalus. 4. T ST-NIPG and t ST-SIPG mtod In tis sction, w prsnt our mtods ST-SDSC-NIPG and ST-SDSC-SIPG for solving t comprssibl Navir-Stoks quations in two spac dimnsions. Rlatd vrsions in on spac dimnsion av bn prsntd in [9]. In t following, w will focus on t dscription of t mtods in t intrior of t spac domain. Ncssary modifications to account for boundary conditions ar discussd in sction 4.3. 4.. T IP formulation W introduc t following notation: F rfrs to t collction of all dgs of t triangulation T wit F i rfrring to t collction of intrior dgs and F Γ rfrring to t collction of boundary dgs. For ac dg F i w assign a unit normal ν = ν, ν T,.g., to point from to. For an dg F Γ, ν is assumd to coincid wit t xtrior unit normal vctor. W dfin t avrag and jump for an dg F i sard by triangls and by { } =, +, and ρ [ ] =,,. For an dg F Γ, wic blongs to cll, w dfin { } = [ ] =,. For bot ST-SDSC-NIPG and ST-SDSC-SIPG, w sk t discrt solution k suc tat B DG, Φ + B IP SD, Φ + B IP SC, Φ + B IP,ζ, Φ = 0 Φ k. 6 Hr B DG is givn by 5; BSD IP and BIP SC ar modifications of t stramlin diffusion and sockcapturing trms, wic will b dscribd blow. T form B IP,ζ rprsnts t discrtization 0
of t diffusion trm and is givn by B IP,ζ, Φ = A x n, x A{ } n F i + ζ A{ } n F i + σ n F i + IP,ζ, Φ, n F Γ B Γ,n, Φ x Φ x dx dt { } { x } [Φ ]ν x [Φ ]ν { } Φ { x } [ Φ ]ν x [ ]ν A{ [ } ]ν [Φ [ ]ν ]ν [Φ ]ν dσx dt dσx dt dσx dt B B B3 7 wit B Γ,n, IP,ζ, Φ modling t bavior for boundary dgs F Γ and tim intrval. W will dscrib t dtails for Diriclt boundary conditions and adiabatic solid wall boundary conditions in sction 4.3. T paramtr σ > 0 rprsnts a pnalty paramtr and dnots t lngt of t dg tat is intgratd ovr. W not tat t dfinition of B IP,ζ is indpndnt of t coic of t dirction of t normal ν. Notation 4.. T mtod is calld NIPG mtod for ζ = and SIPG mtod for ζ =. 4.. Stramlin diffusion and sock-capturing oprator W adjust t stramlin diffusion and sock-capturing trms in ordr to account for t prsnc of t diffusion trm. Diffrnt to 0, t intra-lmnt rsidual is now givn by Rs IP = U t + F k xk k= A k, x + A k, x k= x k. 8 On could tn dfin a sock-capturing trm BSC IP witout furtr cangs otr tan using Rs IP in t dfinition of Rs n,, cmp. c, instad of Rs. For t stramlin diffusion, on nds to mak t following adjustmnt BSD IP, Φ = U Φ t + F k Φ x k n, k= A k, Φ x + A k, Φ x k= x k DSD n, Rs IP dx dt. 9 Tis adjustmnt is ncssary in ordr to nsur t ntropy stability of t rsulting mtod. If w us ts formulations of BSC IP and BIP SD in 6, w will obsrv suboptimal convrgnc rats of O k for tsts involving smoot flow compar t corrsponding on-dimnsional rsults in [9]. Tis was not t cas for t original scm 4 for consrvation laws wn t artificial diffusion trms wr includd. W bliv tat tis is du to t fact tat now scond-ordr drivativs ntr t computation of t rsidual and trfor rduc t ordr
of convrgnc of t rsidual. In [6], Hartmann prsnts sock-capturing trms for t comprssibl Navir-Stoks quations. If w multiplid BSC IP on a cll-wis lvl wit 0.9, t sock-capturing trms would av crtain similaritis. Howvr, in tis cas our rsulting sock-capturing trm would not rduc to t sock-capturing trm for t comprssibl Eulr quations if t pysical diffusion is not sufficintly rsolvd; as a rsult, oscillations migt not b sufficintly dampd. W trfor adjust t formulation of t stramlin diffusion and sock-capturing trm diffrntly: In [0], t autors introducd a prssur scaling trm in B SC in ordr to captur contact discontinuitis for t comprssibl Eulr quations mor sarply: ty cangd t trm Dn, SC in b in t following way wit D p n, = D SC n, D SC n, D p n, 30 t n t n k= p x k x k dx dt I p dx dt. 3 n T autors did not includ tis trm in tir formulation of B SC for gnral systms of consrvation laws as tis adjustmnt is spcific to t comprssibl Eulr quations. W will us tis formulation in our mtod for t comprssibl Navir-Stoks quations. To b consistnt, w also cang t stramlin diffusion trm and scal D SD n, dfind in wit D p n,. W summariz our cangs compard to t stramlin diffusion and sock-capturing trms of t original scm: us t dfinition of t cll-wis rsidual Rs IP givn by 8 instad of Rs givn by 0; also cang tis in t dfinition of Rs n, in c; us t dfinition of B IP SD givn by 9 instad of B SD givn by 9; multiply D SC n, 4.3. Boundary conditions in b and DSD n, D SC n, D SC n, D p n, in wit t prssur scaling trm dfind in 3: and DSD n, D SD n, D p n,. 