A Parameter Free Generalized Moment Limiter for High-Order Methods on Unstructured Grids

Transcription

1 Advances n Appled Mathematcs and Mechancs Adv. Appl. Math. Mech., Vol. 1, No. 4, pp DOI: /aamm.09-m0913 August 2009 A Parameter Free Generalzed Moment Lmter for Hgh-Order Methods on Unstructured Grds Mchael Yang 1 and Z.J. Wang 1, 1 Department of Aerospace Engneerng and CFD Center, Iowa State Unversty, Ames, IA 50011, USA Receved 09 January 2009; Accepted (n revsed verson) 07 March 2009 Avalable onlne 18 June 2009 Abstract. A parameter-free lmtng technque s developed for hgh-order unstructured-grd methods to capture dscontnutes when solvng hyperbolc conservaton laws. The technque s based on a troubled-cell approach, n whch cells requrng lmtng are frst marked, and then a lmter s appled to these marked cells. A parameter-free accuracy-preservng TVD marker based on the cell-averaged solutons and soluton dervatves n a local stencl s compared to several other markers n the lterature n dentfyng troubled cells. Ths marker s shown to be relable and effcent to consstently mark the dscontnutes. Then a compact hghorder herarchcal moment lmter s developed for arbtrary unstructured grds. The lmter preserves a degree p polynomal on an arbtrary mesh. As a result, the soluton accuracy near smooth local extrema s preserved. Numercal results for the hgh-order spectral dfference methods are provded to llustrate the accuracy, effectveness, and robustness of the present lmtng technque. AMS subject classfcatons: 65M70, 76M20, 76M22 Key words: Lmter, shock-capturng, hgh-order, unstructured grds. 1 Introducton A nonlnear hyperbolc conservaton law can generate dscontnutes even f the ntal soluton s smooth. A sgnfcant computatonal challenge wth a nonlnear hyperbolc conservaton law s the resoluton of such dscontnutes, whch has been a very actve area of research for over four decades. However, any lnear scheme hgher than frst order accuracy cannot generate monotonc solutons, accordng to the Godunov theorem [8]. Ths means lnear schemes of 2nd-order and hgher wll produce spurous oscllatons near dscontnutes due to the so-called Gbbs phenomenon, whch can result n numercal nstablty and non-physcal data, such as negatve pressure Correspondng author. URL: Emal: mhyang@astate.edu (M. Yang), zjw@astate.edu (Z.J. Wang) c 2009 Global Scence Press

2 452 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp or densty. Early research work on shock-capturng reled on numercal dffuson to smear the dscontnutes so that they can be captured as part of the numercal soluton [14, 20, 25, 40]. Besdes the exstence of user-defned parameters, the hstorcal drawback of the artfcal vscosty approach s that the added terms are frequently too dsspatve n certan regons of the flow. Later, another type of approach was developed based on flux lmtng, whch ntroduced numercal dffuson mplctly to reduce or remove spurous oscllatons. Poneerng works n flux lmtng nclude the FCT [3], the MUSCL and related methods [9, 38, 39], and TVD methods [10, 44]. However, the flux-lmtng and TVD methods suffered from accuracy-degradaton to frst-order at local extrema n smooth regons. Hgh-order (3 rd -order and hgher) shock-capturng algorthms have the potental to obtan sharp non-oscllatory shock profle and smultaneously preserve accuracy n smooth regons. The challenge of producng oscllaton-free numercal solutons s tougher for hgh-order methods than for lower order ones because of much reduced numercal dsspaton. The artfcal vscosty method has been mproved [6, 7, 36] to mnmze undesrable dsspaton by usng a spectrally vanshng vscosty approach based on hgh-order dervatves of the stran rate tensor, though there stll exst userdefned parameters that can be mesh or problem dependent. The ENO [9] and WENO methods [15] used the dea of adaptve stencls n the reconstructon procedure based on the smoothness of the local numercal soluton. However, due to a lack of compactness, the mplementaton of both ENO and WENO methods s complcated on arbtrary unstructured meshes, especally for 3D problems. Hgh-order methods desgned for unstructured meshes offer obvous advantages n geometrc flexblty. Examples of such methods nclude the dscontnuous Galerkn (DG) method [4, 5, 30], the mult-doman staggered- grd method [16, 17], the spectral volume method [41, 42], the spectral dfference (SD) method [22, 34]. A revew of these and other unstructured-grd based hgh-order methods can be found n [43]. These hgh-order methods are usually compact, meanng cells are coupled wth ther mmedate face neghbors. Compact hgh-order methods are much more sutable for massvely parallel machnes as the amount of data communcaton s mnmzed. In desgnng lmters for such methods, t s natural to requre that the lmters should be compact. There have been many notable developments n lmters for hgh-order methods n the last decade. Many of the lmters employ the so-called troubled cell (TC) approach, n whch oscllatory cells are marked frst, and the solutons n these cells are re-generated to remove or reduce the oscllatons satsfyng certan crtera such as mean-preservng. The dea s frst developed n [5], and then further extended n [2]. In [5], a lmter developed for the fnte volume method [1] was used. The moment lmter developed n [2] can be vewed as the generalzaton of the mnmod lmter [39] to hgher order dervatves or moments. The central DG scheme proposed n [23] s a further generalzaton of the MUSCL scheme and the moment lmter. Other more recent developments nclude the use of WENO [28] and Hermte WENO [24, 29] schemes to generate the reconstructon n troubled cells. Hgh-order lmters based on artfcal vscosty have also been nvestgated by varous researchers [13,26]. In the

