Quasi-ENO Schemes for Unstructured Meshes. Carl F. Ollivier-Gooch. Mathematics and Computer Science Division. Argonne National Laboratory

Transcription

1 Quas-ENO Schemes for Unstructured Meshes Based on Unlmted Data-Dependent Least-Squares Reconstructon Carl F. Ollver-Gooch Mathematcs and Computer Scence Dvson Argonne Natonal Laboratory Subject classcaton: 65N06 (Boundary value problems n PDE's, Fnte derence/nte volume methods) (Flud mechancs, computatonal methods) 76G99, 76H05 (Flud mechancs, subsonc and transonc compressble ow)

2 Runnng head: Quas-ENO Schemes for Unstructured Meshes Malng address: Carl Ollver-Gooch Mathematcs and Computer Scence Dvson Argonne Natonal Laboratory 9700 South Cass Avenue, MCS/22 Argonne, IL Emal: Phone: (630) FAX: (630)

3 Abstract A crucal step n obtanng hgh-order accurate steady-state solutons to the Euler and Naver- Stokes equatons s the hgh-order accurate reconstructon of the soluton from cell-averaged values. Only after ths reconstructon has been completed can the ux ntegral around a control volume be accurately assessed. In ths work, a new reconstructon scheme s presented that s conservatve, s unformly accurate, allows only asymptotcally small overshoots, s easy to mplement on arbtrary meshes, has good convergence propertes, and s computatonally ecent. The new scheme, called DD-L 2 -ENO, uses a data-dependent weghted least-squares reconstructon wth a xed stencl. The weghts are chosen to strongly emphasze smooth data n the reconstructon, satsfyng the weghted ENO condtons of Lu, Osher, and Chan. Because DD-L 2 -ENO s desgned n the framework of k-exact reconstructon, exstng technques for mplementng such reconstructons on arbtrary meshes can be used. The scheme allows graceful degradaton of accuracy n regons where nsucent smooth data exsts for reconstructon of the requested hgh order. Smlartes wth and derences from WENO schemes are hghlghted. The asymptotc behavor of the scheme n reconstructng smooth and pecewse smooth functons s demonstrated. DD-L 2 -ENO produces unformly hgh-order accurate reconstructons, even n the presence of dscontnutes. Results are shown for one-dmensonal scalar propagaton and shock tube problems. Encouragng prelmnary two-dmensonal ow solutons obtaned usng DD-L 2 -ENO reconstructon are also shown and compared wth solutons usng lmted least-squares reconstructon. 3

4 Introducton A crucal step n obtanng hgh-order accurate steady-state solutons to the Euler and Naver- Stokes equatons s the hgh-order accurate reconstructon of the soluton from cell-averaged values. Only after ths reconstructon has been completed can the ux ntegral around a control volume be accurately assessed. Ideally, the reconstructon should conserve the mean value of the functon n each cell, be unformly accurate wth no overshoots, have good steady-state convergence propertes, be computatonally ecent and be easy to mplement on arbtrary meshes. In partcular, the scheme must be easy to mplement on meshes for whch the local mesh connectvty s varable. The varable connectvty, not the shape of the cells, s the key dstngushng feature of unstructured versus structured meshes n a reconstructon context. Both lmted k-exact reconstructon and essentally non-oscllatory (ENO) reconstructon have many of these desrable propertes. The desgn crtera for k-exact reconstructon gven by Barth and Frederckson [] ensure conservaton of the mean, good accuracy for smooth functons, and computatonal ecency. These schemes are easy to mplement on arbtrary meshes. A xed stencl s used, allowng pre-computaton of certan purely geometrc quanttes used n the reconstructon, wth a correspondng mprovement n ecency. Accuracy near dscontnutes s poor, however, and lmtng s requred to prevent overshoots of O (). Lmters that retan good convergence propertes (e.g., [2]) are often computatonally expensve. By desgn, the ENO reconstructon schemes of Harten et al [3, 4, 5] conserve the mean, are unformly accurate at all ponts for whch a smooth neghborhood exsts, and guarantee that overshoots wll be no larger than the order of the truncaton error of the reconstructon. Unform hgh-order accuracy s obtaned by usng reconstructon stencls that vary n both space and tme. Unfortunately, such stencls can hamper convergence to steady state. The weghted ENO (WENO) famly of schemes overcomes ths dculty by usng all possble stencls wth a data-dependent weghtng [6, 7]. Stencls contanng non-smooth data are not excluded by these schemes, but nstead are gven a weght on the order of the truncaton error. Because the weghts vary smoothly wth the data, these schemes converge well. Wth proper choce of weghts, WENO schemes can attan 2k + order accuracy for smooth functons usng k + -pont stencls [7]. These schemes are one-dmensonal n nature and are appled drecton-by-drecton for reconstructon on mult-dmensonal structured meshes. The mplementaton of stencl-searchng ENO schemes on unstructured meshes s dcult because stencls must be sought for all dmensons smultaneously and because the number of possble stencls s very large [8, 9]. The WENO famly holds out some promse here, n that each of a set of stencls could be ncluded wth a data-dependent weght, elmnatng the need for complex logc to nd a smooth stencl. Nevertheless, such schemes would stll be hampered n practce because there may be no smooth stencl; graceful recovery n ths stuaton s not a subject dscussed n the lterature. In ths work, a new reconstructon scheme s presented that has the best propertes of both lmted k-exact reconstructon schemes and ENO schemes. The new scheme, called DD-L 2 -ENO, uses a data-dependent weghted least-squares reconstructon wth a xed stencl. To ths extent, the scheme resembles exstng k-exact reconstructon algorthms [, 0] and can buld on that technology base. The feature whch dstngushes DD-L 2 -ENO from 4

