of Hydrodynamic Stability Problems* Department of Computer Science, University oftennessee, Knoxville, Tennessee

Transcription

1 Chebyshev tau - QZ lgorithm Methods for Calculating Spectra of Hydrodynamic Stability Problems* J.J. Dongarra è1è, B. Straughan è2è ** & D.W. Walker è2è è1è Department of Computer Science, University oftennessee, Knoxville, Tennessee , U.S.., and Mathematical Sciences Section, Computer Science and Mathematics Division, Oak Ridge ational Laboratory, P.O. Box 2008, Oak Ridge, Tennessee , U.S.. è2è Mathematical Sciences Section, Computer Science and Mathematics Division, Oak Ridge ational Laboratory, P.O. Box 2008, Oak Ridge, Tennessee , U.S.. bstract The Chebyshev tau method is examined in detail for a variety of eigenvalue problems arising in hydrodynamic stability studies, particularly those of Orr-Sommerfeld type. We concentrate on determining the whole of the top end of the spectrum in parameter ranges beyond those often explored. The method employing a Chebyshev representation of the fourth derivative operator, D 4 ; is compared with those involving the second and ærst derivative operators, D 2 ;D, respectively. The latter two representations require use of the QZ algorithm in the resolution of the singular generalised matrix eigenvalue problem which arises. Physical problems explored are those of Poiseuille æow, Couette æow, pressure gradient driven circular pipe æow, and Couette and Poiseuille problems for two viscous, immiscible æuids, one overlying the other. Keywords: Eigenvalue problems, Orr-Sommerfeld equations, Multi-layer æows, Chebyshev polynomials, QZ algorithm. * This work was supported in part by the Oæce of Scientiæc Computing, U.S. Department of Energy under contract DE-C05-84OR21400 with Lockheed Martin Energy Systems, and by the dvanced Research Projects gency under contract DL03-91-C-0047, administered by the rmy Research Oæce. ** Permanent address: Department of Mathematics, The University, Glasgow, G12 8QW, U.K.

2 Chebyshev tau - QZ algorithm methods Page 1 1. Introduction There has been much recent attention directed at solving diæcult eigenvalue problems for diæerential equations like the Orr - Sommerfeld one, with particular interest in the removal of spurious eigenvalues or calculations in high Reynolds number ranges, cf. bdullah & Lindsay ë1ë, Davey ë7ë, Fearn ë13ë, Gardner et al. ë14ë, Huang & Sloan ë17ë, Lindsay & Ogden ë18ë, McFadden et al. ë19ë, Orszag ë22ë, and Zebib ë30ë. Equations of Orr-Sommerfeld type govern the stability of shear and related æows which have important application in many æelds. One such æeld is climate modelling with questions like determining an explanation for the origin of the mid-latitude cyclone which in turn is responsible for producing the high and low pressure regions from which variable weather patterns arise. nother application is to shear æows in electrohydrodynamic èehdè systems which have industrial relevance in the invention of devices employing the electroviscous eæect or those utilizing charge entrainment, such as EHD clutch development, or EHD high voltage generators. Yet other important mundane applications include the prediction of landslides, and æow over an aeroplane wing covered in de-icer. These topics will form part of future research. The goal of this paper is to describe how to implement in an eæcient way a Chebyshev tau - QZ algorithm method for ænding eigenvalues and eigenfunctions in diæcult but practical problems which occur in hydrodynamical contexts. We employ a technique which systematically writes the diæerential equations which occur as systems of second order equations in order to utilise the growth properties of the Chebyshev matrices which arise. One could argue that there are several other methods of ænding eigenvaluesèeigenfunctions and this is true. Indeed, we mention ænite diæerence discretization coupled with a matrix technique such as the QR algorithm. We believe the method advocated here is, in general, more accurate and eæcient. Other options are to employ a ænite diæerence technique followed by inverse Rayleigh iteration, or use of compound matrices; the latter is discussed in e.g. Drazin & Reid ë11ë, Straughan & Walker ë27ë. These are two viable options if one is primarily interested in only one eigenvalue. In certain hydrodynamic stability problems one may beinterested in only one eigenvalue, namely the èdominantè one which is likely to contribute the most destabilizing mode in a linear instability analysis. If one can show a priori that a particular eigenvalue is the dominant one then a method which tracks a single eigenvalue is useful. However, for stability problems where the æuid layer is sheared it is usually not possible to show one eigenvalue is dominant. In addition, recent work has demonstrated the necessity to calculate many eigenvalues, see e.g. Butler & Farrell ë3ë, Reddy & Henningson ë23ë. The Chebyshev tau - QZ algorithm method we describe ænds all the eigenvalues è eigenfunctions we require in a very eæcient manner. Finally, one