3 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp present study, our focus s on the TC approach. There are two major components n the TC approach: the markng or detecton of troubled cells, and the data lmtng (or remappng) n these cells. In developng the present moment-based lmter, we set to acheve several goals: (1) free of user adjustable parameters; (2) capable of preservng accuracy at smooth regons ncludng smooth extrema; (3) compact and effcent for arbtrary unstructured meshes. The requrement of no-user adjustable parameters s very mportant for a general purpose producton-type flow solver, whch can be appled to a wde varety of problems. If a lmter s success hnges on a sutable parameter whch depends on the soluton, the mesh and the order of accuracy, the lmter wll more lkely fal than succeed n real world applcatons. In the present study, we compare several markers nvestgated n [27], namely, the mnmod TVB marker [5], the KXRCF marker developed by Krvodonova et al. n [19], and the Harten marker [11], wth a parameter-free accuracy-preservng TVD marker. For the lmter step, we extend the approach n [2] and [23] to arbtrary unstructured meshes n an effcent manner. Numercal results show that the present lmter can preserve accuracy at smooth regons, whle capturng dscontnutes. The remander of ths paper s organzed as follows. In Secton 2 we revew the formulaton of the spectral dfference method as t wll be used as the carrer of the present lmter. In Secton 3 we compare several markers n the lterature and descrbe n detal the constructon and mplementaton of the present TVD marker. In Secton 4, we formulate the generalzed moment lmter for arbtrary unstructured meshes. In Secton 5 we provde extensve numercal examples to demonstrate the performance of the present marker and lmter wth the SD method. Fnally, conclusons and some possbltes for future work are gven n Secton 6. 2 Revew of the spectral dfference method Consder the followng hyperbolc conservaton law Q t + F = 0, (2.1) on doman Ω [0, T] and Ω R d (d=2 or 3) wth proper ntal condtons wthn Ω and boundary condtons on Ω. The state varable Q can be a scalar or a vector, and the generalzed flux F can be a scalar, vector, or tensor. In the case of the Euler equatons, Q s the vector of conservatve varables. Doman Ω s parttoned nto non-overlappng trangular or quadrlateral cells (or elements). In the SD method, two sets of ponts,.e., the soluton ponts and flux ponts are defned n each element. The soluton ponts are the locatons where the nodal values of the state varable Q are specfed. Flux ponts are the locatons where the nodal values of fluxes are computed. The DOFs n the SD method are the conservatve varables at the soluton ponts. Fg. 1 dsplays the placement of soluton and flux ponts for the thrd-order SD schemes on trangular and quadrlateral cells.

4 454 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 1: Soluton (red sold crcles) and flux ponts (green/blue sold squares). (a) Trangular mesh; (b) Quadrlateral mesh. Gven the soluton Q j, at the j-th soluton pont wthn cell (denoted as r j, ), an element-wse degree p polynomal can be constructed usng Lagrange-type polynomal base,.e., p ( r) = m L j, ( r)q j,, (2.2) j=1 where L j, ( r) are the Lagrange shape functons. Wth (2.2), the solutons at the flux ponts can be computed. Snce the solutons are dscontnuous across element boundares, the fluxes at the element nterfaces are not unquely defned. Obvously, n order to ensure conservaton, the normal component of the flux vector on each face should be dentcal for the two cells sharng the face. A one dmensonal approxmate Remann solver [21, 31] s then employed n the face normal drecton to compute the common normal flux ˆF(Q, Q +, n). Snce the tangental component of the flux does not affect the conservaton property, several choces are possble at the face flux ponts. Let the unt vector n the tangental drecton be l as shown n Fg. 2. Here we offer two possbltes. One s to use a unque tangental component by averagng the two Fgure 2: Flux computaton.

5 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp tangental components from both sdes of the face,.e., F l = F l (Q, Q +, l) = 1 2 {[ F(Q ) + F(Q + )] l}. (2.3) The other opton s to use ther own tangental components separately, allowng dscontnuous tangental components on the element nterfaces. For cell, the tangental component s F(Q ) l, and for ts neghbor F(Q + ) l. For a corner flux pont n cell, two faces (vewed from cell ) share the corner pont, as shown n Fg. 2. The full flux vector at the corner pont can be unquely determned from the two normal Remann flux components ˆF 1 = F n1 and ˆF 2 = F n2. In spte that the fluxes at a cell corner pont do not have the same value for all the cells sharng the corner, local conservaton s guaranteed because neghborng cells do share a common normal flux at all flux ponts. Once the fluxes at all flux ponts are re-computed, they are used to form a p + 1 degree polynomal,.e., P ( r) = m p+1 l=1 Z l, ( r) Fl,, (2.4) where Fl, = F( rl, ) and Z l, ( r) are the set of Lagrange shape functons defned by the flux ponts. The dvergence of the flux at the soluton ponts can be easly computed as, m p+1 P ( r) = Z l, ( r j, ) Fl,. (2.5) l=1 Fnally the sem-dscrete scheme to update the soluton unknowns can be wrtten as dq j, dt m p+1 + l=1 Z l, ( r j, ) Fl, = 0. (2.6) The SD method for quadrlateral or hexahedral grd s dentcal to the staggered grd mult-doman spectral method [16, 17]. It s partcularly attractve because all the spatal operators are one-dmensonal n nature. In the orgnal staggered-grd method [16, 17], the soluton and flux ponts are the Chebyshev-Gauss and Chebyshev-Gauss- Lobatto ponts. Recently, t was found [12, 37] that these flux ponts result n a weak nstablty for hgh-order schemes. New stable fluxes ponts were suggested n [12,37]. In the present study, we employ the Legendre-Gauss ponts plus the two end ponts as the flux ponts, as suggested n [12]. In an actual mplementaton, each physcal element (possbly curved) s frst transformed nto a standard element (square). The governng equatons are also transformed from the physcal space to the computatonal space as follows Q dt + F ξ + G = 0, (2.7) η where [ F G ] [ ξx ξ = J y η x η y ] [ Fx F y ], Q = J Q.