5 standard k-exact schemes s the use of data-dependent weghts. Each pont n a reconstructon stencl s gven a data-dependent weght, chosen to strongly emphasze smooth data n the reconstructon. Speccally, the weght gven to non-smooth data s on the order of the truncaton error. In ths respect, DD-L 2 -ENO s smlar to WENO schemes, except that the pont-wse nature of the weghtng wth DD-L 2 -ENO mparts some addtonal exblty n choosng the smooth data to nclude n the reconstructon. For both DD-L 2 -ENO and WENO schemes, the weghts vary smoothly wth varatons n the data, whch mproves steady-state convergence n comparson to stencl-searchng ENO schemes. In regons where nsucent smooth data s avalable for hgh-order reconstructon, DD-L 2 -ENO automatcally reduces the order of accuracy of the reconstructon locally; ths should mprove robustness n practce. Sectons 2 revews WENO reconstructon schemes n one dmenson. Secton 3 presents the new data-dependent least-squares reconstructon, ncludng dscusson of the computaton of the data-dependent weghts and requrements for the algorthm used to solve the least-squares problem. Secton 4 shows results for reconstructon of smooth and non-smooth functons, ncludng solutons to the scalar wave equaton. In Secton 5, some sample ow solutons are shown. 2 Weghted ENO Reconstructon The crtera set forth by Harten et al. [5] for essentally non-oscllatory reconstructon place a great emphass on accuracy, n that ENO schemes obtan hgh-order accurate reconstructons for all control volumes havng a smooth neghborhood. Also, ENO reconstructon conserves the mean and, n one dmenson, guarantees that the total varaton of the reconstructon wll not exceed that of the orgnal functon by more than O x k+. The weghted ENO (WENO) schemes of Lu, Osher, and Chan [6] take a somewhat derent approach to non-smooth data. ENO schemes try to nd the smoothest possble stencl of k + ponts contanng the control volume n queston; k + such stencls exst. WENO schemes use a convex combnaton of reconstructons usng each possble stencl, wth weghts that can be wrtten as: w = P k =0 () When the functon s smooth, each stencl provdes a k + order accurate reconstructon. In ths case, the can be chosen to gve 2k + order accuracy [7]. On the other hand, when the data n a stencl s non-smooth, the weght for that stencl s the order of the truncaton error. Lu, Osher, and Chan [6] dene ths as the \ENO property" for WENO schemes. For a more complete dscusson of these schemes, see [6, 7]. 3 Data-Dependent Least-Squares ENO Reconstructon Rather than begnnng wth a drecton-by-drecton ENO scheme and adaptng t to unstructured meshes, we take the opposte approach: begn wth a least-squares reconstructon 5

6 scheme sutable for use on unstructured meshes and modfy t to satsfy the ENO property. That s, we seek to solve the followng problem: Consder a doman whch has been tessellated; the tessellaton has a characterstc length scale x, at least locally. The medan dual of the tessellaton denes for each vertex v a surroundng control volume V. For any functon u(~x) dened on and ts control-volume averaged values u, compute an expanson R (~x?~x ) about v that Conserves the mean. Has compact support. Reconstructs exactly polynomals of degree k. Equvalently, R (~x? ~x )? u(~x) = O x k+. Satses the ENO property of Lu, Osher, and Chan [6]. The remander of ths secton wll descrbe ths process n two-dmensons. Reducton to one dmenson, extenson to three dmensons, and applcaton to structured meshes are all straghtforward varatons on the theme. 3. Conservaton of the Mean Conservaton of the mean wthn a control volume requres that Z V R (~x? ~x )da = Z V u(~x)da (2) Ths can be accomplshed by usng zero-mean polynomals n expandng about v. That s, by wrtng: R (~x? ~x ) = u (x? x ) (y? y ) u (x? x ) 2? x 2 2 u ((x? )(y? y )? xy ) 2 (y? y ) 2? y (3) where x n y m A ZV (x? x ) n (y? y ) m da (4) 6

