AIAA-00-058 A HIGH-ORDER SPECTRAL (FIITE) VOLUME METHOD FOR COSERVATIO LAWS O USTRUCTURED GRIDS Z.J. Wang Department of Mechancal Engneerng Mchgan State Unversty, East Lansng, MI 88 Yen Lu * MS T7B-, ASA Ames Research Center Moffett Feld, CA 905-000 ABSTRACT A tme accurate, hgh-, conservatve, fnte volume method named Spectral Volume (SV) method has been developed for conservaton laws on unstructured grds. The concept of a spectral volume s ntroduced to acheve hgh- accuracy n an effcent manner smlar to spectral element and mult-doman spectral methods. Each spectral volume s parttoned nto control volumes (CVs), and cell-averaged data from these control volumes s used to reconstruct a hgh- polynomal approxmaton n the spectral volume. Remann solvers are used to compute the fluxes at spectral volume boundares. Then cell-averaged state varables n the control volumes are updated ndependently. Furthermore, TVD (Total Varaton Dmnshng) and TVB (Total Varaton Bounded) lmters are ntroduced n the SV method to remove spurous oscllatons near dscontnutes. A very desrable feature of the SV method s that the reconstructon s dentcal for cells of the same type wth smlar parttons, and that the reconstructon stencl s always non-sngular, n contrast to the memory and CPU-ntensve reconstructon n a hgh- k-exact fnte volume (FV) method. The hgh- accuracy of the SV method s demonstrated for several model problems wth and wthout dscontnutes.. ITRODUCTIO umercal algorthms for conservaton laws have been extensvely researched n the last three decades because conservaton laws govern many physcal dscplnes such as flud dynamcs, electromagnetcs, aeroacoustcs, to name ust a few. One of the most successful algorthms for conservaton laws s the Godunov method [], whch lad a sold foundaton for the development of modern upwnd methods ncludng MUSCL [0], PPM [0], EO [] and weghted EO (WEO) schemes [6,0]. There are two key components n a Godunov-type scheme. One s data reconstructon, and the other s the Remann solver. The Godunov scheme employed a pece-wse constant data reconstructon, and the resultant scheme was only frst accurate. Van Leer extended the frst- Godunov scheme to second- [0] by usng a pecewse lnear data reconstructon. In addton, lmters were also ntroduced to remove spurous numercal oscllatons near steep gradents. Meanwhle the exact Remann solver used n the Godunov scheme was sometmes replaced by approxmate Remann solvers or flux-splttng procedures for better effcency [9,6,,,5,9]. The dffculty n generatng smooth structured grds for complex geometres has prompted ntensve research and development of unstructured grd algorthms n the last two decades [-,7,,,-]. Most of the unstructured grd methods are second- accurate because they are relatvely easy to mplement, and are qute memory effcent. Several hgh- schemes have been developed for unstructured grds. For example, a hgh- k-exact fnte volume scheme was developed by Barth and Frederckson n [], an EO scheme for unstructured grd was developed by Abgrall n [], and a WEO scheme was developed by Hu and Shu n []. Although hgh- accurate fnte volume schemes can be obtaned theoretcally for an unstructured grd by usng hgh- polynomal data reconstructons, hgher than lnear reconstructons are rarely used n three dmensons n practce. Ths s manly because of the dffculty n fndng vald (nonsngular) stencls, and the enormous memory requred to store the coeffcents used n the reconstructon. For each control volume, the reconstructon stencl s Assocate Professor (zw@egr.msu.edu), AIAA Senor Member * Research Scentst (lu@nas.nasa.gov) Copyrght 00 by Z.J. Wang & Yen Lu. Publshed by the Inc., wth permsson
unque for an unstructured grd as shown n Fgure a. A data reconstructon must be performed at each teraton for each control volume. Ths reconstructon step s the most memory and tme consumng n hgher than second- schemes. In a recent mplementaton of a thrd- FV scheme wth a quadratc reconstructon n three dmensons by Delanaye and Lu [], the average sze of the reconstructon stencls s about 50-70. Stll there are many sngular reconstructon stencls. The sze of the reconstructon stencls usually ncreases non-lnearly wth the of accuracy. For a fourth FV scheme, the average stencl sze s estmated to be at least 0. It s very memory and CPU ntensve to perform the reconstructon. More recently, another hgh- conservatve algorthm called the Dscontnuous Galerkn (DG) method was developed by Cockburn, Shu, et al