6 456 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp The Lagrange nterpolaton shape functons n one drecton for the conservatve soluton varable Q and fluxes can be wrtten as follows, respectvely, h (X) = N ( X Xs ) N ( X Xs+1/2 ), l X s=1,s = X +1/2 (X) = s. (2.8) X s=0,s = +1/2 X s+1/2 The reconstructed soluton for the conservatve varables n the standard element s just the tensor products of the three one-dmensonal polynomals,.e., Q(ξ, η) = N N j=1 =1 Q,j h (ξ)h j (η). (2.9) Smlarly, the reconstructed flux polynomals take the followng form: F(ξ, η) = G(ξ, η) = N N j=1 =0 N N j=0 =1 F +1/2,j l +1/2 (ξ)h j (η), (2.10) G,j+1/2 h (ξ)l j+1/2 (η). (2.11) For the nvscd flux, a Remann solver s employed to compute a common flux at the nterfaces to ensure conservaton and stablty. Tme ntegraton s done by usng ether explct TVD or SSP Runge-Kutta scheme [32,33] or an mplct LU-SGS scheme [35,45]. 3 Comparson of troubled cells markers In ths secton, we frst revew and evaluate several troubled-cell detecton methods found n the lterature [27]. Then we present a parameter-free TVD marker. Qu and Shu [27] nvestgated seven markers currently used n the CFD communty, and found that the mnomd TVB marker [5], the marker developed by Krvodonova et al. named KXRCF n [19], and the Harten [11] marker are the best three among the seven markers they studed based on the amount of spurous oscllatons n the soluton, and the total number of cells marked. These three markers are chosen n the current study, and are evaluated next. Consder the followng 1D scalar conservaton law u t + f (u) x = 0, x Ω, t > 0, (3.1a) u(x, 0) = u 0 (x), x Ω. (3.1b) The computatonal doman Ω s parttoned nto N cells wth p + 1 soluton ponts and p + 2 flux ponts n each cell. In the followng descrpton, h, ū and u j, denote the mesh sze of cell, the average soluton and the value of the reconstructed soluton polynomal at the j-th flux pont of the -th cell, respectvely.

7 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Mnmod TVB marker A user specfed parameter M s chosen, whch s of the order of the soluton s second dervatve n a smooth regon. Then the dfferences between the solutons at the cell nterfaces and the cell-averaged soluton are examned. Denote these dfferences u,l =ū u 1, and u,r =u p+2, ū. If the followng nequaltes are satsfed u,l Mh 2 and u,r Mh 2, (3.2) the soluton n cell s consdered smooth, and thus the cell s NOT a troubled cell. Otherwse, compute the followng quanttes ũ,l = mn mod( u,l, ū ū 1, ū +1 ū ), (3.3) ũ,r = mn mod( u,r, ū ū 1, ū +1 ū ), (3.4) where the mnmod functon s defned as { s mn mn mod(a 1, a 2,, a n ) = a k, f sgn(a 1 ) = = sgn(a n ) = s, 1 k n 0, otherwse. (3.5) If ether u,l or u,r are modfed n (3.3) or (3.4),.e., ũ,l = u,l or ũ,r = u,r, the cell s marked a troubled cell. Eqs. (3.3) and (3.4) are smlar to the MUSCL scheme [39] n sprt, but not as restrctve. In order to explan ths, assume the soluton to be lnear wth a slope of S n cell. Then we have Defne two more slopes usng u,l = u,r = S h /2. (3.6) S +1/2 = ū+1 ū x +1 x, S 1/2 = ū ū 1 x x 1. (3.7) The followng equaton s equvalent to (3.3) and (3.4), ( h S = 1 + h h + h ) +1 mn mod S, S 1/2, S h +1/2, (3.8) h where S s the lmted slope. We have used x +1 x =(h +1 + h )/2 and x x 1 =(h 1 +h )/2 n (3.8). If the mesh s unform, then the factors (h 1 + h )/h and (h +1 + h )/h become 2. If the mesh s extremely non-unform, the factors can approach 1. In practce, we could use a factor between 1 and 2,.e., ( S = mn mod S, β ū ū 1, β ū+1 ū ). (3.9) x x 1 x +1 x The larger β s, the fewer number of cells s marked at the expense of possbly mssng some troubled cells. A good compromse s β=1.5. Obvously f the soluton s locally lnear, the cell s not marked because S = ū ū 1 x x 1 = ū+1 ū x +1 x.

8 458 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Eq. (3.9) wll be used agan to desgn marker and lmters. As ponted out n [27], M>0 s a free parameter, whch depends on the soluton of the problem. For scalar problems t s possble to estmate M f the soluton s smooth [5] (M s proportonal to the second dervatve of the ntal condton at smooth extrema). However t s more dffcult to estmate M for the systems case, such as the Euler and N-S equatons. If M s chosen too small, more cells than necessary wll be marked as troubled cells. If M s chosen too large, spurous oscllatons may appear. 3.2 KXRCF marker In [19] Krvodonova et al. proposed a shock-detecton technque based on DG s superconvergence property at the outflow boundares of an element n smooth regons. Ths method was termed the KXRCF marker. The boundary of a cell, I, can be parttoned nto two portons: the nflow boundary I where flow goes nto the cell, and the outflow boundary I + where flow exts the cell. In the 1D case, f the wave speed f (u) s postve at the left nterface, then the left face (x 1/2 ) s an nflow boundary; otherwse, the left face s an outflow boundary. The rght face can be classfed n exactly the opposte fashon. In an actual mplementaton, we use the averaged wave speed from both sdes of a face to determne f t s an nflow or outflow boundary. The KXRCF marker checks the soluton on the nflow boundary to determne troubled cells. Wthout loss of generalty, let s assume the nflow boundary s the left nterface for cell. Then compute the followng quantty L, L = u 1, u p+2, 1. (3.10) h (p+1)/2 ū If L >1, then cell s marked as a troubled cell. Note that snce DG s super-convergence property occurs only n a smooth regon, the KXRCF marker mght unnecessarly mark some cells n contnuous but not smooth regons. 3.3 Harten/ modfed Harten marker The Harten marker was orgnally developed n [11] and further modfed n [27]. Here s the basc dea. Frst extend the reconstructed soluton polynomals from the neghborng cells u 1 (x) and u +1 (x) nto cell. Then compute the dfferences between the average extended polynomals and the average of cell. In 1D, a jump (dscontnuty) wthn cell can cause one extenson above the current cell average and the other below the current cell average. Therefore the Harten marker can be formulated as follows. Compute F (z) = 1 h { z x+1/2 u 1 (x)dx + x 1/2 z } u +1 (x)dx ū. (3.11)