3 W now prsnt t dtails of B Γ,n, IP,ζ for t cas of Diriclt and adiabatic solid wall boundary conditions. Lt F Γ blong to a cll and dnot by ν t xtrior unit normal of triangl on dg. 4.3.. Diriclt boundary conditions For imposing t Diriclt boundary conditions U = g wakly on an dg F Γ, t trm B Γ,n, IP,ζ in 7 uss t following modifid vrsions of B B3 from 7: Φ, ν B Γ,n, IP,ζ, Φ = + ζ + AS U g x,, x,, Φ AS U g x,, σ AS U g Φ x,, Φ, ν, S Ugν dσx dt B, S U gν, S Ugν, S U gν Φ, ν Φ, ν dσx dt B dσx dt. B3
4.3.. Adiabatic solid wall boundary conditions W nforc on F Γ t conditions u = v = 0 no slip condition and κ θ ν = 0 no at flux condition. To tis nd, w dfin basd on t function valu, on t dg t vctor v Γ = 0, 0, 0, v 4, T. W not tat t ntry v, will not play a rol in t following. Furtr, w dfin A µ v Γ µγ as Av Γ but wit t at conduction trms in ntris Av Γ 4,4 and Av Γ 8,8 γ Pr v4 bing rmovd. Tn, for F Γ, t trm B Γ,n, IP,ζ in 7 wic capturs t appropriat modifications of B B3 is givn by B Γ,n, IP,ζ, Φ = A µ v Γ x,, Φ, ν dσx dt B + ζ + x,, Φ A µ v Γ x,, σ A µ v Γ Φ x,,, v Γ ν Φ, ν, v Γ ν, v Γ ν, v Γ ν Φ, ν Φ, ν dσx dt B dσx dt. B3 W not tat t vctor A µ v Γ v Γ as only zro ntris. W kp it toug for consistncy wit t formulation for intrior dgs and Diriclt boundary conditions. 5. Entropy stability In tis sction, w xamin undr wic conditions t suggstd formulations of t ST-SDSC- NIPG and t ST-SDSC-SIPG mtod ar ntropy stabl for t comprssibl Navir-Stoks quations. For now, w will focus on t cas of bot t discrt and t continuous solution aving compact support. Dtails concrning t ntropy conditions in t prsnc of adiabatic solid wall boundary conditions will b prsntd in sction 5.. Torm 5. Entropy stability for ST-SDSC-NIPG. Considr t comprssibl Navir- Stoks quations 5 and lt t ntropy pair S, Q b givn by 7. For simplicity, assum tat t xact and approximat solution av compact support insid t spatial domain. Lt t final tim b dnotd by t N. Tn, t approximat solutions gnratd by t scm 6 wit ζ = and σ > 0 satisfy SUN, x dx SU0, x dx. Torm 5. Entropy stability for ST-SDSC-SIPG. Lt Assumption 3. and t assumptions of Torm 5. old tru. Tn, t approximat solutions gnratd by t scm 6 wit ζ = satisfy SUN, x dx SU0, x dx, 3
providd σ is cosn sufficintly larg suc tat σ c invλ λ wr λ, Λ ar dfind in Lmma 3.3 and t constant c inv will b spcifid blow in Lmma 5.4. In ordr to prov ts torms, w nd t following auxiliary rsults. Lmma 5.. For t ST-SDSC-NIPG mtod, undr t assumptions of Torm 5., tr olds B IP,, 0. Proof. By dfinition, tr olds for aving compact support B IP,, = A x n, x x x dx dt { } A{ } { x } [ n F ]ν x [ ]ν i + { } A{ } { x } [ n F ]ν x [ ]ν i + σ A{ [ } ]ν [ n [ ]ν ]ν [ ]ν F i dσx dt dσx dt dσx dt. T trms in t scond and tird lin cancl ac otr. As A is positiv smi-dfinit according to Lmma 3. and σ > 0, t trms in t first and last lin ar non-ngativ. Tis implis t claim. Lmma 5.. For t ST-SDSC-SIPG mtod, undr t assumptions of Torm 5., tr olds B IP,, 0. T proof is fairly lngty and givn blow. Wit ts prrquisits, w first want to prsnt t proof of Torms 5. and 5. kping in mind tat Lmma 5. still nds to b sown. Proof of Torms 5. and 5.. Tsting in 6 wit Φ = rsults in B DG, + B IP SD, + B IP SC, + B IP,ζ, = 0. W considr ac of t four trms individually:. Trm B DG, : According to Lmma., tr olds B DG, SUN, x dx SU0, x dx. T proof transfrs dirctly from comprssibl Eulr quations to comprssibl Navir- Stoks quations. 33 4