and others [7-9,]. In the DG method, a hgh- data dstrbuton s assumed for each element. As a result, the state varable s usually not contnuous across element boundares. The fluxes through the element boundares are computed usng an approxmate Remann solver, smlar to a FV method. The resdual s then mnmzed wth a Galerkn approach. Due to the use of Remann fluxes cross element boundares, the DG method s fully conservatve. A dsadvantage of the DG method s that very hgh- surface and volume ntegrals are necessary, whch can be expensve to compute. Another hgh- conservatve scheme for unstructured quadrlateral grds s the mult-doman spectral method on staggered grds developed by Korprva and Kolas [8]. The mult-doman spectral method s smlar to the spectral element method by Patera [5], whch s not conservatve. Although very hgh- of accuracy was achevable wth both methods, the methods are dffcult to extend to other cell types such as trangles, or tetrahedra. In ths paper, a new conservatve hgh- SV method s developed for conservaton laws on unstructured grds. In the next secton, we present the basc framework of the SV method on trangular grds together wth a TVD Runge-Kutta tme ntegraton scheme. In Secton, the reconstructon problem based on CV-averaged solutons s studed, and t s shown that the reconstructon problems on all trangles wth a smlar partton are dentcal. In addton, convergent lnear to cubc SVs are presented. Secton dscusses ssues related to dscontnuty capturng and TVD and TVB lmters are presented. In Secton 5, numercal mplementatons of the SV method for both lnear and non-lnear scalar conservaton laws are carred out, and accuracy studes are performed for both lnear and nonlnear wave equatons to verfy the numercal of accuracy. The shock-capturng capablty of the method s also demonstrated wth the Burger s equaton. Fnally, conclusons and recommendatons for further nvestgatons are summarzed n Secton 6.. SPECTRAL (FIITE) VOLUME METHOD Consder a mesh of unstructured trangular cells. Each cell s called a Spectral Volume, denoted by S, whch s further parttoned nto subcells named Control Volumes (CVs), ndcated by C,, as shown n Fgure b. To represent the soluton as a polynomal of degree m n two dmensons (D) we need = (m+)(m+)/ peces of ndependent nformaton, or degrees of freedom (DOFs). The DOFs n a SV method are the volume-averaged mean varables at the CVs. For example, to buld a quadratc reconstructon n D, we need at least (+)(+)/ = 6 DOFs. There are numerous ways of parttonng a SV, and not every partton s admssble n the sense that the partton may not be capable of producng a degree m polynomal. Such parttons are also called sngular ones. Once mean solutons n the CVs of an admssble SV are gven, a unque polynomal reconstructon can be obtaned from p ( x, = L ( x, u, () =, where p ( x, Pm (the space of polynomals of degree m or less), L ( x, Pm, =,, are the shape functons satsfyng Lk ( x, dv = V, δ k. () C, where V, s the volume of C,. Ths hgh- polynomal reconstructon facltates a hgh- update for the mean soluton of each CV. Consder the followng hyperbolc conservaton law u t + F = 0, () where F s the flux vector. Integratng () n each CV, we obtan du K, V, + ( F n ) da = 0, () dt r= Ar where K s the total number of faces n C,, and u, s the volume-averaged soluton at C,. The flux ntegral n () s then replaced by a Gauss-quadrature formula whch s exact for polynomals of degree m Ar J ( F n ) da w F( u( x y )) n A, (5) q=, where J s the number of quadrature ponts on the r-th face, w are the Gauss quadrature weghts, (x, y ) are the Gauss quadrature ponts. Snce the reconstructed polynomals are pece-wse contnuous, the soluton s r r
usually dscontnuous across the boundares of a SV, although t s contnuous across nteror CV faces. The fluxes at the nteror faces can be computed drectly based on the reconstructed solutons at the quadrature ponts. The fluxes at the boundary faces of a SV are computed usng approxmate Remann solvers gven the left and rght reconstructed solutons,.e., F ( u( x r, y r )) nr (6) FRem ( p ( x r, y r ), p, r ( x r, y r ), nr ) where p, r ( x, y ) s the reconstructed polynomal n a neghborng SV sharng face r wth the SV n consderaton, S. Obvously, the approxmate Remann solver must satsfy FRem ( p ( x, y ), p, r ( x, y ), nr ) = F ( p ( x, y ), p ( x, y ), n ), Rem, r to acheve dscrete conservaton. It has been shown [] that of accuracy of ths SV scheme s (m+)-th. In addton, the scheme s compact