3 Chebyshev tau - QZ algorithm methods Page 2 might wish to use a pseudospectral ècollocationè technique, see e.g. Huang & Sloan ë17ë. gain, this is a very viable alternative. However, we believe the Chebyshev tau technique is easier to implement for problems in a cylindrical or spherical geometry where there are terms like èm=rèd=dr present. lso, the growth properties of the Chebyshev tau method are likely to be better. We describe below three versions of the Chebyshev tau method which we refer to as D 4 ;D 2 and D methods due to the order of the highest derivatives we discretize. It will be seen that we basically recommend the D 2 alternative because it has better growth properties than the D 4 one, whereas it makes more eæcient use of the QZ algorithm than the D method which requires matrices which are twice as large. In order to introduce some of the notation used in the æuid dynamics literature and also to describe the Chebyshev tau technique we consider a very elementary, but illuminating, example. Consider the equation and boundary conditions, Lu ç u 00 + çu =0; uè,1è = uè1è = 0; where the diæerential operator L is deæned as indicated. x2è,1;1è; ow write u as a ænite series of Chebyshev polynomials u = +2 k=0 u k T k èxè; è1:1è è1:2è although the underlying logic is that è1.2è represent truncations of an inænite series. Due to the truncation, the tau method argues that rather than solving è1.1è one instead solves the equation Lu = ç 1 T +1 + ç 2 T +2 ; è1:3è where ç 1 ;ç 2 are tau coeæcients whichmaybe used to measure the error associated with the truncation of è1.1è. In an ordinary diæerential equation setting as opposed to an eigenvalue background, explicit use of the tau coeæcnets as error bounds may be found in Fox ë13ë. To reduce è1.1è to a ænite-dimensional problem the inner product with T i is taken of è1.3è in the weighted L 2 è,1; 1è space with inner product èf;gè= Z 1,1 fg p 1, x 2 dx ; and associated norm kæk:the Chebyshev polynomials are orthogonal in this space, and then from è1.3è we obtain è + 1è equations èlu; T i è=0 i=0;1;:::;: è1:4è

4 Chebyshev tau - QZ algorithm methods Page 3 There are two further conditions which arise from è1.3è, èlu; T +j è=ç j kt +j k 2 ; j=1;2; and these may eæectively be used to calculate the ç's. The two remaining conditions are found from the boundary conditions, which since T n èæ1è = èæ1è n ; yield +2 n=0 è,1è n u n =0; +2 n=0 u n =0: è1:5è Equations è1.4è and è1.5è yield a system of è + 3è equations for the è + 3è unknowns u i ;i=0;:::; +2: The derivative of a Chebyshev polynomial is a linear combination of lower order Chebyshev polynomials, in fact n,1 Tn 0 =2n T k ; k=1 n,1 Tn 0 =2n T k + nt 0 ; k=2 n even; n odd: è1:6è Then è1.4è become where the coeæcients u è2è i u è2è i + çu i =0; i=0;:::;; è1:7è are given by u è2è i = 1 c i p= +2 p=i+2 p+i even pèp 2, i 2 èu p ; è1:8è with the numbers c i being deæned by c 0 =2;c i =1;i=1;2;::: : èctually, è1.8è is really a truncation to the + 2nd polynomial of an inænite expansion.è Equations è1.7è and è1.5è represent a matrix equation x =,çbx; è1:9è with x =èu 0 ;:::;u +2 è T : However, the B matrix is inevitably singular due to the way the boundary condition rows are added to : Indeed, the last two lines of B are composed of zeros, while the upper left è +1èæè+ 1è part is simply the identity.