9 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp If F (x +1/2 ) F (x 1/2 ) 0, (3.12) then a dscontnuty possbly exsts wthn cell. To mprove ts performance at smooth extrema, the cell-averaged degree p dervatves between the neghborng cells and the current cell are also compared. Thus, f F (x +1/2 ) F (x 1/2 ) 0, ū (p) > θ ū (p) 1 and ū(p) > θ ū (p) +1, (3.13) cell s marked as a troubled cell. We take the same value for the constant θ(= 1.5) n the numercal tests as n [27]. We can make the followng observatons regardng the Harten marker. When the polynomal degree p s hgh, the extenson of the reconstructed soluton polynomals from the neghborng cells mght be strange and unexpected near a dscontnuty, and may fal to mark a shock, as shown n Fg. 7. In ths case, the extended polynomals from both sdes have cell averaged solutons larger than the current cell. Therefore ths strategy may fal to mark a dscontnuty n a hgh-order scheme. The Harten marker s dffcult to mplement for unstructured grds n multple dmensons. To llustrate the performance of the above three markers, examples of both smooth and dscontnuous soluton profles have been used: ) A smooth sne functon, u=sn(2πx), 0 x 1; ) A combnaton of smooth and dscontnuous profles: a smooth Gaussan, a square pulse, a trangle and half an ellpse [18], whch s defned as u 0 (x) = ( ) G(x, β, z δ) + G(x, β, z + δ) + 4G(x, β, z) /6, 0.8 x 0.6, 1, 0.4 x 0.2, 1 ( 10(x 0.1), ) 0 x 0.2, F(x, α, a δ) + F(x, α, a + δ) + 4G(x, α, z) /6, 0.4 x 0.6, 0, otherwse, G(x, β, z) = e β(x z)2, F(x, α, a) = where a=0.5, z= 0.7, δ=0.005, α=10, β=log 2/(36δ 2 ). (3.14) max(1 α 2 (x a) 2, 0), (3.15) ) An oscllatory shock profle obtaned when solvng nonlnear hyperbolc equatons. The marked cells for the ntal profles are then plotted. In the followng fgures, the sold black lnes stand for the ntal profle, and the elevated red squares represent the troubled cell. The Mnmod TVB marker works well for the scalar cases as shown n Fg. 3(a) and 3(b), where no cell s marked as troubled cell for the smooth sne wave, and only the cells at the dscontnuty regon are marked as troubled cells. Here we estmated M from [5] by computng the maxmum absolute value of the second dervatves of the ntal soluton n smooth regons for each of the two cases. However, for the complex oscllatory shock profle case n Fg. 4, t s dffcult to estmate M from ths ntal profle. It appears that M=40 works well n that only the two cells at the dscontnuty

10 460 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 3: Mnmod TVB marker by usng M from [5] (p=2). (a) sne wave, 20 cells; (b) dscontnuous profle [18], 200 cells, p=2. Fgure 4: Mnmod TVB marker for the oscllatng shock profle wth dfferent M (5 cells), p=6. are marked as troubled cells. But we found M=40 by ad hoc testng. For the system cases such as Euler and Naver-Stokes equatons, t s more dffcult to estmate M. The KXRCF marker detects the dscontnutes as shown n Fg. 5(b) and Fg. 6. It also works well for the smooth sne wave case n Fg. 5(a) as well as the smooth Gaussan extremum n Fg. 5(b) (see the close-up vew n Fg. 5(c)), where no troubled cell at the local smooth extrema s marked. Ths s expected because the KXRCF marker s exactly based on the super-convergence property on the elements outflow boundares n smooth regons. However, n contnuous but not smooth regons, such as the vcnty of x= 0.8 n Fg. 5(b) or x= 0.16 n Fg. 6, the KXRCF marker can unnecessarly mark the cells n those contnuous regons as troubled cells. The Modfed Harten marker gves good results n the smooth sne wave case, as shown n Fg. 8. The results for the dscontnuous profle case are acceptable, but ts performance s senstve to the nterpolaton order of polynomal as shown n Fg. 9, where some more cells are marked when p=5 than the p=2 case. Ths senstvty can cause a serous problem n the hgh-order cases as shown n Fg. 7, where the necessary condton (3.12) of the Harten marker fals to mark the shock cell. Ths s because that the extensons of the soluton polynomals from the neghborng cells can

11 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 5: KXRCF marker (p=2). (a) sne wave, 20 cells; (b) dscontnuous profle [18], 200 cells; (c) close-up vew for the Gaussan peak (the frst from the left). become large n the current cell, and the ntegral values from the left and the rght cells are all postve,.e., F (x 1/2 ) = 1.886, F (x +1/2 ) = That s why the Modfed Harten marker fals n ths typcal case. 3.4 Accuracy-Preservng TVD (AP-TVD) marker The above examples show that the free parameters n the mnmod TVB marker can have decsve effects on the performance of the marker; the KXRCF marker can mark Fgure 6: KXRCF marker for an oscllatng shock profle (5 cells, p=6).

12 462 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 7: Harten condton (3.12) (5 cells, p=6) cells. Crcle: soluton ponts; Blue lne: extenson from rght; Red lne: extenson from the left. Fgure 8: Modfed Harten marker for a sne wave wth 20 cells. (a) p=2; (b) p=5. Fgure 9: Modfed Harten marker for the dscontnuous profle wth 200 cells. (a) p=2; (b) p=5. too many cells n contnuous regons as troubled cells; the Harten marker can fal to detect a shock at a hgh-order settng, due to the unexpected polynomal extensons from the neghborng cells. In addton, the Harten marker s dffcult to mplement n 2D and 3D. Although t s mpossble to desgn a perfect marker, one desgn goal we hope to acheve s a marker free of user-specfed parameters. In the mnmod TVB marker, f parameter M s 0, t becomes a TVD marker. A well known drawback of the TVD marker s that cells at smooth soluton extrema are marked. In order to unmark these smooth extrema, the frst dervatves of the