n the sense that a hgh- polynomal s reconstructed n each SV wthout usng any data from neghborng SVs. Ths property can potentally translate nto sgnfcantly reduced communcaton cost compared to a k-exact FV scheme (for example) when mplemented on parallel computers. ote that one of the subtle dfferences between a FV method and a SV method s that all the CVs n a SV use the same data reconstructon. As a result, t s not necessary to use a Remann flux or flux splttng for the nteror boundares between the CVs nsde a partcular SV because the state varable s contnuous across the nteror CV boundares. Remann fluxes are only necessary at the boundares of the SV. For tme ntegraton, we use a thrd- TVD Runge- Kutta scheme from [7]. We frst rewrte () n a concse ODE form du = Rh (u), (7) dt Then the thrd- TVD Runge-Kutta scheme can be expressed as: ( ) n n u = u + tr ( u ) ; () n () () u = u + [ u + trh ( u )]; (8) n+ n () () u = u + [ u + trh ( u )]. The SV method shares many advantages wth the DG method [7-9] n that t s compact whch s sutable for parallel computng, hgh- accurate, conservatve, and capable of handlng complex geometres. The SV method s expected to be more effcent than the DG method because hgh- volume ntegrals are avoded, and lower surface Gauss ntegral h r formula can be used (m-th vs. m-th ). In addton, the SV method should have hgher resoluton than the DG method for dscontnutes because of the avalablty of local cell-averaged state varables at the CVs.. DATA RECOSTRUCTIO The reconstructon problem reads: Gven a contnuous functon n S, u ( S ) (the space of contnuous functons n S ), and a partton Π m of S, fnd p Pm, such that p ( x, dv = u( x, dv,,, C =. (9), C, To actually solve the reconstructon problem, we ntroduce the complete polynomal bass, el ( x, Pm, where Pm = span{ el ( x, } l=. Therefore p can be expressed as p or n the matrx form = l= a e ( x,, (0) p = e a, () where e s the bass functon vector e,..., e ] and a l l [ s the reconstructon coeffcent vector [ a,..., a ]. Substtutng (0) nto (9), we then obtan al el x y dv u V (, ) =,, =,, () C,, l= T Let u denote the column vector [ u,,..., u, ], Eq. () can be rewrtten n the matrx form R a = u, () where the reconstructon matrx e( x, dv e ( x, dv V, C, V, C, R =. (, ) (, ) e x y dv e x y dv V, C,, C, V The reconstructon coeffcents a can be solved as a = R u, () provded that the reconstructon matrx R s nonsngular. Substtutng Eq. () nto Eq. (), p s then expressed n terms of cardnal bass functons (or shape functons) L = L,..., L ] [ p = L ( x, u = L u. (5) = Here L s defned as L e R. (6), T
Equaton (5) gves the functonal representaton of the state varable u wthn the SV. The functon value of u at a quadrature pont or any pont ( x, y ) wthn the SV s thus smply p ( xxq, y ) = L ( x, y ) u,. (7) = The above equaton can be vewed as an nterpolaton of a functon value at a pont usng a set of cell averaged values wth each weght equal to the correspondng cardnal bass functonal value evaluated at that pont. In the case of trangular SV, t can be proven that the reconstructon coeffcents n (7) are dentcal for all trangles wth smlar parttons [5]. We thus have a unversal reconstructon formula, Eq. (7), for evaluatng the state varable u at smlar ponts. Ths also mples that the reconstructon needs to be carred out only once, and that can be performed usng any shape of trangle. In our study, we use Mathematca to carry out the reconstructon analytcally. The exact ntegratons of polynomals over arbtrary polygons can be found n []. Snce the reconstructon problem s equvalent for all trangles, we focus our attenton on the reconstructon problem n an equlateral trangle E. In parttonng E nto non-overlappng CVs, we further requre that the CVs satsfy the followng three condtons:. The CVs are "symmetrc" wth respect to all symmetres of the trangle;. All CVs are convex;. All CVs have straght sdes,.e., the CVs are polygons. We beleve the symmetry and convexty requrement s mportant for achevng the best possble accuracy and robustness. The requrement of polygons smplfes the formulaton of the SV method. It s then obvous that a CV contanng the centrod of E must be symmetrc wth respect to the three edges and vertces, and at most one such CV can exst. Ths CV, f t exsts, s thus sad to possess degree symmetry (or symmetry, n short). Smlarly, CVs wth degree and 6 symmetres can also be defned. For example, f a CV s sad to possess degree symmetry, then two other symmetrc CVs