Legendre-Tau Spectral Elements fr Incmpressible avier-stkes Flw Kelly Black* Abstract A spectral multi-dmain methd is intrduced and examined. After dividing the cmputatinal dmain int nnverlapping subdmains a Legendre-Tau apprximatin is cnstructed within each subdmain. Unlike the standard Legendre-Tau apprximatin a variatinal apprximatin is cnstructed and the result is that nly simple cntinuity is required at the interfaces between the subdmains. The methd is intrduced fr a simple 1D Helmhltz equatin and tw examples are given: a 1D Burger's Equatin with a small viscsity and avier-stkes incmpressible flw ver a backstep. Key wrds: spectral elements, Helmhltz equatin, Burger's equatin, avier-stkes equatin. AMS subject.classificatins: 6530, 6535, 76D05. I Intrductin A spectral multi-dmain apprximatin using a Lanczs- Tau apprximatin [4, p. 79] is examined and is implemented by enfrcing a variatinal apprximatin. The methd is cnstructed s as t cnstruct a lcal spectral basis within each subdmain [3]. The methd is first intrduced fr a simple Helmhltz equatin, a 1D Burger's equatin with a small viscsity, and finally the avier- Stkes incmpressible flw ver a backstep is examined. Anther technique which yields an apprximatin in the Furier dmain has been prpsed by Israeli, et al in [8], hwever the apprach prpsed here has mre in cmmn with the cllcatin methd prpsed by Patera [11]. A cllcatin methd using the same test functins fr the 'Department f Mathematics, University f ew Hampshire, blackcvidalia.unh.edu ICOSAHOM'95: Prceedings f the Third Internatinal Cnference n Spectral and High Order Methds. ( )1996 Hustn Jurnal f Mathematics, University f Hustn. variatinal frm that is examined here has been prpsed in [2]. Because a variatinal apprach is emplyed the methd has the advantage that the interface can be easily calculated by requiring nly C O cntinuity. Unlike ther variatinal appraches an apprximatin is fund in the Spectral dmain. This apprach easily takes advantage f the accuracy f the spectral multi-dmain methds as well as an advantage in the rbustness f the Galerkin schemes. The numerical schemes generated frm such an apprach can easily accmmdate cmplicated bundary cnditins that depend n the spectrum. One disadvantage is that any nnlinear terms are mre expensive t cmpute xvhen cmpared t the cllcatin apprach. Because this apprach is designed fr use with multiple cmputatinal subdmains the size f the apprximating space n each subdmain is kept small. and the expense f calculating the nnlinear terms can be kept lw. Because the scheme is essentially a Legendre-Tau methd, the system f equatins is cnstructed in the same manner as is dne with Galerkin methds. Since the apprximating functins d nt necessarily satism,the bundary cnditins the bundaries are directly enfrced. This is dne either by enfrcing the bundaries at specific grid pints r by a minimizatin f the difference between the true bundaries and the apprximatin. 2 Multi-dmain tau methd T take advantage f the high accuracy and relatively curse discretizatins ffered by spectral methds and avid restrictins placed n the cmputatinal dmain, the dmain is subdivided int nn-verlapping subdmains. On each subdmain an apprximatin is fund that is a linear cmbinatin f the Legendre plynmials up t a given degree. T present the methd withut the burden f t many technical details, first a simple 1D Helmhltz equatin is exanfined, uxx + Au = f, x (0,1). u(0) = u(1) = 0. 255
256 Kelly Black The apprximatin is cnstructed by integrating against a sequence f test functins and building a system f algebraic equatins. In the standard implementatin f the Tau methd the test functins are chsen t be the same as the trim functins. In such a case bth the test func- tins and the trim functins are fund frm the sequence f Legendre plynmials. Since a multi-dmain apprxi- matin is sught the test functins, j(x), are defined a different xvay, (2) ½j(x) = (1- x2)lj(x), l+x Viv-(x) = 2 ' 1--x ½(X) : 2 O_<j<-1, As defined the test functins are zer n the endpints fr j = 0...