13 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp solutons are examned to see f they are locally monotonc. Ths marker s dvded nto the followng steps. 1. Compute the cell averaged solutons at each cell. Then compute the mn and max cell averages for cell from a local stencl usng ū max, = max(ū 1, ū, ū +1 ) and ū mn, = mn(ū 1, ū, ū +1 ). (3.16) If u j, > ū max, ū max, or u j, < ū mn, ū mn,, (j = 1, p + 2), (3.17) the cell s consdered as a possble troubled cell, whch s further examned n the next step. 2. Ths step s amed to unmark those cells at local extrema that are unnecessarly marked as troubled cells n the frst step (3.17). If an extremum s smooth, the frst dervatve of the soluton should be locally monotonc. Therefore, a mnmod TVD marker s appled to see f the second dervatve s bounded by the slopes computed wth the cell-averaged frst-dervatves. Compute ũ (2) = mnmod (ū (2), β ū(1) +1 ū(1), β ū (1) ū (1) 1 ). (3.18) x +1 x x x 1 If ũ (2) =ū (2), the cell s unmarked,.e., not a troubled cell anymore. Otherwse, the cell s confrmed as a troubled cell. Obvously, ths marker works for p>1. The coeffcent n (3.17) s not problem-dependent free parameters. It s used to overcome machne error when comparng two real numbers so as to avod the trval case that the soluton s constant n the neghborhood. In order to compare the performance of the new AP-TVD marker wth the Mnmod TVB marker, the KXRCF marker, and the Modfed Harten marker, we use the same three testng cases as before. Fg. 10 shows that the present AP-TVD marker performs consstently well at the local extrema regons for all the polynomal order p>1. No cell s marked as a troubled cell n ths smooth case as expected. Fg. 11 shows that the present marker detects the dscontnutes wthout unnecessarly markng other cells n smooth regons. It also shows the consstently good performance of detectng dscontnuty for all the polynomal order p>1. Fg. 12 shows the present marker only marks the two cells at the dscontnuty as expected, none elsewhere, n contrast to the other markers. Comparng wth the three preferred markers from [27], the present accuracy preservng TVD marker has shown the advantages that (1) no free-parameter and problem-ndependent; (2) s effcent n terms of the number of marked cells over the total number of cells and t performs well n markng the dscontnutes; (3) t s compact and easy to mplement for arbtrary unstructured meshes. 4 Formulaton of the generalzed moment lmter Next we present a p-exact hgh-order accuracy-preservng lmter based on the moment lmter [2, 23] and cell averages. The present lmter uses a Taylor-seres-lke

14 464 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 10: AP-TVD marker for the sne wave, 20 cells. expanson for the reconstructon, whch s smlar to that n [23]. The dfference s that the expanson s performed wth respect to the cell-averaged dervatves, rather than the dervatves at a specfc pont such as the cell centrod. Then these cellaveraged dervatves are lmted n a herarchcal manner startng from the hghest dervatve. Combned wth the AP-TVD marker, ths new lmtng technque exhbts the followng propertes: (1) free of problem-dependent parameters; (2) unstructuredgrd based, easy to mplement for 3D arbtrary meshes, and compact for parallel computng; (3) capable of suppressng spurous oscllatons near soluton dscontnutes wthout loss of accuracy at the local extrema n the smooth regons. We wll call ths lmtng technque parameter-free generalzed moment lmter (or termed as PFGM lmter ). In the SD method the soluton ponts are used to construct a degree p polynomal that can recover the conservatve varables at the flux ponts. Ths reconstructon can produce spurous oscllatons near a shock wave. Therefore a new non-oscllatory reconstructon s needed n the troubled cells. The followng dea s followed. Frst the orgnal degree p soluton polynomal wthn a troubled cell s replaced wth an equvalent polynomal based on the cell-averaged dervatves up to degree p. Then the

15 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 11: AP-TVD marker for the dscontnuous profle [18], 200 cells. Fgure 12: AP-TVD marker for the oscllatng shock profle (5 cells, p=6). hgh-order dervatves are herarchcally lmted usng the cell-averaged dervatves of one degree lower. In case that the hghest dervatve s not altered, the orgnal polynomal s preserved. Ths procedure can be easly mplemented for unstructuredgrd based hgh-order methods. Let s consder the 1D case frst. Let the orgnal soluton polynomal before lmtng be u (x), and the lmted polynomal be Y (x) wthn cell. Frst we express u (x) n

16 466 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp terms of the cell-averaged dervatves up to degree p,.e., u (x) = ū + ū (1) (x x ) + 1 2ū(2) ( (x x ) 2 1 ) 1 12 h ū(4) 1 6ū(3) ( (x x ) h2 (x x ) ( (x x ) h2 (x x ) ) 240 h4 ) + (4.1) where x represents the cell centrod coordnate. Next the cell-averaged dervatves are lmted n a herarchcal manner by usng a mnmod-type lmter. Startng from the hghest-order dervatve, ū (p) s lmted usng, Ȳ (p) ( = mn mod ū (p), β ū(p 1) +1 ū (p 1) x +1 x, β ū (p 1) ū (p 1) ) 1. x x 1 (4.2) If Ȳ (p) =ū (p), the hghest dervatve s not altered. No further lmtng s requred, and soluton remans the same. Otherwse, the lmtng process proceeds to the next lower dervatve n a smlar fashon,.e., Ȳ (k) If Ȳ (p 1) ( = mn mod ū (k), β ū(k 1) +1 ū (k 1) x +1 x, β ū (k 1) ū (k 1) ) 1, k = p 1. (4.3) x x 1 =ū (p 1), none of the lower dervatve are further lmted,.e., Ȳ (k) = ū (k), (k = p 2,, 1). (4.4) Otherwse, the process contnues n a smlar fashon herarchcally untl the frst dervatve s lmted. In order to preserve the mean, the zero-th dervatve (the mean) s retaned,.e., Ȳ =ū. Fnally the lmted polynomal s wrtten as Y (x) = Ȳ + Ȳ (x x ) + 1 2Ȳ(2) ( (x x ) 2 1 ) 1 12 h2 + 6Ȳ(3) ( (x x ) h2 (x x ) ) Ȳ(4) ( (x x ) h2 (x x ) h4 ) + (4.5) Note that ths lmter s compact, only nvolvng data from ts mmedate neghbors, and easy to mplement. Next we present an effcent extenson to mult-dmensonal unstructured grds. Smlar to the 1D case, we frst express the soluton polynomal wth respect to the cell-averaged dervatves, u (x, y) = ū + ū x, x + ū y, y + 1 2ūxx,( x 2 I xx ) + 1 2ūyy,( y 2 I yy ) + ū xy, ( x y I xy ), (4.6) where x = x x, y = y y, x 2 = (x x ) 2, y 2 = (y y ) 2, x 1 xdv, y 1 ydv, V V V V I xx 1 x 2 dv, I yy 1 y 2 dv, I xy 1 x ydv. V V V V V V