must exst n the same partton. We shall denote n, n and n 6 the number of degree, and 6 symmetry groups n a partton wth n = 0 or. Then the total number of CVs n the partton s then n + n + 6n 6. In to support the unque reconstructon of a degree m polynomal, the total number of CVs must be dentcal to the dmenson of the polynomal space,.e., ( m + )( m + ) n + n + 6n6 =. (8) The solutons of (8) can be used to gude the partton of E once m s gven. Some possble parttons of the standard trangle correspondng to these solutons for m =,, are shown n Fgures -. ext the queston of how these parttons perform n a data reconstructon needs to be answered. In [], the frst paper on the SV method, t was shown that not all non-sngular reconstructons are convergent. For example, hgh- polynomal reconstructons based on equdstant CVs n one dmenson are not convergent although the reconstructons are nonsngular. We beleve ths s the drect consequence of the Runge phenomenon. Therefore some means to quantfy the qualty of the reconstructons needs to be dentfed. For any u (E), there exsts a unque degree m polynomal p whch satsfes (9) for any admssble partton. Denote p = Γ Π (u), where Γ Π s a proecton operator, whch maps (E) onto P m (E). When both spaces (E) and P m (E) are equpped wth the unform norm,.e., = = max, the norm of ths proecton operator can be defned as ΓΠ u ΓΠ = sup. (9) u 0 u It can be shown that Γ Π = max L ( x,. (0) ( x, E = Γ Π s called the Lebesgue constant, whch s of nterest for the followng reason [5-6]: If * p m s the best unform approxmaton to u on E, then * ( + Γ ) u. u ΓΠ u Π () Thus Γ Π gves a smple method of boundng the nterpolaton polynomal. It s obvous from () that the smaller the Lebesgue constant, the better the nterpolaton polynomal s to be expected. Therefore the problem becomes fndng the partton wth a small Lebesgue constant, f not as small as possble. In ths paper, our focus s to construct convergent SV parttons when the SV s refned. The Lebesgue constant s used as the crteron to udge the qualty of the parttons. The optmzaton of the parttons wll be the subect of a future publcaton. Through extensve testng, the followng lnear to cubc SVs have been found. p m
Lnear Spectral Volume (m = ) Two parttons are possble, as shown n Fgure a and b, whch are named Type and Type parttons. Snce the centrods of the CVs are non-co-lnear, both parttons are admssble. ote that the CVs n both parttons possess a degree symmetry. The Lebesgue constants are / (.) and /5 (.8667) for Type and parttons, respectvely. ote that the Type SV has a much smaller Lebesgue constant than the Type SV, ndcatng that the error wth the Type SV should be smaller than the error wth the Type SV. Quadratc Spectral Volume (m = ) Two possble parttons for m = are shown n Fgure. The partton presented n Fgure a s not unque n the sense that the poston of one vertex on an edge of the trangle can change,.e., the length d shown n Fgure a can be any real number n (0, 0.5) assumng the length of the edge s. It seems that wth any d, the partton s admssble. In our numercal studes, two dfferent values of d were tested, namely d = / and d = /, whch are called Type and Type parttons, respectvely. The Lebesgue constant for the Type partton s 9., and for the Type partton s 8. Therefore, the Type partton s expected to yeld more accurate numercal results. Although the partton shown n Fgure b looks reasonable, t s not admssble. Cubc Spectral Volume (m = ) Three possble parttons for m = are shown n Fgure. For the partton shown n Fgure b, the parameter d can be changed to obtan dfferent parttons. In fact, the Lebesgue constants for parttons wth a set of d values are presented n Table. Among ths set of d values, t s nterestng to note that the Lebesgue constant reaches a smallest value of.85 at d = /5 from a value of 8.99 at d = /6. When d s smaller than /5, the Lebesgue constant starts to ncrease. For presentaton purpose, we call the partton shown n Fgure a the Type partton. The partton shown n Fgure b wth d = /6 s called the Type partton, and wth d = /5 the Type partton. It s expected that the Type partton should gve the most accurate numercal soluton n the unform norm. The Lebesgue constant for the Type partton s 67/ (.9), whch s sgnfcantly larger than those for the Type and parttons. umercal results to be presented later confrm that the Type partton s not convergent wth grd refnement. The partton shown n Fgure c s sngular. Table Lebesgue Constants