7 By nspecton, the expanson of Equaton 3 satses Equaton 2. As a practcal matter, the ntegral of Equaton 4 s most easly computed by usng Green's Theorem to convert t to a boundary ntegral around V. Z x n y m = (x? x ) n+ (y? y ) m dy (5) (n + Ths ntegral may be evaluated exactly usng a Gaussan quadrature of approprate order along the boundary of the control volume. 3.2 Compact Support Compact support mples that the reconstructon R wll not requre data that s too far removed physcally from ~x. Ths n turn mples, for reasonable meshes, that data used n R wll not be topologcally far from control volume ; that s, there wll be a small number of levels of neghbors used to compute R. Conversely, usng only a small topologcal neghborhood of a vertex n the reconstructon ensures compact physcal support as well. For rst- and second-order accuracy (k = 0; ) n two dmensons, control volumes adjacent to V (the rst neghbors) are used to form the set fv j g of control volumes n the reconstructon stencl. For thrd- and fourth-order accuracy (k = 2; 3), second neghbors are also ncluded n fv j g. Because stencl determnaton s done as a pre-processng step, t s possble to add members to the stencls of control volumes whch are too near the boundary to have sucent support for the desred order of accuracy of reconstructon. 3.3 Accuracy for Smooth Functons Accuracy of the reconstructon for smooth functons can be stated n two equvalent ways. The reconstructon can be sad to be k-exact for some k f, when reconstructng P (~x)fx m y n : m + n kg, R (~x? ~x ) P (~x) (6) Alternatvely, one can say that for any u(~x), R (~x? ~x ) = u(~x) + O x k+ In practce, ths accuracy requrement means that the moded Taylor seres expanson of R gven n Equaton 3 must be carred out through the k th dervatves. To compute these dervatves, we seek to mnmze the error n predctng the mean value of the functon for control volumes n the stencl fv j g. The mean value, for a sngle control volume V j, of the reconstructed functon R s Z R (~x? ~x )da = u A j V j Z! (x? )da? x A j V j Z! (y? )da? y A j V j 7 (7)

8 u Z (x? x ) 2 2 2A j V j 2 x2 u Z! (x? )(y? y )da? xy A j V j 2 2A j ZV j (y? y ) 2 da? 2 y2!! + (8) To avod computng moments of each control volume n fv j g about v, replace x? x and y?y wth (x?x j )+(x j?x ) and (y?y j )+(y j?y ) respectvely. Expandng and ntegratng, Z R (~x? ~x ) = u (x A j V j + (x j? x )? x ) j + (y j? y )? y u x 2 j + 2x j (x j? x )? (x j? x ) 2 + x 2 2 u j + x j (y j? y ) + (x j? x )y j +(x j? x )(y j? y )? xy ) u y 2 j + 2y j (y j? y )? (y j? y ) 2 + y The geometrc terms n ths equaton are of the general form d x n y m j A j ZV j ((x? x j ) + (x j? x )) n = ((y? y j ) + (y j? y )) m da? x n y m mx nx m n (x j? x ) k l k l=0 k=0 (y j? y ) l x n?k y m?l j? xn y m (0) In these terms, we can wrte Z R (~x? ~x ) = u bx A j V j j u x c cxy j 2 c y 2 j 2 + () Equaton evaluates the mean value of the reconstructon R (~x?~x ) for a control volume j, gven the low-order dervatves of the soluton at v and low-order moments of the control volumes. The derence between ths predcton and the actual control-volume average u j s easy to assess. The dervatves at v are chosen to mnmze ths error over the stencl fv j g n a least-squares sense. Geometrc weghts w j are used to specfy the relatve mportance of 8

9 good predcton for varous control volumes n the stencl, wth the weghts based on dstance between vertces. The resultng least-squares problem s where and L L 2 L 3. L N 0 3 B u C C A = 0 B w (u? u ) w 2 (u 2? u ) w 3 (u 3? u ). w N (u N? u ) L j = w j bx j w j by j w j c x 2 j w j cxy j w j c y 2j w j = C C A (2) (3) j~x j? ~x j 2 (4) The least-squares problem of Equaton 2 s solved usng Householder transformatons to reduce the left-hand sde of Equaton 2 to upper-trangular form. After the uppertrangularzaton s complete, back-substtuton yelds the requred dervatves. There are several good reasons to use ths approach nstead of the smpler normal equaton soluton to the least-squares problem. Usng Householder transformatons gves a more accurate soluton to the least-squares problem than usng normal equatons, especally for ll-condtoned matrces. The error n the soluton s O (K) usng Householder transformatons and O (K 2 ) for normal equatons, where K s the condton number of the non-square matrx and s machne precson []. Ths also mples greater robustness. As a further mprovement n robustness, the Householder transform approach can detect sngular and nearly-sngular matrces on the y. If the least-squares problem s (nearly) sngular, a column wth (nearly) zero elements on and below the dagonal wll be encountered durng Householder trangularzaton. Ths falure occurs because the stencl s nadequate to support the requested number of dervatves. To resolve ths, ether more ponts must be added to the reconstructon stencl or the reconstructon must be moded to nclude fewer dervatves. The latter course s adopted n ths work. Dervatves are computed only to the hghest order for whch all dervatves can be computed; the addtonal ncomplete set of dervatves s dscarded, because no ncrease n order of accuracy s possble by retanng them. After the upper-trangularzaton of the least-squares problem s complete, the resdual for the soluton s avalable at vrtually no cost. Before back-substtuton, the problem 9