17 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp We proceed to lmt the cell-averaged dervatves nvolved n (4.6) for the troubled cells. In multple dmensons, especally n 3D, the effcency of the lmter s a very mportant crteron. In order to acheve the hghest effcency, we decde to lmt the dervatves of the same degree altogether wth a scalar factor between 0 and 1,.e., the lmted polynomal can be wrtten as Y (x, y) = ū + α (1) (ū x, x + ū y, y) + α (2){ 1 2 2ūxx,( x I xx ) + 1 } 2 2ūyy,( y I yy ) + ū xy, ( x y I xy ), (4.7) where α (1) and α (2) are the scalar lmters for the frst and second dervatves n [0, 1]. The essental 1D dea s then generalzed nto 2D and 3D. The lmter s conducted n the followng steps assumng p=2: 1. Compute the cell averaged 2 nd order dervatves n the troubled cell, and the cell-averaged 1 st order dervatves n the troubled cell and ts mmedate face neghbors, as shown n Fg. 13. Snce the 2 nd order dervatves are constants, and the 1 st order dervatves are lnear, we can assume that the cell-averaged 1 st order dervatves are the frst order dervatves at the cell centrods,.e., ū x, =u x, (x, y ) and ū y, =u y, (x, y ). 2. Assume one of the face neghbors s cell j. Defne the unt vector connectng the centrods of cell and cell j as l. The 2 nd order dervatve n ths drecton s examned next to determne whether lmtng s necessary. Compute ths second-order dervatve accordng to u ll, = ū xx, l 2 x + ū yy, l 2 y + 2ū xy, l x l y. 3. Compute the frst dervatve n l at the centrods of for both cell and j, Estmate the 2 nd -dervatve usng u l, (x, y ) = ū x, l x + ū y, l y, u l,j (x j, y j ) = ū x,j l x + ū y,j l y. u ll, = u l,j(x j, y j ) u l, (x, y ). r r j 4. Fnally the scalar lmter for ths face s computed accordng to α (2) j = mnmod(1, βũ ll, /u ll, ). (4.8) The process s repeated for the other faces. Fnally, the scalar lmter for the cell s the mnmum of those computed for the faces,.e., α (2) = mn j (α (2) j ). (4.9) If α (2) =1, the 2 nd order dervatves are not altered. The soluton polynomal remans

18 468 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 13: Schematc of multdmensonal lmtng. the same. Otherwse, the 1 st order dervatves are lmted n a smlar fashon to compute α (1),.e., α (1) j = mn mod(1, βũ l, /u l, ), (4.10) where ũ l, (ū j ū )/ r r j. Fnally, α (1) = mn j (α (1) j ). (4.11) As can be seen, ths generalzed moment lmter keeps ts compactness for arbtrary unstructured meshes, and can preserve a locally degree p polynomal, therefore satsfyng the p-exact property. 5 Numercal tests In ths secton we provde extensve numercal expermental results to demonstrate the performance of the PFGM lmter descrbed n Secton 4. In the numercal tests, the three-stage explct TVD Runge-Kutta scheme [22] was used for tme ntegraton, unless otherwse noted. 5.1 Accuracy study wth smooth problems Lnear scalar wave equaton Consder the 1D lnear wave equaton u t + u x = 0, (5.1) wth ntal condton u(x, 0)=sn(2πx), and perodc boundary condtons. The CFL number (CFL= f (u) x = t/ x) used for each case s as follows: (1) f p=1, 2, 3, CFL=0.01; (2) f p=4, 5, CFL= These CFL numbers are small enough so that the error s domnated by the spatal dscretzaton. In ths test, we dd not use the AP- TVD marker so that the generalzed moment lmter s appled to every cell n order

19 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 14: Grd convergence of the present lmter. (a) Lnear advecton equaton (5.1) for sne wave at t=1; (b) Non-lnear Burgers equaton (5.2) at t=0.1. to test the performance of the lmter alone. Otherwse, none of the cells would be marked and the results would be the same as the unlmted schemes. The L 1 error at t=1 for varous schemes wth and wthout the lmter are shown n Fg. 14(a). We can see that the present lmter preserves the desgned order of accuracy of the orgnal SD method, although the magntude of the error s larger than the unlmted schemes Nonlnear Burgers equaton Next consder the 1D non-lnear Burgers equaton u t (u2 ) x = 0, (5.2) wth ntal condton u(x, 0)=1 + sn(πx), perodc boundary condtons. The CFL number used for each case s as follows: (1) f p=1, 2, 3, CFL=0.01; (2) f p=4, 5, CFL= The soluton errors at t=0.1 (when the soluton s stll smooth) are shown n Fg. 14(b). Agan we can see that the present lmter preserves the desgned order of accuracy of the orgnal SD method. The results are qute smlar to the lnear scalar wave case D problems wth dscontnutes Combned smooth and dscontnuous waves We solve the 1D wave equaton (5.1) at t=8 wth the ntal condton [18] as plotted n Fg. 15. Perodc boundary condtons were used. A unform mesh s used wth a total of 200 cells. The CFL number used for each case s as follows: (1) f p=1, 2, CFL=0.01; (2) f p=3, 4, 5, CFL= The long tme evoluton (t=8) was consdered n order to check the performance of the hgh-order schemes. The numercal soluton s plotted at each soluton pont (red square). As seen that the present PFGM lmter yelds good results at both the smooth regon (as for the local extrema of the frst jump) and dscontnutes.