for the Partton Shown n b d Lebesgue constant /6 8.99 /7 6.778 /8 5.790 /0.9 /5.85 /0.57595 /5.6598 /00.95. MULTI-DIMESIOAL LIMITERS A lmter n a hgh- numercal method such as the SV method should satsfy the followng two requrements: ) non-oscllatory, sharp resoluton of dscontnutes, and ) recover the full formal of accuracy away from the dscontnutes. To that end, a TVB lmter [8] has been mplemented. Refer to (), whch s used to update the CV-averaged state varable. Denote u = p ( x, y ) u,, r =,, K; q =,, J. Followng the TVB dea, f u Mh, r =,, K; q =,, J, () t s not necessary to do any data lmtng. In (), M represents some measure of the second dervatve of the soluton, and h s the dstance from pont ( x, y ) to the centrod of C,. If for any value of r and q, () s volated, t s assumed that C, s near a steep gradent and data lmtng s necessary. Instead of usng the polynomal p ( x, n C,, we assume lnear data dstrbuton n C,,.e., u, ( x, = u, + u, ( r - r, ), r C,, () where r, s the poston vector of the centrod of C,. In to acheve the hghest resoluton, we need to maxmze the magntude of the soluton gradent u,. At the same tme, we requre that the reconstructed solutons at the quadrature ponts of C, satsfy the followng monotoncty constrant: mn max u, u, ( x, y ) u,, () mn max where u, and u, are the mnmum and maxmum cell-averaged solutons among all ts neghborng CVs sharng a face wth C,. A very effcent approach can be used to compute the gradent. In ths approach, we avod a separate data reconstructon by reusng the polynomal reconstructon already avalable for the SV. For each CV, we use the gradent of the reconstructed polynomal at the CV centrod,.e., 5
p p u, =,. (5) x y r, Ths gradent s then lmted f necessary to satsfy (). If any of the reconstructed varable at the quadrature mn max ponts s out of the range [ u,, u, ], the gradent s lmted,.e., u, ϕ u,, where ϕ [0, ] s calculated from: u mn, f u 0 max > u, u, u mn ϕ =, f u < 0. mn u, u, otherwse ote that f parameter M = 0, the TVB lmter s TVD (total varaton dmnshng), whch strctly enforces monotoncty by sacrfcng accuracy near extrema. The avalablty of cell-averaged data on the CVs nsde a SV makes ths CV-based data lmtng possble, whereas n the DG method, one can only perform an element based data lmtng. Due to the ncreased local resoluton, the SV method has been shown to have hgher resolutons for dscontnutes than the DG method []. 5. UMERICAL TESTS Accuracy Study wth D Lnear Wave Equaton In ths case, we test the accuracy of the SV method on the two-dmensonal lnear equaton: u u u + + = 0, t x y x, y, u( x, y,0) = u ( x,, perodc b.. 0 c The ntal condton s u 0( x, = snπ ( x +. A fourth- accurate Gauss quadrature formula [] s used to compute the CV-averaged ntal solutons. These CV-averaged solutons are then updated at each tme step usng the thrd- TVD Runge-Kutta scheme presented earler. The numercal smulaton s carred untl t = on two dfferent trangular grds as shown n Fgure 5. One grd s generated from a unform Cartesan grd by cuttng each Cartesan cell nto two trangles, and s named the regular grd. The other grd s generated wth an unstructured grd generator, and s named the rregular grd. ote that the cells n the rregular grd have qute dfferent szes. In Table, we present the L and errors n the CVaveraged solutons produced usng second to fourth SV method schemes wth SVs shown n Fgures - on the regular grd. The errors presented n the table are tme-step ndependent because the tme step t was made small enough so that the errors are domnated by the spatal dscretzaton. In ths test, all SVs except the Type cubc SV (shown n Fgure a) are convergent wth grd refnement on ths regular grd. It s obvous that the expected of accuracy s acheved by all the convergent SVs. It s not surprsng that the Type cubc SV s not convergent because of ts rather large Lebesgue constant of.9. In contrast, the Type and cubc SVs have Lebesgue constants of 8. and. respectvely. It s nterestng to note that the Type lnear SV gves more accurate results n both the L and norms than the Type lnear SV even f the Type SV has a larger Lebesgue constant of. versus that of.87, of the Type SV. Ths ndcates that the Lebesgue constant cannot serve as an absolute error estmator, but rather an estmate of the upper bound of the error. For the quadratc and cubc SVs, the parttons wth smaller Lebesgue constants do gve more accurate numercal solutons, as shown n Table. ext, the L and errors n the numercal