10 looks lke: x x x x x x x x x... x x 0 m n B B 2 u 2 u 2. C C A = 0 B r r 2. r m r m+ Clearly, the rst m equatons wll be satsed exactly. The remanng n? m equatons wll not be; the resdual R, whch s the same as the resdual for the orgnal problem, s R = vu u t n X l=m+. r n? r n C C A (5) r 2 l (6) Ths resdual wll be put to use n computng data-dependent weghts to enforce the ENO property, after scalng by x 2 to remove the eects of the geometrc weghtng term. 3.4 ENO Property The reconstructon scheme descrbed above s desgned for smooth functons. For nonsmooth functons those wth O () dscontnutes such a scheme allows overshoots of O (). Ths s not desrable for ether functon approxmaton or scentc computaton, where such overshoots can easly produce aphyscal values. Ths problem has typcally been addressed by performng a reconstructon wth geometrc weghts and preventng overshoots by heurstcally lmtng, or reducng, the dervatves (e.g. [0, 2]). Whle ths approach s not unsuccessful, t provdes only a mechancal soluton to an underlyng theoretcal problem. Speccally, the stencl for a control volume near a dscontnuty wll nclude control volumes j whch le across the dscontnuty. Because the functon s not smooth, approxmatng data n V j by a moded Taylor seres around v s napproprate. Ignorng ths mathematcal fact causes the unphyscally large dervatves that lmtng seeks to reduce. A better alternatve s to reconstruct usng only data that s smoothly connected to data n. Ths approach s taken drectly by typcal ENO schemes, whch by desgn search for a smooth stencl and completely exclude non-smooth data from the reconstructon. WENO schemes work less drectly, weghtng all possble stencls and weghtng those contanng non-smooth data wth a weght that s of the order of the truncaton error. In a weghted least-squares context, weghts are assgned to control volumes rather than to stencls. Nevertheless, the goal s to weght non-smooth data wth O x k+ to satsfy the ENO condton of Lu, Osher, and Chan [6]. We seek to construct a data-dependent weght, whch wll multply the prevously-calculated geometrc weght. Ths constructon s based on two observatons.. If the functon s non-smooth, then f the neghborhood of V crosses a dscontnuty, then a moded Taylor expanson does not adequately descrbe the functon locally 0

11 and there wll be one or more control volumes j for whch Z R (~x? ~x )da? u j = O () (7) A j V j Ths means that the resdual R of the least-squares problem wll be O (). On the other hand, for stencls whch cover only smooth regons of the functon, R (~x? ~x )? u(~x) = O x k+ and R s also O x k+. Therefore, R can be used as a gauge of the smoothness of the data for the entre stencl fv j g. 2. Whether the data n V j s smoothly connected to data n V can be determned asymptotcally by evaluatng ( u j? u j~x j? ~x = O () smoothly connected (9) j O (x? ) not smoothly connected We seek a data-dependent weghtng whch uses R to detect stencls wth non-smooth data and u j?u j~x j?~x to determne whch data wthn that stencl should be excluded and whch j ncluded. Put another way, we seek to construct a smoothness ndcator analogous to the dvded derence ndcators of [6, 7] whch s approprate for unstructured meshes and leastsquares reconstructon. One approprate weghtng s W DD j = + Rl 2 u j?u j~x j?~x j (8) (k+) (20) where = 0?0 prevents dvson by zero; l, a length scale for the control volume, s ncluded to remove the eects of the geometrc weghtng on R; Rl 2 u j?u j~x j?~x (k+) j ndcator. Asymptotcally, the behavor of ths weghtng for the three mportant cases s W DD j = 8 >< >: max (? ; O x?(k+) ) smooth functon s the smoothness max (? ; O x?(k+) ) smooth data n stencl wth non? smooth data O () non? smooth data In summary, for stencls contanng only smoothly-connected data, the data-dependent weghts are all of same order, ensurng that the good qualtes of the data-ndependent reconstructon wll be preserved for smooth functons. For stencls that are not entrely smooth, the data-dependent weght for non-smooth data s smaller than that for smooth data by a factor of the order of truncaton error. These weghtngs satsfy the ENO condton of Lu, Osher, and Chan [6]. Once the data-dependent weght has been computed for each control volume j n the stencl, the j th row n the least-squares problem s scaled by t. Wthout any computaton (2)