-2. Fr j = - 1 andj = the test functins are linear plynmials, and the span f all f the test functins is I I I L R = L R*' Figure 1: A Multi-Dmain Example in 1-D fr subdmains 1 and r. The cmputatinal dmain is t be divided int M nnverlapping subdmains. Fr a given subdmain, r, the apprximatin is written as u'/v(x ) 6 P2v and the left and right endpints are L" and R, respectively (see Figure 1). On each subdmain the dmain is mapped t the unit square [-1,1] using a simple linear transfrmatin: (3) X-- L r The apprximatin n subdmain r is written as a linear cmbinatin f the Legendre plynmials and has supprt nly n subdmain r: (4) u* (x) --{ Y] i5øct}'li( r)' 0 therwise. x [Lr, r] In this example x is in the cmputatinal dmain defined in equatin (1) and is fund frm equatin (3). Fr 0 _< j <_, the test functins fr subdmain r are define, therwise. x L t R t = L,' R,- Figure 2: Trial functins _ (x) and ¾' ;v(x) cmbine n adjacent subdmains t assemble an "hat" functin. With these definitins the variatinal apprximatin can be examined. Except fr the linear functins, the test functins are zer n the subdmain bundaries. and each test functin has supprt n nly ne subdmain. When integrating against the apprximatin the nly integral that need be fund is that part within the individual subd- nain. The linear test functins, hwever, d nt have zer bundaries n each subdmain. If the linear functins n adjacent subdmains are examined the result is a simple hat functin (see Figure 2). This cmpsite test functin is used t insure that the flux is balanced acrss the sub- dmain interface. By splitting the trial functins int the plynmials that are zer at the bundaries and thse that are nt, the methd is a p-versin finite element methd. Unlike mre cnventinal p-versin schemes, by cnstructing the apprximatin as a linear cmbinatin f the Legendre plynmials it becmes quite easy t increase the rder f the apprximatin xvithin each subdmain (p-refinement). Mrever, the resulting matrices share a similar structure t thse fund in the cnventinal single subdmain Legendre-Tau methd[4]. Fr example, within each subdman the entries fr the stiffness natrix crespnding t trial functins ;vhich are zer at the bundaries represesent a sparse upper-triangular matrix. Once the apprximatin and the test functins are de- fined n the tw subdmains the variatinal apprximatin f equatin (1) is cnstructed. Fr j = 0,..., - 2 the fllwing equatins are enfrced n each subdmain: Assuming that subdmain I is the subdmain t the left and adjacent t subdmain r, the equatins fr the linear test functins are cnstructed by integrating against the
Legendre- Tau Spectral Elements 257 hat functin: R* d l First, the sum f Legendre plynmials is substituted fr the apprximatin U v (x). Fr j = 0... - 2 the result is (10) R R r The bundary cnditins are enfrced as they are dne in the standard Tau techniques. Assuming that subdmain 0 is the far left subdmain and M is the subdmain n the far right, the bundaries are directly enfrced in a pintwise manner, (s) tt = (-1)' =, = = = 0. u:¾(1) '¾ The subdmain interfaces are enfrced by simply requiring C ø cntinuity: (9) U =0 I The slutin t the system f equatins in equatins (6) thrugh (9) is the apprximatin t equatin (1). 2.1 The stiffness matrix Equatins (6) thrugh (9) are used t cnstruct an apprximatin t equatin (1). By substituting U v(x ) frm equatin (4) the stiffness and the mass matrices can be cn- structed. Here we will cncentrate n the secnd derivative peratr and find the entries fr the stiffness matrix. The entries fr the stiffness matrix are derived and the mass matrix can be fund in a similar prcess. The stiffness matrix is fund by examining the variarihal frm fr the secnd derivative. After a substitutin t allw fr integratin acrss [-1, 1] the variatinal frm f the secn derivative in equatin (6) is derived fr ;=0...-2: i=o ' i----o ai R _--L 1 d ((1 - The system f equatins can be cnstructed thrugh the use f a stiffness matrix, $, by setting ($. ')ji - R - L' 1 L i(x) ((1- x2)lj(x)) dx (11) = R _L fro_<j<-1, 2( 4% + 2j(1-2j+l j) 6ji) ' ando<i<%,isgivenbv (12) eji = 01 therwise. i+jeven, i_>j+2 This yields a sparse upper-triangular matrix and is similar t the result fr the standard single dmain tau methd [4]. The equatins fr the interface are fund by integrating against the hat functin ver the tw subdmains as given in equatin (7), (13) R d l n d d -2 R" - L ai R l _ L t + i----1 i=1 i dd i dd % R, _ L, The entries fr the stiffness matrix are fund by enfrcing the variatinal frm f the secnd derivative as well as the bundaries. The mass matrix, A42v, can be cnstructed using the same apprach.