results computed on the rregular grd usng second- fourth SVs are shown n Table. Ths should be a much tougher test case because of the truly unstructured nature of the computatonal grd. What s strkng s that the Type lnear SV faled to acheve second- accuracy on ths grd. As a matter of fact, t s only frst accurate. Ths may be contrbuted to the acute angles of the CVs n the partton. ote that both quadratc SVs are convergent, and gve smlar results. Thrd accuracy s acheved by both types of quadratc SVs n the L norm although the numercal of accuracy n the norm s only slghtly over second-. We beleve ths s due to the nonsmoothness of the computatonal grd. The Type cubc SV also showed a non-convergent behavor n the norm on the fnest grd. It s nce to see that the Type cubc SV s not only convergent, but also fourth accurate n both the L and norms. Accuracy Study wth D Burgers Equaton In ths case, we test the accuracy of the SV method on the two-dmensonal non-lnear wave equaton: u u / u / + + = 0, t x y x, y, u( x, y,0) = + snπ ( x +, perodc b. c. The ntal soluton s smooth. Due to the non-lnearty of the Burgers equaton, dscontnutes wll develop n the soluton. Therefore we also test the capablty of the 6
method to acheve unform hgh- accuracy away from dscontnutes. At t = 0., the exact soluton s stll smooth, as shown n Fgure 6a. The numercal smulaton s therefore carred out untl t = 0. wthout the use of lmters on the rregular grd as shown n Fgure 5b. The numercal soluton on the 0x0x rregular grd computed wth the Type quadratc SV (thrd- accurate) s dsplayed n Fgure 6b. otce that the agreement between the numercal and exact solutons s excellent. In Table, the L and errors on the rregular grd are presented. The performance of the SV method on the non-lnear Burgers equaton s qute smlar to the performance on the lnear wave equaton, although there s a slght loss of accuracy (from 0. 0.6 s) especally on the rregular grd n the norm, probably due to the non-lnear nature of the Burgers equaton. Once agan, the Type lnear SV has dffculty n achevng second- accuracy on the rregular grd n both norms. At t = 0.5, the exact soluton has developed two shock waves as shown n Fgure 7a. A lmter s necessary to handle the dscontnutes. Shown n Fgure 7 are the exact soluton, and the computed numercal solutons wth the Type quadratc SV on the 0x0x rregular grd usng the TVD lmter,.e., M was taken to be 0. ote that the lmter produced a very good soluton. In to estmate the numercal of accuracy for the soluton away from the dscontnutes, L and L errors n the smooth regon [-0., 0.]x[-0., 0.] are computed. Computatons were carred out on the rregular grd only. Wthout the use of the lmter, the soluton quckly dverged after shock waves were developed n the soluton. The parameter M was set to be 00 n the computaton. If M s too small, the accuracy n the smooth regon s degraded probably because lmtng was carred out n the smooth regon as well as near the shock. The L and errors wth the bset performng SVs for a gven of accuracy are presented n Table 6. Obvously, wth ths choce of M, the desgned of accuracy was acheved away from dscontnutes. 6. COCLUSIOS A hgh- Spectral (Fnte) Volume method has been developed for two-dmensonal scalar conservaton laws on unstructured trangular meshes. Each mesh cell forms a spectral volume, and the spectral volume s further parttoned nto polygonal control volumes. Hgh schemes are then bult based on the CVaveraged solutons. It was shown that a unversal reconstructon can be obtaned f all spectral volumes are parttoned n a smlar manner [5]. However, as n the one-dmensonal case, the way n whch a SV s parttoned nto CVs affects the convergence property of the resultant numercal scheme. A crteron based on the Lebesgue constant has been developed and used successfully to determne the qualty of varous parttons. Symmetrc, stable, and convergent lnear, quadratc and cubc SVs have been obtaned, and many dfferent types of parttons are evaluated based on the Lebesgue constants and ther performance on model test problems. Accuracy studes wth D lnear and non-lnear scalar conservaton laws have been carred out, and the of accuracy clam has been numercally verfed on both smooth and non-smooth trangular grds for convergent SVs. TVD and TVB lmters have been developed for non-oscllatory capturng of dscontnutes, and found to perform well. The TVB lmters wth a properly selected parameter (M) are capable of mantanng unformly hgh- accuracy