12 other than ths scalng, the least-squares problem of Equaton 2 can be moded to where and L 0 L 0 2 L 0 3. L 0 N u C A = 0 B w 0 (u? u ) w 0 2 (u 2? u ) w 0 3 (u 3? u ). w 0 N (u N? u ) L 0 j = w 0 j bx j w 0 j by j w 0 j c x 2 j w 0 j cxy j w 0 jc y 2 j C C A (22) (23) w 0 j = j~x j? ~x j 2 W DD j (24) As data-dependent weghts are assgned, the number that are numercally large s counted. If too few control volumes are assgned hgh data-dependent weghts, nsucent smoothlyconnected data s avalable to compute the desred number of dervatves. Smlarly to the data-ndependent case, ths results n a lowerng of the nomnal order of accuracy of the reconstructon locally. Ths loss of accuracy s most lkely to occur for control volumes straddlng dscontnutes; near the ntersecton of two dscontnutes; or near the mpngement of a dscontnuty on a boundary. Because no non-smooth data has a sgncant mpact on the reconstructon, however, large overshoots n the reconstructon are not expected. Ths would not be true f non-smooth data were ncluded. 3.5 Comparson wth WENO Schemes Conceptually, there s a great deal of smlarty between WENO schemes and the new DD-L 2 -ENO schemes. Both seek to produce a hgh-order accurate, essentally non-oscllatory reconstructon of a functon, elmnatng the deleterous eects of usng non-smooth data n reconstructng a functon by gvng non-smooth data a weght no larger than the order of the truncaton error. In both cases, the weghts vary smoothly wth changes n the controlvolume averaged functon, elmnatng the \swtchng" behavor found n stencl-searchng ENO schemes. There are, however, several key derences between these two famles of schemes.. DD-L 2 -ENO schemes are nherently mult-dmensonal; WENO schemes are desgned to be appled dmenson-by-dmenson. DD-L 2 -ENO schemes are therefore suted for use wth unstructured as well as curvlnear structured meshes, whle WENO schemes can be used only wth structured meshes. Ths does not preclude the constructon of a mult-dmensonal WENO reconstructon scheme capable of reconstructng on unstructured meshes. In outlne, one would need to construct for each control volume V a set of canddate stencls; reconstruct usng each stencl; compute weghts for each stencl based on the apparent smoothness of the data; and compute a weghted sum of the stencl reconstructons to obtan the nal reconstructon. The most dcult step 2

13 here s the rst: constructon of each canddate stencls requres a systematc search for a xed number of nearby neghbors. Whether these stencls are stored or recomputed for each use s an mplementaton ssue of great consequence for the CPU and memory resources requred by such a scheme. 2. WENO schemes can be desgned to produce very hgh-order (2k + ) accurate reconstructon for smooth data n a sngle dmenson [7]. Ths s done by choosng weghts for each of k + k + -pont stencls so that successve truncaton error terms are annhlated. Ths possblty s latent n DD-L 2 -ENO schemes; gven a stencl that contans all the ponts used n such a hgh-order WENO reconstructon, an equal number of dervatves could be accurately calculated. However, the emphass n developng DD-L 2 -ENO schemes has been on applcaton to rregular, mult-dmensonal geometres. For these cases, dmenson-by-dmenson reconstructon fals, and because the number of dervatves grows as k D, requred stencl szes for such hgh accuracy grow very rapdly n multple dmensons. For ths reason, DD-L 2 -ENO was not desgned wth such hgh accuracy n mnd. 3. DD-L 2 -ENO schemes weght ponts ndvdually, whle WENO schemes apply weghts to stencls. In one dmenson, ths derence s probably moot; the same ponts are lkely to have hgh weghts ether way. In multple dmensons, the derence between schemes wll depend on how mult-dmensonal WENO stencls are chosen. It s probable that some smoothly-connected data would be gnored by a mult-dmensonal WENO scheme. Whether ths would make a practcal derence n the reconstructon s far less clear. 4. DD-L 2 -ENO schemes have a bult-n mechansm to allow accuracy degradaton when nsucent smooth data exsts. Speccally, f too lttle smoothly-connected data exsts, the degree of the reconstructon s automatcally reduced to a level for whch there s sucent smooth data. WENO schemes, lke stencl-searchng ENO schemes, work from the premse that there wll be sucent smooth data f only t can be found. In cases where functon dscontnutes are very closely spaced, ths assumpton fals. WENO schemes can produce poor reconstructons here, whle DD-L 2 -ENO schemes wll produce a low-order accurate reconstructon usng only smoothly-connected (= relevant) data. 3.6 Summary The data-dependent least-squares approach can be used to produce functon reconstructons whch satsfy the ENO condton. The least-squares hertage of these DD-L 2 -ENO schemes allows them to be appled easly to functon reconstructon on unstructured meshes n multple dmensons. The algorthm can be summarzed as follows: Input: A computatonal mesh, structured or unstructured. For each control volume, the average value of a functon to be reconstructed. 3