258 Kelly Black Burger's equatin with a small viscsity An example f the discretizatin fr Burger's equatin with a small viscsity is examined, ut q- uu - unix, x (-1, 1), t>0, (+l,t) = 0, t>0, u(x,o) = -sin(7rx). This equatin develps a steep gradient arund x = 0 and has been examined in Basdevant, et al [1]. The tempral discretizatins emplyed here clsely parallel thse fund in [1]. The previusly described spatial discretizatin is emplyed in the apprximatin t equatin (14). The tempral discretizatin is cnstructed frm a finite difference apprximatin and emplys an Adams-Bashfrth/Crank- ichlsn Scheme (ABC) [1]. The apprximatin at the n th time step is dented tt v. The cnvective term is apprximated using the explicit Adams-Bashfrth discretizatin and the diffusive term is apprximated using the implicit Crank-ichlsn discretizatin, (14) U At n n 1 ^ --1 and the resulting natrix equatins are given by (15) " = u (u )x, The true slutin that is used fr reference is apprximated frm the cnvlutin prduct given in Cle's transfrmatin[5]. An apprximatin f the true slutin ;vas calculated using Gauss-Hermite integratin with 9 digits f accuracy [6, 12]. (Fr v = i5-67 the gradient achieves its maximum near t = 0.5 [1].) The L errrs fr these values are given in Figures 3 thrugh 5. Fr this test case the L 2 errrs are presented fr the times t= 1/, t= 2/. and t= 3/7r [1]. Fr each f the three trials the errr reprted is nt the percentage errr. In the test case a steep gradient ccurs arund x = 0 and nce this gradient is reslved the tw multi-dmain methds ffer a mre accurate apprximatin. Figures 3 thrugh 5 demnstrate that the multi-dmain techniques can yield a mre accurate apprximatin when cmpared t a single dmain apprximatin. Because the steep gradient ccurs arund a subdmain interface the tw multi-dmain techniques are better able t reslve the gradient. The apprximatins at the times 2,/7r and 3/7r demnstrates the rbustness f the Tan apprximatins. When the gradient is nt adequately reslved the cllcatin scheme actually diverges while bth Tau methds maintain their stability. The multi-dmain Tau methd maintains the advantages f bth the Tan methd and the multi-dmain methd. 4 avier-stkes flw incmpressible The incmpressible avier-stkes flw equatin. (16) ut + (u' V) + Vp = i V2u, Re subject t V. = O. with n slip bundaries are examined [7]. The gexnetries examined are fr flw within a driven cavity as well as flw ver a backstep. The spatial discretizatin mnplyed is the The nnlinear terms can be calculated as a cnvlutin same as examined in sectin 2. The tempral discretizatin sum [4. p. 82] r using cllcatin n the abscissa km the Gauss-Lbatt quadrature as was dne here [4, p. 83]. A cmparisn between three different methds is examined. The first is a single dmain Chebychev-Tau scheme is based n the the splitting methd [10] and the methds prpsed by Karniadakis, et al [9]. The splitting methd is a cnvenient scheme t separate the actins f the tw spatial peratrs acting n the velcity, [4. p. 80], a Chebychev-Galerkin-Cllcatin h n [11], 1 and a multi-dmain Legendre-Tan scheme. (A cmparisn (17) = 5 (½. + v. between a finite difference apprximatin and the spectral techniques can be fund in [1].) Fr the tw spectral 1 Van. œ(u) = Re element apprximatins fur equally spaced subdmains 1 are implemented. In the examples the values v - 00=, (The implementatin emplys the skew-symmetric frm f 1 At = 2 are emplyed. the nnlinear peratr).