away from dscontnutes. The extenson of the method to one and two dmensonal hyperbolc systems s under way, and wll be reported n future publcatons. ACKOWLEDGEMET The frst author grateful acknowledges the start-up fundng provded by the Department of Mechancal Engneerng, College of Engneerng of Mchgan State Unversty. REFERECES. R. Abgrall, On essentally non-oscllatory schemes on unstructured meshes: analyss and mplementaton, J. Comput. Phys., 5-58 (99).. T.J. Barth and D.C. Jespersen, The desgn and applcaton of upwnd schemes on unstructured meshes, AIAA Paper o. 89-066.. T.J. Barth and P.O. Frederckson, Hgh- soluton of the Euler equatons on unstructured grds usng quadratc reconstructon, AIAA Paper o. 90-00, 990.. F. Bass and S. Rebay, Hgh- accurate dscontnuous fnte element soluton of the D Euler equatons, J. Comput. Phys. 8, 5-85 (997). 5. L.P. Bos, Boundng the Lebesgue functons for Lagrange nterpolaton n a smplex, J. Approx. Theo. 9, (98). 6. Q. Chen and I. Babuska, Approxmate optmal ponts for polynomal nterpolaton of real functons n an nterval and n a trangle, Comput. Methods Appl. Mech. Engrg, 8, 05-7 (995). 7
7. B. Cockburn and C.-W. Shu, TVB Runge-Kutta local proecton dscontnuous Galerkn fnte element method for conservaton laws II: general framework, Mathematcs of Computaton 5, - 5 (989). 8. B. Cockburn, S.-Y. Ln and C.-W. Shu, TVB Runge-Kutta local proecton dscontnuous Galerkn fnte element method for conservaton laws III: one-dmensonal systems, J. Comput. Phys. 8, 90- (989). 9. B. Cockburn, S. Hou and C.-W. Shu, TVB Runge- Kutta local proecton dscontnuous Galerkn fnte element method for conservaton laws IV: the multdmensonal case, Mathematcs of Computaton 5, 55-58 (990). 0. P. Colella and P. Woodward, The pecewse parabolc method for gas-dynamcal smulatons, J. Comput. Phys. 5, (98).. M. Delanaye and Y. Lu, Quadratc reconstructon fnte volume schemes on D arbtrary unstructured polyhedral grds, AIAA Paper o. 99-59-CP, 999.. S.K. Godunov, A fnte-dfference method for the numercal computaton of dscontnuous solutons of the equatons of flud dynamcs, Mat. Sb. 7, 7 (959).. A. Harten, B. Engqust, S. Osher and S. Chakravarthy, Unformly hgh essentally non-oscllatory schemes III, J. Comput. Phys. 7, (987).. C. Hu and C.-W. Shu, "Weghted essentally nonoscllatory schemes on trangular meshes," J. Comput. Phys. 50, 97-7 (999). 5. A. Jameson, Analyss and desgn of numercal schemes for gas dynamcs : artfcal dffuson and dscrete shock structure, Int. J. of Comput. Flud Dynamcs 5, -8 (995). 6. G. Jang and C.-W. Shu, Effcent mplementaton of weghted EO schemes, J. Comput. Phys. 6, 0 (996). 7. Y. Kallnders, A. Khawaa and H. McMorrs, Hybrd prsmatc/tetrahedral grd generaton for complex geometres, AIAA Journal,, 9-98 (996). 8. D.A. Koprva and J.H. Kolas, A conservatve staggered-grd Chebyshev multdoman method for compressble flows, J. Comput. Phys. 5, (996). 9. M.-S. Lou, Mass flux schemes and connecton to shock nstablty, J. Comput. Phys. 60, 6-68 (000). 0. X.D. Lu, S. Osher and T. Chan, Weghted essentally non-oscllatory schemes, J. Comput. Phys. 5, 00- (99).. Y. Lu and M. Vnokur, Exact ntegraton of polynomals and symmetrc quadrature formulas over arbtrary polyhedral grds, J. Comput. Phys. 0-7 (998).. H. Luo, D. Sharov, J.D. Baum and R. Lohner, On the computaton of compressble turbulent flows on unstructured grds, AIAA Paper o. 000-097, Jan. 000.. D.J. Mavrpls and A. Jameson, Multgrd soluton of the aver-stokes equatons on trangular meshes, AIAA J. 8, 5-5 (990).. S. Osher, Remann solvers, the entropy condton, and dfference approxmatons, SIAM J. on umercal Analyss, 7-5 (98). 5. A.T. Patera, A Spectral element method for flud dynamcs: lamnar flow n a channel expanson, J. Comput. Phys. 5 68-88 (98). 6. P.L. Roe, Approxmate Remann solvers, parameter vectors, and dfference schemes, J. Comput. Phys. 57-7 (98). 7. C.-W. Shu, Total-Varaton-Dmnshng tme dscretzatons, SIAM Journal on Scentfc and Statstcal Computng 9, 07-08 (988). 8. C.-W. Shu, TVB unformly hgh- schemes for conservaton laws, Math. Comp. 9, 05- (987). 9. J.L. Steger and R.F. Warmng, Flux vector splttng of the nvscd gasdynamcs equatons wth applcaton to fnte dfference methods, J. Comput. Phys. 0 6 (98). 0. B. van Leer, Towards the ultmate conservatve dfference scheme V. a second sequel to Godunov s method, J. Comput. Phys., 0-6 (979).. B. van Leer, Flux-vector splttng for the Euler equatons, Lecture otes n Physcs, 70, 507 (98).. V. Venkatakrshnan and D. J. Mavrpls, Implct solvers for unstructured meshes, J. Comput. Phys. 05, o., 8-9 (99).. Z.J. Wang and R.F. Chen, Ansotropc Cartesan Grd Method for Vscous Turbulent Flow, AIAA Paper o. 000-095, 000.. Z.J. Wang, Fnte spectral volume method for conservaton laws on unstructured grds: basc formulaton, submtted to J. Comput. Phys. 5. Z.J. Wang and Yen Lu, Spectral volume method for conservaton laws on unstructured grds II: extenson to D scalar equaton, submtted to J. Comput. Phys. 8