14 Output: An ENO reconstructon of the functon, n the form of a moded Taylor seres expanson about each vertex, vald wthn the control volume surroundng the vertex. Preprocessng: For each control volume V, nd a sucently large set of nearby control volumes fv j g for reconstructon of the desred order. Generally, ths requres ndng rst or second neghbors for each vertex. Preprocessng: Compute all control volume moments that wll be needed n the least-squares problem of Equaton 2. For each control volume each tme a functon s reconstructed:. Compute geometrc weghts and construct the arrays needed for the data-ndependent least-squares problem of Equaton Solve the least-squares problem usng Householder transformatons. The soluton algorthm should not destroy the orgnal arrays and should return both the soluton and the resdual of the least-squares problem, along wth nformaton about how many dervatves were actually calculable. 3. Usng the resdual of the least-squares problem, compute a data-dependent weght for each control volume n the stencl and multply the approprate row of the orgnal least squares problem by ths weght. Keep track of how many hgh weghts there are, as ths lmts the number of dervatves whch can be plausbly calculated. 4. Re-solve the least-squares problem. The result gves the dervatve terms to be used n the reconstructon of Equaton 3. 4 Functon Reconstructon Because the new DD-L 2 -ENO scheme s desgned for use on unstructured meshes, results wll be compared both to ENO reconstructons and to a standard algorthm for unstructured meshes, least-squares reconstructon wth Venkatakrshnan's lmter [2]. Reconstructon accuracy for smooth functons Fgure?? shows the L error n the reconstructon of a snusod on a perodc doman usng four reconstructon technques: data-ndependent least squares, data-ndependent least squares wth Venkatakrshnan's lmter, RP-ENO, and DD-L 2 -ENO. In each case, the nomnal order of accuracy of the reconstructon s attaned, except for the lmted reconstructon, whch s only rst order because of the reducton of gradents near smooth extrema. 4

15 Solutons of the scalar wave equaton Solutons to the scalar wave equaton n one dmenson were computed usng a complex ntal condton. The scalar wave equaton s u t + u x = 0? < x < u(x; 0) = u 0 (x) Perodc BC where u 0 s a functon gven by Jang and Shu [7]: u 0 (x) = 8 >< >: G(x;z?)+4G(x;z)+G(x;z+)?0:8 x?0:6 6?0:4 x?0:2? j0(x? 0:)j 0 x 0:2 F (x;a?)+4f (x;a)+f (x;a+) 0:4 x 0:6 6 0 otherwse (25) (26) wth a = 0:5 G(x; z) q= exp(?(x? z) 2 ) z =?0:7 F (x; a) = max(0;? 2 (x? a) 2 ) = 0:005 = 0 = log (27) Fgure 2 shows the result of propagatng ths functon to t = 8 on a mesh wth 200 ponts usng fourth-order Runge-Kutta tme advance wth a CFL number of 0.4. Only thrd- and fourth-order soluton are shown; the second-order solutons are all hghly smeared. The thrdorder DD-L 2 -ENO scheme performs nearly dentcally to the thrd-order RP-ENO scheme; each s much better than the DI-L 2 result, whch contans sgncant overshoots. For fourthorder nomnal accuracy, RP-ENO s slghtly better that DD-L 2 -ENO for all four proles. Non-smooth functon reconstructon n two dmensons The new reconstructon scheme was tested for non-smooth functon reconstructon n two dmensons. The functon chosen s that of Abgrall [9]. wth u(x; y) = 8 < : f(r) = q cos y f(x? cot y) x 2 2 f(x + cot q 2 8 >< >: y) + cos (2y) x > cos y 2 3?r sn r? 3 jsn (2r)j jrj < 3 2r? + sn (3r) 6 r 3 2 r2 Contour plots of ths functon are shown n Fgure 3 Because the reconstructon scheme s k-exact, we know that the accuracy of the reconstructon n smooth regons of the functon Several typographcal errors n the denton of ths functon n [9] cause a msmatch wth the plotted functon theren; the functon shown here matches the plots n [9]. (28) (29) 5

16 wll be of order k+. Norms of the error n the reconstructon are meanngless n ths context, because the derence between the actual functon and ts reconstructon s guaranteed to be O () n control volumes that are crossed by a dscontnuty. Instead, for nomnal thrdand fourth-order accuracy respectvely, Fgures 4{5 show the control volumes for whch the nomnal order of accuracy was not obtaned. Traces of the dscontnutes can be clearly seen n both gures, wth control volumes showng degraded accuracy usually losng only one order (84% for thrd order, 59% for fourth). Also, full accuracy s generally obtaned on the boundary untl the fourth order, for whch some control volumes lack enough second neghbors for the reconstructon. No attempt has been made to further extend these stencls nto the nteror of the mesh to remedy ths problem. 5 Invscd ow solutons The reconstructon of Secton 3 can easly be extended to systems. The approach taken here s to reconstruct all components smultaneously. In ths case, the least squares problem of Equaton 2 has as many columns on the rght-hand sde as there are components n the system. The expresson for the resdual R s extended to be v u R = t C CX nx c= l=m+ r 2 l;c (30) where C s the number of columns. In the present work, prmtve varables ( u P ) T are used for reconstructon throughout, and densty s chosen to compute the smoothness ndcator. As n the scalar case, the least-squares problem s moded by multplcaton of each row by the data-dependent weght for that control volume, and a second least-squares problem s solved. Sod's problem The rst result presented s the soluton of Sod's shock tube problem n one dmenson. The ntal condtons are u P (x; 0) = 8 < : x < 0 x > 0 Fourth-order Runge-Kutta tme advance was used, wth a CFL number of 0.6, on a 400 pont mesh. Note that reconstructon for ths problem used prmtve varables, not characterstc varables. Densty at t = 2 s shown n Fgure 6. The data-dependent schemes provde better resoluton near the contact dscontnuty and the ends of the expanson fan whle keepng overshoots acceptably small. The contact dscontnuty becomes progressvely sharper wth ncreasng order of accuracy and accuracy near the head of the expanson mproves smlarly. (3) 6