. Trm B IP SD, : Claim: Tr olds B IP SD, 0. Proof: W ssntially follow t proof of Torm 3. in [0]. Basd on our nw dfinition of t stramlin diffusion trm givn by 9, tr olds by cain rul BSD IP, = Rs IP DSD n, Rs IP dx dt. n, Wit t dfinition of D SD n, givn by 3 and and du t ntropy S bing strictly convx, tis implis BSD IP, 0. 3. Trm B IP SC, : Claim: Tr olds B IP SC, 0. Proof: By dfinition compar a and sction 4. B IP SC, = n, D SC n, t U Ṽn,t + k= t n x k U Ṽn,x k dxdt wit Dn, SC bing givn by 3 and b but wit Rs n, bing basd on Rs IP instad of bing basd on Rs. Du to t strict convxity of t ntropy function S, bot U and U ar strictly positiv dfinit. Tis implis DSC n, 0. Tis also dirctly implis BSC IP, 0. 4. Trm B IP,ζ, : Basd on Lmmata 5. and 5. tr olds for bot t ST-SDSC- NIPG and t ST-SDSC-SIPG mtod undr t rspctiv assumptions B IP,ζ, 0. Summarizing t stimats for t four trms rsults in 0 = B DG, + BSD IP, + BSC IP, + B IP,ζ, SUN, x dx SU0, x dx + 0 + 0 + 0, wic implis t claim. Tis concluds t proof of ntropy stability for ST-SDSC-NIPG. In ordr to sow ntropy stability for ST-SDSC-SIPG, it rmains to prov Lmma 5.. To do so, w nd t following lmma. Lmma 5.3. Lt t matrix C : R m R m b symmtric positiv smi-dfinit. Tn tr olds for arbitrary vctors v, w R m and δ > 0 w T Cv δw T Cw + δ vt Cv. 5
Proof. T proof follows dirctly from 0 δ δw v T Cδw v = δw T Cw w T Cv + δ vt Cv. In t proof of Lmma 5., w also nd to apply t following invrs trac stimat [33, 38]. Lmma 5.4. Tr olds for p P k p x dσx c inv p x dx wit c inv = c k and wit dnoting t diamtr of t cll. W can now procd to proving Lmma 5.. Proof of Lmma 5.. W xploit tat t solution as compact support and tat trfor t contributions from dgs F Γ drop out. For simplicity, w will just writ in t following wit t maning of F i. Using tat A is symmtric, tr olds B IP,, = x x A x n, x dx dt { } [ ]ν [ ]ν A{ } { x } n, dσx dt x + σ [ ]ν [ ]ν A{ [ } ]ν n, [ ]ν dσx dt. Applying Lmma 5.3 wit arbitrary δ > 0 to t trms in t middl lin givs { } [ ]ν [ ]ν A{ } { x } x { } { } { } δ x x A{ } { x } + [ ]ν [ ]ν A{ [ } ]ν x δ [ ]ν wit δ to b dtrmind latr. W not tat for σ δ, t scond trm can trivially b boundd by t pnalty trm. Lt us trfor focus on t first trm. Applying Lmma 3. and using t trin dfind matrics R and  implis x x A x dx dt n, n, = n, n, x δ { x } { x } A{ } x x R T  R x dx dt δ { x } x { x } R T Â{ }R { } { x } dσx dt x { } { x } dσx dt. x 6
W not tat  is positiv dfinit and tat by Assumption 3. its ignvalus ar uniformly boundd. Trfor, t abov trm can b boundd from blow by λ x x R T R x dx dt n, n, = n, n, Λδ { x } x { x } R T R { } { x } dσx dt x λ v x + v 3 x + v x + v 4 x + v 3 x + v 4 x dx dt Λδ {v x } + {v 3 x } + {v x } + {v 3 x } + {v 4 xk } dσx dt. W want to transform t boundary intgral to a domain intgral. Lt b t dg btwn clls and and not tat {v j xd } = v j xd, + v j xd, k= v j x d, + v j x d,. 34 Dnot by,, and 3 t tr dgs of a triangl. Furtr not tat for fixd t v j xd t, x, x is a polynomial of dgr k in x, x. W can apply t invrs trac stimat from Lmma 5.4 to gt Tis implis n, n, = n, n, 3 k= k v j x d, dσx = v j x d, dσx c inv Λδ {v x } + {v 3 x + v x } + {v 3 x } + Λδ 3 k= k v j x d dx. {v 4 xk } dσx dt v x, + v x, +... + v 4 x, + v 4 x, dσx dt k= Λδ v k x, +... + v 4 x, dσx dt c invλδ k v x + v 3 x + v x + v 3 x + v 4 x k dx dt, wr w av rordrd t sum ovr dgs as sum ovr triangls and av ignord contributions from t domain boundary Γ. As t lngt of ac triangl dg k can b boundd k= 7
by t diamtr of t cll, tis implis λ x x R T R x dx dt n, n, n, Λδ { x } λ c invλδ x { x } R T R { } { x } dσx dt x v x + v 3 x + v x + v 3 x + v 4 x k k= dx dt. To summariz rsults, tr olds B IP,, λ n, c invλδ v x + v 3 x + v x + v 3 x + + σ δ [ ]ν [ ]ν A{ [ } ]ν n, [ ]ν dσx dt, v 4 x k k= dx dt wic implis t claim if δ by 33. λ c inv Λ and σ δ, rsulting in t condition σ c invλ λ, as givn 5.. Boundary conditions W now xamin wtr Torms 5. and 5. still old tru in t prsnc of adiabatic solid wall boundary conditions, wic ar commonly usd for t comprssibl Navir-Stoks quations. For ts boundary conditions, tr sould old d dt SUx, t dx 0. W assum t boundary of to b split into parts Γ adia, t part of t boundary on wic w nforc adiabatic solid wall boundary conditions, and Γ rmaindr, t rmaining part of t boundary. W assum t two parts to b sparatd. W will ignor t boundary part Γ rmaindr by assuming compact support of t solution insid Γ adia. A classic application for tis stup is flow around