20 470 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 15: Soluton of lnear advecton problem at t=8, N=200 cells, p=1, 2, 3, 4, 5. Sold lne: exact soluton; square: soluton ponts Burgers equaton wth shock In ths example the Burgers equaton (5.2) was solved wth the same ntal condtons and perodc boundary condtons as n 5.2.2, but untl t=0.8 when a shock appears. The CFL number used for each case s as follows: (1) f p=1, 2, CFL=0.01; (2) f p=3, 4, 5, CFL= A mesh wth 100 unform cells was used wth varous schemes. The shock was captured well wthout oscllatons, as shown n Fg Sod shock-tube problem Sod shock-tube problem was solved to test the lmter for the Euler equatons u t + f (u) x = 0, (5.3)

21 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 16: Soluton of Burgers equaton (5.2) at t=0.8, N=100 cells, p=1, 2, 3, 4, 5. Sold lne: exact soluton; red square: soluton ponts. where u = (ρ, ρv, E) T, f (u) = (ρν, ρv 2 + p, v(e + p)) T, E = p γ ρv2, γ = 1.4, and ρ, v, E, p are the densty, velocty, total energy, and pressure, respectvely. The ntal condton s { (1, 1, 0) for x < 0, (ρ, p, v) = (0.125, 0.1, 0) for x 0. In Fg. 17, the computed densty at t=2 wth the present PFGM lmter s compared wth the exact soluton for p=1, 2, 3, 4. The tme step sze used for each case s as follows: (1) f p=1, 2, dt=0.001; (2) f p=3, 4, dt= Note that the solutons appear oscllaton-free, and both the shock and contact were well captured. The hgher-order scheme appears to yeld better results.

22 472 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 17: Sod problem, t=2, N=200 cells, p=1, 2, 3, 4. Sold lne: exact soluton; Red square: numercal soluton Shock acoustc-wave nteracton The problem of shock-acoustc wave nteracton [15] was solved to show the advantage of the present hgh-order lmter for the problems wth both shock waves and complex smooth features. We solved the Euler equatons (5.3) wth a movng Mach=3 shock nteractng wth a sne wave n densty,.e., ntally, (ρ, p, v) = { ( , , ) for x < 4, ( sn(5x), 1, 0) for x 4. (5.4) For comparson, a converged soluton usng a second-order MUSCL scheme on a grd of 3200 cells s used as the exact soluton. In Fg. 18, the soluton of densty at t=1.8 s compared wth the exact soluton for p=1, 2, 3 wth the present PFGM lmter on a medum mesh of N=400 cells and tme step sze dt= It shows that the smooth local extrema are better recovered f usng the present lmter n hgher-order form (p=2, 3). A close-up vew of the complex smooth regon s also shown for each case n Fg D test cases Next we test the present lmter for 2D nvscd flow problems wth dscontnutes. The conservatve form of the 2D Euler equaton can be wrtten as Q t + F x + G y = 0, (5.5)

23 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 18: The shock-acoustc nteracton problem, t=1.8, N=400 cells, p=1, 2, 3. Sold lne: exact soluton; red square: computed soluton at the soluton ponts. Rght: close-up vew for the complex smooth regon n the left graphs. where Q s the conservatve soluton varables, F, G are the nvscd flux gven below, Q = (ρ, ρu, ρv, E) T, F = (ρu, ρu 2 + p, ρuv, u(e + p)) T, G = (ρv, ρuv, ρv 2 + p, v(e + p)) T. Here ρ s the densty, u, v are the velocty components n x and y drectons, p s the pressure, and E s total energy. The pressure s related to the total energy by E = wth rato of specfc heat γ=1.4. p γ ρ(u2 + v 2 ),

24 474 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 19: Pressure contour for 2D shock-vortex nteracton, t= Shock vortex nteracton Ths problem descrbes the nteracton between a statonary shock wave and a vortex, and s a good test for the PFGM n resolvng both dscontnutes and mportant smooth features. The flow condtons are the same as n [15]. The computatonal doman s taken to be [0, 2] [0, 1]. A statonary shock wth a pre-shock Mach number of M s =1.1 s postoned at x=0.5 and normal to the x-axs. Its left state s (ρ, u, v, p)=(1, γ, 0, 1). An sentropc vortex p/ρ γ =const. s superposed to the flow left to the shock and centers at (x c, y c )=(0.25, 0.5). Therefore the flow varables on the left sde of the shock are as follows: u = M s γ + ετe α(1 τ2) sn θ, v = ετe α(1 τ2) cos θ, ρ = ( 1 γ 1 4αγ ε2 e 2α(1 τ2 ) ) 1 (γ 1), p = ( 1 γ 1 4αγ ε2 e 2α(1 τ2 ) ) γ (γ 1), where τ = r/r c and r = (x x c ) 2 + (y y c ) 2. Here ε denotes the strength of the vortex, α s the decay rate of the vortex; and r c s the crtcal radus for whch the vortex has the maxmum strength. They are set to be ε=0.3, α=0.204, r c =0.05. The 3 rd order SD scheme was employed n the smulaton on a coarse mesh of cells n order to have almost the same numbers of degree of freedom as n [15] (where the WENO method was used) for comparson purposes. The tme step sze used s dt= The grds are unform n y-drecton and clustered near the shock n x-drecton. The boundary condtons for the top and bottom boundares are set to symmetry, or slp wall. The pressure contours computed wth the present PFGM lmter and a lnear lmter (n whch the soluton at the troubled cells s assumed lnear) at t=0.05, t=0.20, and t=0.35 are shown n Fgs. 20 and 21, respectvely. It appears the present smulaton captures the shock waves wth a hgher resoluton than