Legendre-Tau Spectral Elements 259 Apprximatin t Burger's Equatin - L 2 Errrs 10-2 v = 1/(100 ), at = 1/(200 ) Time = 1/ 0 0 0 a 10 '3 0 0 10-4 B Tau 0 Galerkin 0 Petrv-Tau 0 0 <3 6 50 70 90 110 130 150 Degrees f Freedm Figure 3: The L 2 errrs fr the apprximatin t Burger's Equatin with a small viscsity at the time Fllwing the methd prpsed by Karniadakis, et al [9], the pressure is nt calculated, rather the time averaged pressure is apprximated. The three relavent spatial peratrs can then be islated in three separate steps, (is) - = -[t' + jv(u) dt, dtn = -X7i0, subject t K7. =0, = œ(u) dt. In the first time step the nnlinear term is integrated thrugh the use f an explicit methd such as thse frm the Adams-Bashfrth family f schemes while the third step emplys an implicit step such as thse fund in the Adams-Multn family f schemes. Because an explicit step is taken there is a restrictin n the time step. Hwever, the mre stringent restrictin n the time step cmes frm the Stkes peratr. This is mitigated thrugh the use f the implicit step in the final equatin. Fr the 2D equatins the bth the apprximating and trial functins are taken as tensr prducts f thse fund in the 1D case. Within each subdmain an apprximatin is cnstructed which is a linear cmbinatin f the Legendre plynmials, fr (,!/) in subdmain r, (19) u (x, y) = E E øzi Li( )Lj( r)' j=0 i=0 The test functins are als fund as a simple tensr prduct, Cntinuity acrss the submain interfaces are enfrced by minimizing the difference between the apprximatins n adjacent subdmains. Fr example, if subdmain r is t the right f subdmain 1 then n subdmain r the bundary ---- -1 is adjacent t the bundary n subdmain 1
260 Kelly Black Apprximatin t Burger's Equatin - L Errrs v = 1/(100r ), 5t = 1/(200r 0 Time = 2/r 10 '2 10 '3 ø øøo /x /x /x 10-4 Tau /x Galerkin Petrv-Tau O O OO O 50 70 90 110 130 150 Degrees f Freedm Figure 4: The L 2 errrs fr the apprximatin t Burger's Equatin with a small viscsity at the time t -- - 2 w' when. : 1. The cntinuity acrss this interface is en- and, frced bv requiring that the difference between the tw apprximatins be rthgnal t the space f plynmials (22) f degree - 2, 1 (20) (u%(1, y)- u (-1, y)) Lj(y)dy --- 0, j=0...-2. Cntinuity is ensured with the final requirement that the subdmains be cntinuus at the crners which is directly enfrced as it is dne with cllcatin type methds, (21) l U(1,1) = u (-1,1), =0 j=o i=0 j=0 4.1 Flw ver a Backstep In the secnd trial, the avier-stkes incmpressible flw ver a backstep, the dmain is divided int 30 subdmains (see Figure 7). On each subdmain the apprximatin utilizes a plynmial f degree 6 in bth the x and the y-directins. The initial cnditin is zer velcity with a time step f At = 10-3. The height f the backstep is 1 and the maximum velcity at the inlet is 1. The implicit step that is taken in equatin (18) is inverted thrugh the use f the GMRES methd [13, 14].
Legendre- Tau Spectral Elements 261 Apprximatin t Burger's Equatin - L 2 Errrs v= 1/(100 ), 8t = 1/(200 ) Time = 3/ 10' 10 '2 10 '3 0 0 0000 A 0 0 A A 0 10 '4 r Tau 0 A Galerkin A 0 Petrv-Tau 0 0 A 0 0 50 70 90 110 130 150 Degrees f Freedm Figure 5: The L 2 errrs fr the apprximatin t Burger's Equatin with a small viscsity at the time t - 71' 3. In this example tw different Reynlds numbers are ex- amined, Re= t and Re=4-0 and the velcity fields are 1 shwn in Figure 8. Fr the situatin fr Re-2-- the ini- tial velcity was taken t be zer and the velcity field shwn was fund after 6000 time steps. Fr the situatin 1 fr Re-4- - the initial cnditin emplyed was the velcity field btained in the previus situatin. The velcity field shwn was fund after 2300 time steps. Figures 8 and 9 shw the velcity fields fr bth trials. The first figure, Figure 8, demnstrates the velcity field fr the area arund the inlet and the backstep. The secnd figure, Figure 9, is a mre detailed view f the area directly behind the backstep itself and shws the area f recirculatin. References [1] Basdevant, C., M. Deville, P. Haldenwang, J.M. Lacrix, J. Ouazzani, R. Peyret, P. Orlandi and A.T. Patera, Spectral and Finite Difference Slutins f the Burger's Equatin, Cmputers and Fluids, 14 (1986), pp. 23-41. [2] Black, K., A Petrv-Galerkin Spectral Element Technique fr Hetergeneus Prus Media Flw, Internatinal Jurnal f Cmputers in Mathematics and Applicatins, 29 (1995), pp. 49-65. [3] Black, K., Legendre-Tau Spectral Elements fr Incmpressible avier-stkes Flw, 1995. Accepted fr the prceedings f the ICOSAHOM 95 Cnference.