an airfoil compar sction 6.5, wr Γ adia corrsponds to t airfoil boundary and Γ rmaindr corrsponds to t far fild boundary. Tr olds t following torm. Torm 5.3 Entropy stability for adiabatic solid wall boundary conditions. Considr t comprssibl Navir-Stoks quations. Lt t assumptions of Torms 5. and 5. old tru, but rquir σ c invλ λ for t ST-SDSC-SIPG mtod and only assum compact support of t solution insid Γ adia. For t numrical nforcmnt of boundary conditions on Γ adia follow sction 4.3. for t B IP,ζ -trm and t dscription blow for t B DG -trm. Follow [] for t dfinition of t ntropy consrvativ flux F k, and us Rusanov diffusion for t oprator D. Tn, t approximat solutions gnratd by t scm 6 wit ζ = or ζ = satisfy SUN, x dx SU0, x dx. 8
Lik in t proof of Torms 5. and 5., w again nd to xamin all four quasilinar forms apparing in 6: B DG, BSD IP, BIP SC, and B IP,ζ. Examining t dfinitions of t artificial diffusion trms BSD IP and BIP SC, w find tat only spactim domain trms ntr t formulation, no spatial boundary trms. Trfor, t proof of Torms 5. and 5. dirctly transfrs. Nxt, w considr t trm B DG. In t proof for t cas of compact support insid, w rfrrd to Lmma. wic is sown in Hiltbrand and Misra [0] for an stimat for tis trm. T sam stimat as claimd in Lmma. can b sown for t cas considrd in Torm 5.3. As t focus of tis contribution is on t viscous trms, w provid t corrsponding rsult as Lmma A. in appndix A. Hr, w sortly dscrib ow w nforc t boundary conditions: lik for intrior dgs, w comput a numrical flux for dgs on Γ adia. On input argumnt is t stat from t intrior of t flow domain, dnotd by,. W dfin t otr input argumnt for t numrical flux F as T,+ = v,, v,, v3,, v4,. 35 Tis corrsponds to invrting t vlocity vctor and nsurs tat t scond and tird componnt of, +,+ vanis. Finally, w nd to sow in t prsnc of adiabatic solid wall boundary conditions B IP,ζ, 0. 36 W will do tat in t following by xamining t spcial cas of a cll tat as dgs F Γadia, on wic adiabatic solid wall boundary conditions ar nforcd. 5... ST-SDSC-NIPG Lt us focus on t cas of a boundary dg F Γadia. Tr olds A µ v Γ v Γ = 0. Trfor, du to t symmtry of A µ, t boundary trms B and B in t dfinition of B Γ,n, IP,ζ cancl ac otr for ζ =. Furtr, du to A µ bing positiv smi-dfinit, t trm B3 is non-ngativ. Togtr wit t considrations for intrior dgs, tis implis 36. 5... ST-SDSC-SIPG For intrior dgs F i, w procd as dscribd in t proof of Lmma 5.. Lt us trfor focus on t boundary dg F Γadia. Diffrnt to t ST-SDSC-NIPG mtod, w nd to includ t domain trm x x A x dx dt x in our considrations. Using A µ v Γ v Γ = 0, w considr x x A x dx dt + x,, x,, σ, ν, ν x A µ v Γ, ν, ν A µ v Γ, ν, ν dσx dt dσx dt. 9
W procd as in t proof of Lmma 5. and apply Young s inquality to split t trm in t middl lin. Tn, w bound on part by t pnalty trm. T otr part can b boundd by t domain trm following t proof of Lmma 5. togtr wit t boundary trms from intrior dgs. Not tat w did not cang t matrix occurring in t domain trm. Trfor, w can bound tis trm from blow using t lowr bound for t ignvalus λ providd Assumption 3. olds tru. Not furtr tat t ignvalus of A µ v Γ can still b boundd from abov by Λ from Lmma 3.3. Comparing wit t proof of Lmma 5., w find tat tr is a factor of wic was prsnt for intrior dgs, compar 34 missing. Trfor, w nd to incras t pnalty paramtr σ by a factor of on boundary clls to guarant ntropy stability for adiabatic solid wall boundary conditions compard to t cas considrd in Torm 5.. 6. Numrical rsults In t following, w prsnt numrical xampls, mostly comparing wit standard tsts from t litratur, to tst our claims about ig-ordr accuracy in smoot flow and robustnss of t scm. Following 8, t numrical flux F is split into two parts: W follow [] for t dfinition of F k, t ntropy-consrvativ flux in x k -dirction for t comprssibl Eulr quations and us Rusanov diffusion for t oprator D [0]. Also, w will sligtly cang our notation and us x, y to dnot coordinats instad of x, x for our tsts in two spac dimnsions. W will start wit t Sod tst in on spac dimnsion in ordr to confirm t appropriat bavior of our artificial