17 Two-dmensonal nvscd arfol problems To demonstrate the soluton accuracy for two-dmensonal aerodynamcs problems, two nvscd ow results wll be shown. For both cases, the ow solver s a multgrd scheme usng three coarse meshes. Multstage tme advance s used, along wth local precondtonng [2, 3]. The CFL number s 0.8, and the multstage coecents are f0.532,.37, g. The rst test case s a hgh angle of attack ow around a NACA 002 arfol, at M = 0:302 and = 9:86 o. Fgure 7 shows the ne mesh used for ths case, whch contans 3323 vertces. The soluton was computed by usng unlmted DI-L 2, lmted DI-L 2 (Venkatakrshnan's lmter [2]), and unlmted second-order DD-L 2 -ENO. The solutons are vrtually dentcal except near the sucton peak on the upper surface. A detal of the surface pressure coecent n ths regon s shown n Fgure 8; the gure also ncludes a soluton from INS2D to whch the Karman-Tsen pressure correcton has been appled. The two unlmted reconstructon schemes gve dentcal results to plottng accuracy; the lmted scheme comes slghtly closer to capturng the sucton peak. The lmted resoluton near the leadng edge prevents any of the three solutons from matchng the sucton peak computed by INS2D on a ne mesh. AGARD test case [4] was used to test the new scheme for ows wth shock waves. Ths case computes ow around a NACA 002 arfol at a Mach number of 0.8 and an angle of attack of.25 o. A mesh wth 456 vertces was used (see Fgure 9). The soluton was computed by usng both lmted DI-L 2 reconstructon and second-order DD-L 2 -ENO reconstructon. The surface pressure coecents for these two solutons and for the accepted AGARD soluton [4] are gven n Fgure 0. An nset shows a close-up of the upper-surface shock, whch s shfted by one mesh vertex between the two solutons. The qualty of the solutons usng lmted DI-L 2 reconstructon and unlmted DD-L 2 -ENO reconstructon are comparable. 6 Conclusons A new method for functon reconstructon has been descrbed that combnes the best features of least-squares and ENO reconstructons. Ths new scheme, DD-L 2 -ENO, uses a datadependent weghtng n least-squares reconstructon to satsfy the ENO condton of Lu, Osher, and Chan [6]. Lke other ENO schemes, DD-L 2 -ENO s unformly accurate, even n the presence of dscontnutes, and allows only asymptotcally small overshoots. Because DD-L 2 -ENO s a least-squares reconstructon procedure, exstng ecent algorthms for least-squares reconstructon can be easly moded for DD-L 2 -ENO. An algorthm for computng DD-L 2 -ENO reconstructons of arbtrary order on unstructured meshes s gven. The data-dependent weghts used n the scheme vary contnuously wth the data and consequently good convergence behavor s antcpated for steady-state calculatons. The stencl for DD-L 2 -ENO s larger than that requred by conventonal ENO schemes, but no larger than requred by k-exact least-squares reconstructons of the same order of accuracy. The asymptotc behavor of the scheme n reconstructng smooth and pecewse smooth functons has been demonstrated. DD-L 2 -ENO produces unformly hgh-order accurate reconstructons, even n the presence of dscontnutes. The new scheme behaves comparably to RP-ENO n solvng the scalar wave equaton n one dmenson. 7