25 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 20: Pressure contours for the 2D shock-vortex nteracton by the usng 3 rd -order PFGM lmter 61 contours from (a) t=0.05; (b) t=0.2; (c) t=0.35. Fgure 21: Pressure contours for the 2D shock-vortex nteracton by the usng lnear lmter 61 contours from (a) t=0.05; (b) t=0.2; (c) t=0.35. [15], Fg. 15. The color plots clearly show the regons before and after the shock. A black and whte contour plot s gven n Fg. 19 (for t=0.2). Comparng Fgs. 20 and 21, we can see that the PFGM lmter recovers the vortex much better than the lnear lmter, and the shock dscontnuty has been sharply captured as well by the present lmter. Fgs. 22 and 23 shows snapshots for later moments, t=0.6 and t=0.8 usng the 3 rd - order PFGM lmter and lnear lmter, respectvely. We can see here that the reflectve boundary takes effects as tme goes long enough when one of the shock bfurcatons reaches the top boundary and s reflected. Fg. 22(b) shows that the reflecton s well captured. Agan the 3 rd -order PFGM lmter gves better results than the lnear lmter n terms of less numercal nose and better-resolved vortex Oblque shock reflecton by a wedge Ths example consders a Mach 2 flow passng a wedge of 20. Notce that n the normal shock s algned wth the grd, whle n ths example we don t have ths luxury. The state ahead of the shock s set to be (ρ, u, v, P)=(1.4, 2, 0, 1). The boundary condtons are as follows: (1) supersonc nlet at the nlet on the left sde; (2) nvscd wall boundary condton for the wall; (3) smple extrapolaton boundary condton for

26 476 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 22: Pressure contours for the 2D shock-vortex nteracton by the usng 3 rd -order PFGM lmter 90 contours from (a) t=0.6; (b) t=0.8. Fgure 23: Pressure contours for the 2D shock-vortex nteracton by the usng lnear lmter 90 contours from from (a) t=0.6; (b) t=0.8. the upper boundary and the outlet on the rght end. A coarse mesh (400 elements, 20 boundary elements) was used for ths case, as shown n Fg. 24(b). The densty contours n Fg. 24(a) shows that the present 3 rd -order PFGM lmter captured the shock sharply (wthn one element). Only the cells at the shock are marked (n red), and the typcal marked cells when the shock s formed are shown n Fg. 24(b). As we can see, the AP-TVD marker works well as expected Transonc flow over NACA0012 arfol Ths example s the transonc flow over a NACA0012 arfol at Mach 0.85 and an angle of attack α=1, characterzed by the exstence of two shocks, one on the upper surface and one on the lower surface. To demonstrate the advantage of the present hgh-order lmter, we used a relatvely coarse mesh (1584 hexahedral elements, 52 elements on the upper and lower wall surfaces) as shown n Fg. 25. An mplct LU-SGS scheme was used for tme ntegraton n ths case. Fg. 26(a) shows the Mach contours obtaned wth the 3 rd -order PFGM lmter, and Fg. 26(b) gves a snapshot of the typcal dstrbuton of the marked cells. It s shown that the present lmter s ndeed able to elmnate the spurous oscllatons and capture the shock dscontnutes sharply whle mantanng the hgh-order accuracy

27 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp Fgure 24: Mach 2 flow past a wedge of 20 by usng the 3 rd -order PFGM lmter wth the SD method (400 elements, 20 boundary elements). (a) Densty contour; (b) Marked cells. Fgure 25: The unstructured hexahedral meshes for the NACA0012 arfol n transonc flow (1584 elements, 52 wall boundary elements). (a) the whole doman; (b) close-up vew around the arfol. Fgure 26: The transonc flow over NACA0012 arfol (M =0.85, α=1 ) by usng the 3 rd -order PFGM lmter n the SD method. (a) Mach contours; (b) the marked cells (red) at the 1000 th mplct tme step.

28 478 M. Yang, Z.J. Wang / Adv. Appl. Math. Mech., 4 (2009), pp at smooth regons. It was notced that the marked cells are located just n the vcnty of the upper and lower shock dscontnutes, and the average number of the marked cells durng the LU-SGS mplct tme teratons s a very small percentage (about 2%) of the total number of cells. Therefore t shows that the present AP-TVD marker works well and effcently for multdmensonal cases. 6 Conclusons Three desgn crtera have been set for a general purpose lmter: (1) free of userspecfed parameters, (2) capable of preservng a local degree p polynomal, (3) applcable to arbtrary unstructured meshes. The parameter-free generalzed moment (PFGM) lmter developed n the present study appears to meet all of the crtera. The lmter s composed of two components: an effcent accuracy preservng TVD marker for troubled cells based on cell averaged state varables, and a herarchcal generalzed moment lmter capable of handlng arbtrary unstructured meshes. The PFGM lmter has been mplemented and tested for a hgh-order SD method, although t can be easly appled to all other smlar hgh-order methods. The AP-TVD marker s based on the cell-averaged solutons and soluton dervatves, and s qute effcent to mplement. It appears that smooth extrema are not marked, whle the dscontnuous cells are consstently marked, wthout the use of any user-specfed parameter. The AP-TVD marker compares favorably aganst several markers n the lterature, such as the TVB marker, KXRCF marker, or the Harten marker. Accuracy studes confrmed that the lmter s capable of preservng accuracy n smooth regons. Numercal tests for a wde varety of problems n 1D and 2D wth both dscontnutes and smooth features demonstrated the capablty and usefulness of the PFGM lmter. A remanng challenge s to enhance the convergence property of the lmter for steady state problems. Acknowledgments The study was funded by AFOSR grant FA , and DOE grant DE-FG02-05ER The vews and conclusons contaned heren are those of the authors and should not be nterpreted as necessarly representng the offcal polces or endorsements, ether expressed or mpled, of AFOSR, DOE, or the U. S. Government. References [1] T. J. BARTH AND D. C. JESPERSON, The desgn and applcaton of upwnd schemes on unstructured meshes, AIAA Paper No , [2] R. BISWAS, K. D. DEVINE AND J. FLAHERTY, Parallel, adaptve fnte element methods for conservaton laws, Appl. Numer. Math., 14 (1994), pp

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