diffusion trms clos to socks and contact discontinuitis. In many tst cass, t rsults for ST-SDSC-SIPG and ST-SDSC-NIPG ar vry similar to ac otr. W will trfor only prsnt rsults for ST-SDSC-NIPG unlss otrwis spcifid. W us σ = 0 and σ = 0 for our computations for ST-SDSC-NIPG and ST-SDSC- SIPG, rspctivly. For our on-dimnsional tst, w us a uniform ms. For our tsts in two dimnsions, w us a structurd or unstructurd triangl ms in spac. Altoug not ndd for stability, w typically us for our tim-dpndnt tst problms a CFL condition wit CFL numbr 0.5 takn wit rspct to t convctiv part of t quations for accuracy rasons. 6.. Sod tst in d W start wit a vrsion of t Sod tst, similar to t tst in [37, 9]. Altoug t focus of tis contribution is on d, w start wit tis d tst problm as it is vry wll-undrstood and vry suitabl for tsting t bavior of our artificial diffusion trms. W considr initial data {.0, 0.0,.5 if x < 0, ρ, m, E = 0.5, 0.0, 0.5 if x > 0, on t domain = 0.5, 0.5. T viscosity ν =.5 0 6 is fairly small. Figur sows t rsult for dnsity for t final tim T = 0. for polynomial dgrs on, two, and tr and for two diffrnt grid rsolutions =.0 0 and =.5 0 4. Altoug w solvd t comprssibl Navir-Stoks quations, t small diffusion trms could not b rsolvd on t coars grid wit ms widt =.0 0 and trfor t solution bavs similar to t solution of t comprssibl Eulr quations: w obsrv oscillations around contact discontinuity and sock wn not using artificial viscosity trms. T oscillations ar 0
. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. w/o w/ 0. 0.5 0 0.5. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. w/o w/ 0. 0.5 0 0.5. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. w/o w/ 0. 0.5 0 0.5 a =.0 0, b =.0 0, c =.0 0, 3. w/o w/. w/o w/. w/o w/ 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.6 0.6 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0. 0. 0. 0. 0.5 0 0.5 0. 0.5 0 0.5 0. 0.5 0 0.5 d =.5 0 4, =.5 0 4, f =.5 0 4, 3 Figur : Sod tst: Rsult for dnsity for varying polynomial dgr and grid rsolution. T plots sow t solution for using t mtods bot witout BSD IP and BIP SC trms w/o and wit BSD IP and BIP SC trms w/.
mostly gon wn t BSD IP and BIP SC trms includd. Tis sows tat a stabilization trms ar ndd in undrrsolvd rgions of t flow and b tat our BSD IP and BIP SC trms ar suitabl for tat purpos. If w solv t comprssibl Navir-Stoks quations on a finr grid wit widt =.5 0 4 t pysical diffusion will srv as stabilization. W obsrv in Figur tat wit incrasing polynomial dgr t ovrsoot dcrass du to a bttr rsolution of t pysical diffusion. Tus artificial stabilization is not rally ndd. But t rsults also sow tat our artificial trms still sligtly nanc t solution for tis scnario. 6.. Manufacturd solution To tst t accuracy of our scm in smoot flow w us t tst of a manufacturd solution providd in t litratur [5] but w us a squar domain 3, 3 witout t cylindr cut out. As our artificial diffusion trms ar not mant to dal wit sourc trms w do not includ tm in t simulation. T rsults for t L rror ovr all 4 componnts at t final tim T = 0. for ν = 0.0 for ST-SDSC-SIPG and ST-SDSC-NIPG witout BSD IP and BIP SC ar sown in Figur. W obsrv convrgnc rats of O k+ for all scnarios. Also, t actual rrors for ST-SDSC- SIPG and ST-SDSC-NIPG ar almost idntical. W xpctd to s suboptimal convrgnc rats of O for ST-SDSC-NIPG for. W attribut t surprisingly good convrgnc rat to t fact tat t tst problm is too simpl and uss u = v =. 0.0 0.0 3 0 4 4.0 0 4 4.0 3.0 3.0 0 6 0 6 0 0 0 0 0 0 a ST-SIPG b ST-NIPG Figur : Rsults for manufacturd solution: L rror masurd ovr all componnts. x-axis dnots t ms widt, t y-axis t L rror. T 6.3. Stady stat tst wit smoot solution To b abl to also includ our artificial diffusion trm, w construct a smoot solution of t comprssibl Navir-Stoks quations ourslvs. W rstrict ourslvs to a stady stat, axisymmtric cas suc tat w ar abl to construct a rfrnc solution by numrical intgration of a systm of ODEs. Mor prcisly, considr t stady stat comprssibl Navir-Stoks Not tat tr is a typo in [5, p. 76, q 75]: t trm 3ω in s and s 3 nds to b rplacd by 3k.