18 A feature of the new scheme s ts ablty to detect and control stuatons where nsucent smooth data exsts for the desred order of reconstructon. In such cases, the scheme automatcally determnes the amount of smooth data present and computes a reconstructon of approprately reduced accuracy. Ths capablty was demonstrated by reconstructng a non-smooth functon; control volumes for whch nsucent smooth data exsted were shown to have a lower-order reconstructon computed nstead. Extenson of the reconstructon scheme to systems was dscussed. As examples of the behavor of such reconstructons, solutons were shown for the Sod shock tube problem n one dmenson and for two aerodynamc problems n two dmensons. These early twodmensonal results are very encouragng. In all cases, soluton accuracy was comparable to exstng schemes sutable for unstructured meshes and overshoots n the soluton were of acceptable magntude. Acknowledgments The author would lke to thank Tm Barth for numerous helpful dscussons on k-exact reconstructon, Mchael Aftosms for readng and commentng on an early draft of ths paper, and the revewers for helpful comments whch ponted out several weaknesses n the exposton. Ths work was supported n part by the Mathematcal, Informaton, and Computatonal Scences Dvson subprogram of the Oce of Computatonal and Technology Research, U.S. Department of Energy, under Contract W-3-09-Eng-38; and n part by the Natonal Research Councl whle the author was a Research Assocate at NASA Ames Research Center. References [] Barth, T. J. and Frederckson, P. O., \Hgher Order Soluton of the Euler Equatons on Unstructured Grds Usng Quadratc Reconstructon." AIAA paper , Jan [2] Venkatakrshnan, V., \On the Accuracy of Lmters and Convergence to Steady-State Solutons." AIAA paper , Jan [3] Harten, A. and Osher, S., \Unformly Hgh-Order Accurate Nonoscllatory Schemes," SIAM Journal on Numercal Analyss, vol. 24, pp. 279{309, Apr [4] Harten, A., Osher, S., Engqust, B., and Chakravarthy, S. R., \Some Results on Unformly Hgh-Order Accurate Essentally Non-Oscllatory Schemes," Appled Numercal Mathematcs, vol. 2, pp. 347{377, 986. [5] Harten, A., Enqust, B., Osher, S., and Chakravarthy, S. R., \Unformly Hgh Order Accurate Essentally Non-oscllatory Schemes, III," Journal of Computatonal Physcs, vol. 7, pp. 23{303, Aug [6] Lu, X.-D., Osher, S., and Chan, T., \Weghted Essentally Non-oscllatory Schemes," Journal of Computatonal Physcs, vol. 5, pp. 200{22,

19 [7] Jang, G.-S. and Shu, C.-W., \Ecent Implementaton of Weghted ENO Schemes," Journal of Computatonal Physcs, vol. 26, pp. 202{228, June 996. [8] Durlofsky, L. J., Enqust, B., and Osher, S., \Trangle Based Adaptve Stencls for the Soluton of Hyperbolc Conservaton Laws," Journal of Computatonal Physcs, vol. 98, pp. 64{73, Jan [9] Abgrall, R., \On Essentally Non-oscllatory Schemes on Unstructured Meshes: Analyss and Implementaton," Journal of Computatonal Physcs, vol. 4, no., pp. 45{58, 994. [0] Barth, T. J., \Recent Developments n Hgh Order K-Exact Reconstructon on Unstructured Meshes." AIAA paper , Jan [] Golub, G. H. and van Loan, C. F., Matrx Computatons. Baltmore, Maryland: The Johns Hopkns Unversty Press, 983. [2] Ollver-Gooch, C. F., \Multgrd Acceleraton of an Upwnd Euler Solver on Unstructured Meshes," AIAA Journal, vol. 33, pp. 822{827, Oct [3] Ollver-Gooch, C. F., \Towards Problem-Independent Multgrd Convergence Rates for Unstructured Mesh Methods I: Invscd and Lamnar Vscous Flows," n Proceedngs of the Sxth Internatonal Symposum on Computatonal Flud Dynamcs, Japan Socety of Computatonal Flud Dynamcs, Unversty of Calforna at Davs, Sept [4] AGARD Flud Dynamcs Panel, Test Cases for Invscd Flow Feld Methods. AGARD, May 985. AGARD Advsory Report AR-2. 9

20 L reconstructon error DI-L2-2 Lmted DI-L2-2 RP-ENO-2 DD-L2-ENO N Second Order DI-L2-3 RP-ENO-3 DD-L2-ENO-3 L reconstructon error N Thrd Order L reconstructon error DI-L2-4 RP-ENO-4 DD-L2-ENO N Fourth Order Fgure : Reconstructon of Smooth Functon 20

21 u DI-L2-3 RP-ENO-3 DD-L2-ENO x Thrd Order u DI-L2-4 RP-ENO-4 DD-L2-ENO x Fourth Order Fgure 2: Lnear Wave Propagaton 2

22 Fgure 3: Contours of Functon Dened by Equaton

23 .0 Frst order Second order 0.5 y x Fgure 4: Control Volumes Whch Fal to Attan Thrd-Order Accuracy (k = 2) 23

24 .0 Frst order Second order Thrd order 0.5 y x Fgure 5: Control Volumes Whch Fal to Attan Fourth-Order Accuracy (k = 3) 24

25 Exact Lmted DI-L2 DD-L2-ENO-2 DD-L2-ENO-3 DD-L2-ENO-4 Densty x Fgure 6: Densty plots for Sod's shock tube problem 25

26 Fgure 7: Mesh for Hgh Case Vertces. 26

27 Surface pressure coeffcent Unlmted DIL2 Lmted DIL2 Unlmted DDL2 Corrected INS2D soluton x/c Fgure 8: Surface C p near Sucton Peak 27

28 Fgure 9: Mesh for AGARD Test Case. 456 Vertces 28

29 AGARD 2 DDL2 Lmted DIL Fgure 0: Surface Pressure Coecent for AGARD Test Case 29