262 Kelly Black True Slutin t Burger's Equatin With a Viscsity v= 1/(100 ) 0.5 > 0.0-0.5 t=l t=2/,-: t=3/,.0-0.5 0.0 0.5 1.0 Figure 6: The true slutin t Burger's Equatin t =, t = -}, and t = _3. Canut, C., Hussaini, M.Y., Quarterni, A., and Zang, T.A., Spectral Methds in Fluid Dynamics, Springer-Verlag, ew Yrk, 1988. [5] Cle, Julian D., On a Quasi-Linear Parablic Equatin Occuring in Aerdynamics, Quarterly f Applied Mathematics, IX (1951), pp. 225-236. [6] Davis, Philip J. and Rabinwitz, Philip, Methds f umerical Integratin, Academic Press, inc. (Harcurt Brace Jvanvich), ew Yrk, secnd ed., 1984. [71 Gresh, Philip M., and Rbert L. Sani, On Pressure Bundary Cnditins fr the Incmpressible avier- Stkes Equatins, Internatinal Jurnal fr mnerical Methds in Fluids, 7 (1987), pp. 1111-1145. [8] Israeli, M., L. Vzvi and A. Averbach, Spectral Multidmain Technique with Lcal Furier Basis, Jurnal f Scientific Cmputing, 8 (1993), pp. 135-149. [9] Karniadakis, G., M. Israeli, and S. Orszag, High- Order Splitting Methds fr the Incmpressible amer-stkes Equatins, Jurnal f Cmputatinal Physics, 97 (1991), pp. 414-443. [10] Maday, Y., A. Patera, and E. Rnquist, An Operatr- Integratin-Factr Splitting Methd fr Time-Dependant Prblems: Applicatin t Incmpressible Flw, Jurnal f Scientific Cmputing, 5 (1990), pp. 263-292. [11] Patera, Anthny T., A Spectral Element Methd fr Fluid Dynamics: Laminar Flw in a Channel Expansin, Jurnal f Cmputatinal Physics, 54 (1984), pp. 468-488.
Legendre- Tau Spectral Elements 263 Figure 7: Graph f the gemetry fr the backstep. The backstep has height 1 and the maximum velcity at the inlet is 1. The dmain is divided int 30 subdmains and a spectral apprximatin is cnstructed within each subdmain. Within each subdmain a Legendre plynmial apprximatin is emplyed with the degree f the plynmial 6 in the.r-directin and 6 in the y-directin. [12] Press, Wiliiam, S. Teuklsky, W. Vetterling, and B. Flannery, umerical Recipes in C, Cambridge University Press, ew Yrk, secnd ed., 1992.!13] Saad, YuceL and Martin Schultz, GMRES: A Generalized Minimal Residual Algrithm fr Slving nsymmetric Linear Systems, Siam Jurnal f Scientific and Statistical Cmputing, 7 (1986), pp. 856-869. Walker, Hmer F., Implementatin f the GMRES Methd Using Husehlder Transfrmatins, Siam Jurnal f Scientific and Statistical Cmputing, 9 (1988), pp. 152-163.
264 Kelly Black Re=200 ß Re=400 Figure 8: Vectr plt fr the inlets fr tw simulatins. The tp si nulatin is frm the test case Re=200 and the bttm simulatin is frm the test case Re;400.
Legendreø Ta u Spectral Elements 265 Re=200 Re=400 Figure 9: Vectr plt fcusing n the regin behind the backstep fr tw simulatins. The tp simulatin is frm the test case Re-200 and the bttm simulatin is frm the test case Re=400.
266 Kelly Black