0 0 4.0 0 0 4.0 3 0 6 0 6 3.0 4.0 3.0 4.0 0 8 0 8 0 0 a ST-SDSC-SIPG 0 0 b ST-SDSC-NIPG Figur 3: Rsults for stady stat tst wit smoot solution: L rror masurd ovr all componnts. T x-axis dnots t ms widt, t y-axis t L rror. Dasd lins corrspond to rsults wit BSD IP and BIP SC, solid lins to rsults witout BIP SD and BIP SC. quations in polar coordinats r, φ wit vlocity componnts u r and u φ. In t following, suprscripts r and φ dnot t rspctiv componnts of u, subscripts dnot partial drivativs. For simplicity, w st tangntial vlocity u φ = 0. Tis rsults in ρu r r = r ρur ρu r + p r = r ρur + r rτ rr r r τ φφ 37 u r E + p r = r ur E + p + r rτ rr u r r + κ r rθ r r wit τ rr = µ + λu r r + λ ur r, τ φφ = µ + λ ur r + λur r, E = p γ + ρur, θ = p Rρ. Hr, unknowns dnsity ρ, radial vlocity u r, and prssur p ar functions of t radius r only. W solv t first quation in 37 for ρ analytically using intgration. W solv t rmaining two quations wit unknowns u r and p numrically wit ig accuracy. W us t initial conditions ρ =, u r =, p =, u r r = 0., and p r = 0. and paramtr valus µ =.5/ 0 and κ/r =.875 0. W solv for r [, 3]. W us t rsult as a rfrnc solution tat w compar our numrical solution wit. For our numrical tst, w us t following data: t initial and boundary data ar givn by t rfrnc solution, and w solv on t domain /, + / until w av racd stady stat. Figur 3 sows t L rror masurd ovr all componnts for t ST-SDSC-SIPG and t ST-SDSC-NIPG scm for bot options of using t trms BSD IP and BIP SC dasd lin and not using tm solid lin. W ssntially obsrv optimal convrgnc rats of O k+ for all scnarios. In particular, 3
w do not obsrv a dcay of convrgnc ordr for t ST-SDSC-NIPG scm for ; our artificial diffusion trms BSD IP and BIP SC do not dtriorat t convrgnc ordr of our scm; also in trms of t actual siz of t rrors, t rsults wit BSD IP and BIP SC ar fairly clos to t rsults witout artificial diffusion trms; it is not clar wy for t solution wit artificial diffusion sows smallr rrors tan witout artificial diffusion; for on grid lvl finr, t rsults wit BSD IP and BIP SC ar still sligtly bttr tan witout but t quotint is clos to. B IP SD and BIP SC 6.4. Blasius boundary layr Nxt, w considr t classic Blasius boundary layr tst for low-spd laminar flow along an adiabatic plat. Undr t assumptions tat t flow is incomprssibl and tat t Rynolds numbr is sufficintly larg, on assums t solution of t comprssibl Navir-Stoks quations in t boundary layr to b clos to t solution of t Prandtl boundary layr quations tat w want to compar our rsults wit. 0.8 0.6 0.4 Rf sol 0. 3 0 0 3 4 5 6 7.8.6.4. 0.8 0.6 0.4 Rf sol 0. 3 0 0 3 4 5 6 7 a η vs. ũ b η vs. ṽ Figur 4: Rsults for Blasius boundary layr: scald vlocitis ũ and ṽ. W solv t comprssibl Navir-Stoks quations wit Mac numbr M = 0., Rynolds numbr R = 0 5, and Prandtl numbr Pr = 0.7. Our grid uss 544 clls and is cosn suc tat tr ar sufficint clls in t boundary layr to rsolv it. Trfor, w do not includ artificial viscosity trms in tis tst. W solv on t domain 0.5, 0, 0.5 and assum t flat plat to b locatd at [0, ] [0]. W valuat scald vlocitis ũ and ṽ at x = 0.5 and plot η vs. ũ and η vs. ṽ wit η = y x Rx, R x = u x ν, ũ = u u, ṽ = v u Rx, wit u dnoting t far fild vlocity in x-dirction, wic satisfis R = u L ν wit L dnoting t plat lngt. Figur 4 sows t rsults for t scald vlocitis ũ and ṽ, bot plottd against η, for polynomial dgrs two and tr. T rsults for ũ matc vry wll wit t rfrnc solution. Evn for coarsr grids, it was vry straigt-forward to captur t scald u-vlocity wll. A 4
0.5 0 0.5 0.5 0 0.5.5 Figur 5: Airfoil tst: Bas ms M wit 78 clls zoomd around t airfoil. mor callnging tst is t approximation of t vlocity ṽ for wic w also obsrv a good agrmnt wit t analytic solution. W notic t following toug: as w comput t discrt solution mor accuratly, t agrmnt wit t rfrnc solution sms to bcom sligtly wors, i.., t computd solution tn lis sligtly abov t rfrnc solution. W attribut tis to t fact tat our rfrnc solution is not t tru solution of t comprssibl Navir-Stoks quations for t cosn stting but for t boundary layr quations. 6.5. Flow around NACA 00 airfoil W conclud our numrical rsults wit a standard tst in t litratur: flow around a twodimnsional NACA 00 airfoil. W us t following airfoil gomtry y = ±0.5946898 [0.98773 x 0.753x 0.357907906x + 0.998497x 3 0.0574606x 4]. Our bas ms M as 78 triangls and uss a far fild radius of 50 cords. It as bn gnratd using DistMs [3]. Figur 5 sows t clos nigborood of t airfoil. W also us onc and twic globally rfind mss, dnotd by M and M, wit 69 and 7648 clls, rspctivly. W not tat t rsults of airfoil tsts strongly dpnd on t quality of t ms: t goal is to av noug clls in t boundary layr wil aving as littl clls as possibl in total. Our ms as not bn optimizd in tat rspct. T focus r is on validation of our mtod, for wic tis ms turnd out to b sufficint. W us picwis cubic polynomials for our tsts and Pr = 0.7. W us a igr-ordr boundary approximation along t airfoil boundary to b consistnt wit using igr-ordr polynomial spacs. 5