Table. Accuracy on ut + ux + uy = 0, wth u0 ( x, = snπ ( x +, at t = (regular grds) Order of Accuracy Grd L error (Type SV) (Type SV) (Type, d = /) (Type, d = /) (Type SV) (Type, d = /6) (Type, d = /5) L error 0x0x.0e- -.97e- - 0x0x 7.68e-.98.e-.00 0x0x.9e-.00.0e-.00 80x80x.8e-.00 7.75e-.00 60x60x.0e-.00.9e-.00 0x0x.0e- - 6.68e- - 0x0x.06e-.9.78e-.9 0x0x.7e-.97.5e-.97 80x80x 6.8e-.99.e-.99 60x60x.7e-.00.87e-.99 0x0x.8e- - 7.76e- - 0x0x 5.e-.97.0e-.9 0x0x 6.7e-5.99.5e-.0 80x80x 8.5e-6.99.55e-5.0 60x60x.06e-6.99.9e-6.00 0x0x.7e- - 7.88e- - 0x0x.77e-. 9.8e-.00 0x0x 6.0e-5.98.e-.00 80x80x 7.58e-6.99.5e-5.0 60x60x 9.57e-7.99.9e-6.00 0x0x.8e- -.86e- - 0x0x 8.6e-6.00.98e-5.6 0x0x 5.7e-7.98.5e-6.7 80x80x.6e-8.98.7e-7.69 60x60x.9e-8 egatve 5.5e-7 egatve 0x0x 9.e-5 -.7e- - 0x0x 5.86e-6.99.9e-5.0 0x0x.70e-7.99.e-6.95 80x80x.e-8.00 7.78e-8.99 60x60x.5e-9.00.8e-9.0 0x0x 7.6e-5 -.5e- - 0x0x.5e-6.0.6e-5.96 0x0x.8e-7.0.0e-6.99 80x80x.75e-8.0 6.0e-8.00 60x60x.0e-9.99.9e-9.0 (a) (b) Fgure. (a) A Possble Stencl of Cells Used to Buld a Quadratc Reconstructon n an Unstructured Grd; (b) A Quadratc Trangular Spectral Volume 9
Table. Accuracy on ut + ux + uy = 0, wth u0 ( x, = snπ ( x +, at t = (rregular grds) Order of Accuracy Grd L error (Type SV) (Type SV) (Type, d = /) (Type, d = /) (Type, d = /6) (Type, d = /5) L error 0x0x.0e- -.60e- - 0x0x 6.66e- 0.96.9e- 0.9 0x0x.5e- 0.9 9.8e- 0.96 80x80x.85e- 0.9.9e-.00 60x60x 9.7e- 0.9.86e- 0.78 0x0x 6.7e- -.6e- - 0x0x.8e-.87.e-.6 0x0x.7e-.96.5e-.9 80x80x.9e-.98.9e-.97 60x60x.00e-.99 8.85e-.7 0x0x 9.7e- -.67e- - 0x0x.5e-.87 5.8e-.80 0x0x.6e-.9 8.e-.67 80x80x.5e-5.9.8e-.8 60x60x.79e-6.95.05e-5.8 0x0x 8.6e- -.76e- - 0x0x.5e-.86 5.6e-.7 0x0x.5e-.9.00e-.9 80x80x.0e05.9.e-. 60x80x.6e-6.9 5.e-5.0 0x0x.0e- -.58e- - 0x0x.0e-5.9.7e-.90 0x0x.e-6.9.e-5.6 80x80x 9.6e-8.80.0e-6.79 60x60x.0e-8.06.e-6 egatve 0x0x.7e- -.5e- - 0x0x.6e-5.07.e-.7 0x0x 9.9e-7.0 8.8e-6.78 80x80x 6.7e-8.0 5.0e-7.9 60x60x.87e-9.99.79e-8.8 (a) (b) Fgure. Possble Parttons of a Lnear Spectral Volume d (a) (b) Fgure. Possble Parttons of a Quadratc Spectral Volume 0
Table. Accuracy on ut + uux + uuy = 0, wth u0 ( x, = + snπ ( x +, at t = 0. wth rregular grd Order of Accuracy Grd error error (Type SV) (Type SV) (Type, d = /) (Type, d = /) (Type, d = /6) (Type, d = /5) L L 0x0x.6e- -.e- - 0x0x.5e-.9.e- 0.88 0x0x 5.8e-.8.e- 0.9 80x80x.e-.7 6.09e-.00 60x60x.06e-.9.99e-.0 0x0x 5.79e- -.96e- - 0x0x.6e-.99 9.5e-.69 0x0x.67e-.99.87e-.67 80x80x 9.0e-5.97 8.78e-.7 60x60x.9e-5.98.5e-. 0x0x 6.7e- -.7e- - 0x0x.e-.0.6e-.90 0x0x.0e-5.58.5e-.8 80x80x.e-6.65 8.e-5.0 60x60x 5.0e-7.68.66e-5. 0x0x 6.8e- -.9e- - 0x0x.7e-..09e-.85 0x0x.9e-5.6.05e-.8 80x80x.0e-6.67 7.6e-5.09 60x60x.6e-7.70.e-5. 0x0x 7.87e-5 -.0e- - 0x0x 6.07e-6.70.00e-.5 0x0x.55e-7.7 9.6e-6.8 80x80x.e-8.7 8.55e-7.9 60x60x.79e-9.6 8.75e-8.9 0x0x 9.7e-5 -.9e- - 0x0x 7.7e-6.76.e-.8 0x0x 5.0e-7.79.07e-5.5 80x80x.79e-8.79 9.e-7.5 60x60x.88e-9.7 8.e-8.9 d (a) (b) (c) Fgure. Possble Cubc Trangular Spectral Volumes
Table 5. Accuracy on ut + uux + uuy = 0, wth u0 ( x, = + snπ ( x +, at t = 0. 5 n [-0., 0.]x[-0., 0.] on rregular grd, TVB Lmter wth M = 00 Order of Accuracy Grd L error (Type SV) (Type SV) (Type SV) L error 0x0x.68e- - 5.e- - 0x0x.9e-5.0.65e-.69 0x0x 9.66e-6.0.8e-.77 80x80x.e-6.99.58e-.6 60x60x 6.0e-7.0.9e-5.8 0x0x 6.e-5-6.57e- - 0x0x 6.5e-6. 5.86e-.9 0x0x 6.06e-7.7 7.e-5.0 80x80x 7.0e-8.0.9e-5.8 60x60x 9.7e-9.97.68e-6.7 0x0x 7.8e-5 -.9e- - 0x0x 6.78e-7 6.85.e- 5.07 0x0x 6.8e-9 6.7.97e-6 8.78 80x80x.85e-0.05 6.65e-8 5.8 60x60x.8e-.76.6e-9.9 (a) (b) Fgure 5. Regular and Irregular "0x0x" Computatonal Grds (a) Exact Soluton (b) umercal Soluton Fgure 6. Exact and Computatonal Solutons of the Burgers Equaton at t = 0. on the 0x0x Irregular Mesh Usng the Type Quadratc Spectral Volume (Thrd-Order Accurate)
(a) Exact Soluton (b) umercal Soluton Fgure 7. Exact and Computatonal Solutons of the Burgers Equaton at t = 0.5 on the 0x0x Irregular Grd Usng the Type Quadratc SV (Thrd-Order Accurate)