Tabl : Airfoil tst: Rsults for R=5000, Ma=0.5, α = 0. c p D c v D c p L c v L ST-SIPG, ms M 0.0409 0.0337-3.33-03 -.-04 ST-NIPG, ms M 0.0385 0.03367-3.38-03 -.05-04 ST-SIPG, ms M 0.03 0.0338.50-04.3-05 ST-NIPG, ms M 0.09 0.0339.44-04.8-05 Hartmann & Houston [7] 0.09 0.0354 Swanson & Langr [35] 0.079 0.0379 Tory 0 0 6.5.. R=5000, Ma=0.5, α = 0 W start wit on of t most popular tsts and coos R=5000, Ma=0.5, α = 0. W do not includ artificial diffusion trms for tis tst. W valuat t functionals [7] c p D = ρ u pn ψ d ds, c p L = S ρ u pn ψ l ds, S c v D = ρ u τn ψ d ds, c v L = S ρ u τn ψ l ds, S wit S dnoting t airfoil surfac, ρ t far-fild dnsity, τ t viscous strss tnsor, and for 0 angl of attack ψ d = 0 T and ψl = 0 T. Not tat du to 0 angl of attack, c p L = cv L = 0 for t xact solution. Tabl rports t rsults for mss M and M using bot SIPG and NIPG discrtization for t pysical diffusion trm. T rsults for SIPG and NIPG ar vry similar. Tabl also includs rfrnc valus rportd by otr rsarcrs. Our rsults for t ms M ar in vry good agrmnt wit ts valus, wras t rsults for ms M ar sligtly off. Tis is consistnt wit our xamination of ow many clls on nds to rsolv t boundary layr. Basd on our rsults from t Blasius tst, w nd at last rougly 3 clls in t boundary for picwis cubic polynomials to rsolv t layr. Tis is satisfid for tis tst for ms M but not for ms M. 6.5.. R=000, Ma=., α = 0 In our final numrical tst w combin flow around an airfoil wit a sock. W follow Hartmann [6] for t tst stup. In tis tst w compar t prformanc of our mtod wit and witout t artificial diffusion trms BSD IP and BIP SC. Figur 6 sows t Mac contour lins for bot vrsions for ms M. Witout artificial diffusion trms w obsrv oscillations in t nigborood of t sock. Ts ar mostly rmovd wn t BSD IP and BIP SC ar mployd. Away from t sock, t rsults ar vry similar. 6
.5.5 0.5 0.5 0 0 0.5 0.5.5.5 0.5 0 0.5.5 0.5 0 0.5.5 a witout B IP SD and B IP SC b wit B IP SD and B IP SC Figur 6: Airfoil tst: Rsults for Mac contours for ms M for R=000, Ma=., α = 0. 7. Conclusions In tis papr, w prsntd two scms, ST-SDSC-SIPG and ST-SDSC-NIPG, for solving t comprssibl Navir-Stoks quations. T scms ar basd on a spactim DG approac and us ntropy variabls as dgrs of frdom. T scms includ stramlin diffusion and sock-capturing trms tat vanis wit t corrct ordr of convrgnc in smoot flow. For t discrtization of t pysical diffusion trms t NIPG and t SIPG formulation ar usd, rspctivly. T rsulting scms satisfy ntropy stability stimats. T providd numrical rsults sow tat t scms also prform wll numrically. Possibl futur dirctions ar t xtnsion to tr dimnsions or to goal-orintd adaptivity, compar,.g., [8]. Acknowldgmnts T autors tank Siddarta Misra for many lpful discussions and for providing t rsourcs to mak tis work possibl. S. M. also tanks Miloslav Fistaur for inviting r to a vry intrsting and stimulating wk at t Carls Univrsity, Pragu. Tis work was supportd by ERC STG. N 30679, SPARCCLE. A. Proof of ntropy stability for inviscid trm Lmma A.. Lt t assumptions of Torm 5.3 old tru. And lt t input argumnt,+ in t computation of t numrical inviscid flux at t wall boundary b givn by 35. Tn, tr olds B DG, SUN, x dx SU0, x dx. 7
Proof. T proof of tis statmnt is vry similar to t proof of Lmma., wic as bn sown as part of Torm 3. in [0]. As tat proof is quit long, w do not rviw t full proof r. Instad w focus on t diffrncs. In [0], B DG using upwind flux in tim is split into tmporal trms BDG t, = U t dx dt n, + Un+, n+, dx Un, n,+ dx n, n, and spatial trms B s DG, = n, + n, k= N F k x k dx dt F,,,+; ν, dσx dt. As BDG t dos not includ spatial boundary trms, on can follow t proof providd in [0] to sow BDG t, SUN, x dx SU0, x dx. W ar now going to sow tat B s DG, 0 38 still olds tru for t cas considrd in Torm 5.3. For t considrd ntropy pair S, Q, dnot by ψ k = F k Q k, k =,, t corrsponding ntropy potntial. Tn, using ψx k k = xk F k [0, p.5], w gt by mans of t divrgnc torm Tis implis k= F k x k dx dt = = k= N ψ k xk dx dt k= ψ k, ν k dσx dt. BDG s, = n, N ψ k, ν k + F,,,+; ν, k= dσx dt. 39 W not tat w sligtly misusd notation in t abov considrations by assuming tat vry cll as potntially fictitious cll nigbors. In [0, p.5/6], it is sown undr t assumption of compact support insid tat 39 implis 38. Hr, w sow t sam claim 8
for t spcial cas of an dg F Γadia, wic blongs to cll. Using t dfinition 8 of t numrical flux, w considr ψ k, ν k + k= k= F k,,,,+, ν k D,+,, dσx dt. 40 According to 35, w st on for givn,,+ = v,, v,, v 3,, v 4, T. T ntropy consrvativ flux tat w us [] s [4] for a two-dimnsional vrsion is basd on valuating various aritmtic and logaritmic mans. Du to t spcific rlation of, and,+, most trms cancl; a sort computation sows F,,,,+ = Tis implis using 8 k= T T 0, p, 0, 0 and F,,,,+ = 0, 0, p, 0. F k,,,,+, ν k = p v, ν + p v3, ν = ρ u ν + ρ v ν. Furtrmor, tr olds [4, p. 567] ψ, = ρ u, ψ, = ρ v. Trfor, t trms in t first lin of 40 cancl ac otr. It rmains to sow tat t diffusion oprator in t scond lin rsults in a non-ngativ valu. W us Rusanov diffusion and considr t following formulation [9] Ua + Ub Da, b; ν = max λ max Ua, ν, λ max Ub, ν U S U wit λ max U, ν dnoting t maximum ignvalu, takn in absolut valu, of t Jacobian F U Uν + F U Uν. A sort computation sows tat t 4 4 matrix U Ṽ wit Ṽ = S U U, + U,+ only as non-zro ntris on t diagonal and for indics, 4 and 4,. Tn, wit pṽ dnoting t prssur corrsponding to ntropy variabl Ṽ T, U Ṽ,+, = pṽv, + pṽv 3, 0, wic concluds t proof. Rfrncs [] D. N. Arnold, F. Brzzi, B. Cockburn, and L. D. Marini. Unifid analysis of discontinuous Galrkin mtods for lliptic problms. SIAM J. Numr. Anal., 39:749 779, 00. 9
[] T. J. Bart. Numrical mtods for gasdynamic systms on unstructurd mss. In D. rönr, M. Olbrgr, and C. Rod, ditors, An Introduction to Rcnt Dvlopmnts in Tory and Numrics of Consrvation Laws, volum 5 of Lctur Nots in Computational Scinc and Enginring volum 5, pags 95 85. Springr, 999. [3] T. J. Bart. Spac-tim rror rprsntation and stimation in Navir-Stoks calculations. In S.C. assinos, C.A. Langr, G. Iaccarino, and P. Moin, ditors, Complx Effcts in Larg Eddy Simulations, volum 56 of Lctur Nots in Computational Scinc and Enginring volum 5, pags 9 48. Springr, 007. [4] F. Bassy and S. Rbay. A ig-ordr accurat discontinuous finit lmnt mtod for t numrical solution of t comprssibl Navir-Stoks quations. J. Comput. Pys., 3:67 79, 997. [5] F. Bassy, S. Rbay, G. Mariotti, S. Pdinotti, and M. Savini. A ig-ordr accurat discontinuous finit lmnt mtod for inviscid turbomacinry flows. In Procdings of t nd Europan Confrnc on Turbomacinry Fluid Dynamics and Trmodynamics, pags 99 08, 997. Antwrp, Blgium. [6] C. E. Baumann and J. T. Odn. A discontinuous p finit lmnt mtod for t Eulr and Navir-Stoks quations. Int. J. Numr. Mtods Fluids, 3:79 95, 999. [7] S. Brdar, A. Ddnr, and R. löfkorn. Compact and stabl discontinuous Galrkin mtods for convction-diffusion problms. SIAM J. Sci. Comput., 34:A63 A8, 0. [8] J. Čsnk, M. Fistaur, and A. osík. DGFEM for t analysis of airfoil vibrations inducd by comprssibl flow. ZAMM Z. Angw. Mat. Mc., 93:387 40, 03. [9] B. Cockburn and C.-W. Su. T local discontinuous Galrkin mtod for tim-dpndnt convction-diffusion systms. SIAM J. Numr. Anal., 35:440 463, 998. [0]. Doljši. On t discontinuous Galrkin mtod for t numrical solution of t Navir- Stoks quations. Int. J. Numr. Mtods Fluids, 45:083 06, 004. []. Doljši. Smi-implicit intrior pnalty discontinuous Galrkin mtods for viscous comprssibl flows. Commun. Comput. Pys., 4:3 74, 008. []. Doljší and M. Fistaur. Discontinuous Galrkin Mtods. Springr, 05. [3] M. Fistaur,. Doljší, and. učra. On t discontinuous Galrkin mtod for t simulation of comprssibl flow wit wid rang of Mac numbrs. Comput. isual. Sci., 0:7 7, 007. [4] U. S. Fjordolm, S. Misra, and E. Tadmor. Arbitrarily ig-ordr accurat ntropy stabl ssntially nonoscillatory scms for systms of consrvation laws. SIAM J. Numr. Anal., 50:544 573, 0. [5] G. Gassnr, F. Lörcr, and C.-D. Munz. A discontinuous Galrkin scm basd on a spac-tim xpansion II. iscous flow quations in multi dimnsions. J. Sci. Comput., 34:60 86, 008. 30