5 Chebyshev tau - QZ algorithm methods Page 4 To clarify this point, we observe +2 u 0 = u s Tsèxè 0 s=0 +2 = = ç +2 u s s=0 r=0 +2ç +2 r=0 s=0 D rs T r ç D rs u s çt r and so we may make the identiæcation +2 u è1è r = D rs u s : Similarly, and, therefore, u 00 = +2 r=0 +2 u è2è r = s=0 +2 = s=0 +2 = s=0 s=0 ç +2 s=0 D rs u è1è s +2 D rs +2 k=0 ç D rs u è1è s T r k=0 D sk u k D rs D sk u k : This allows us to introduce the diæerentiation matrix D; and second diæerentiation matrix D 2 which are shown to have components D 0;2j,1 =2j,1; jç1; D i;i+2j,1 =2èi+2j,1è; i ç 1;j ç1; D 2 0;2j = 1 2 è2jè3 ; j ç 1; è1:10è or D = 0 D 2 i;i+2j =èi+2jè4jèi + jè; i ç 1;j ç1; ::: ::: ::: ::: ::: ::: ::: ::: ::: ::: ::: ::: ::: ::: ::: ::: 1 C

6 Chebyshev tau - QZ algorithm methods Page 5 D 2 = ::: ::: ::: ::: ::: ::: ::: ::: ::: ::: ::: where we observe D 2 = D æ D in the sense of matrix multiplication. These matrices are started at è0; 0è and truncated at column +2:However, from è1.5è, we easily eliminate u +1 ;u +2 : To do this, suppose for deæniteness is odd, then 1 C u +1 =,èu 0 + u 2 + :::+u,1 è u +2 =,èu 1 + u 3 + :::+u è è1:11è and thus the + 1 and +2 rows of D 2 may be removed and the +1; +2 columns eliminated using è1.11è. This yields an è +1èæè+ 1è matrix D 2, and the matrix problem which results from è1.9è does not suæer from B being singular due to zero boundary condition rows. The outcome is that equation è1.1è is replaced by a system x =,çx è1:12è where x =èu 0 ;:::;u è; and where is now the D 2 matrix with the boundary condition rows removed as described above. If we instead consider the solution to è1.1è by writing as a system of ærst order equations then we must solve u 0 = v; v 0 =,çu; uè,1è = uè1è = 0: Regard u and v as independent variables and write u = +1 k=0 u k T k èxè; v = +1 k=0 v k T k èxè; and then we see that solving è1.13è by a tau method requires us to solve è1:13è L 1 èu; vè ç u 0, v = ç 1 T +1 ; L 2 èu; vè ç v 0 + çu = ç 2 T +1 ; è1:14è and thus we obtain 2è + 1è equations, L1 èu; vè;t i æ =0; i=0;:::;;, L2 èu; vè;t i æ =0; i=0;:::;; è1:15è

7 Chebyshev tau - QZ algorithm methods Page 6 and two equations for the tau coeæcients,, Lq èu; vè;t +1 æ = çq kt +1 k 2 ; q =1;2: è1:16è The boundary conditions in è1.13è are again equivalent to è1.5è. The ænite dimensional system to be solved is then u è1è i, v i =0; v è1è i + çu i =0; i=0;:::;; è1:17è and where +1 n=0 è,1è n u n =0; u è1è i = 2 c i p= +1 p=i+1 p+i odd +1 n=0 pu p ; u n =0; è1:18è è1:19è with a similar expression for v è1è i : ote that in è1.19è all the boundary conditions refer to u i and none to v i : Hence, we require the solution of the matrix problem 0 D,I BC1 0 :::0 0 D BC2 0 :::0 1 C ç ç u v =ç :::0 0:::0,I 0 0:::0 0:::0 1 C ç ç u v è1:20è where BC1;BC2 refer to the conditions on the u i in è1.18è. We are unable to remove the boundary condition rows as before since we do not have conditions on v i : It is typical of the discretizations obtained in this paper that we have to solve a generalised eigenvalue problem like è1.20è and we refer to such problems in the form x = çbx; è1:21è where B is, in general, singular. While the scheme leading to è1.12è yields all the eigenvalues accurately with the aid of the QZ algorithm, it is reported in Straughan & Walker ë27ë that è1.20è leads to the production of a spurious eigenvalue. By this we mean a number which is seen in the eigenvalue list but is not a solution to the diæerential equation. To elucidate on this, the QZ algorithm of Moler & Stewart ë20ë does not produce the eigenvalues ç i but reduces and B to upper triangular form with diagonal elements æ i and æ i : The eigenvalues are ç i = æ i æ i

8 Chebyshev tau - QZ algorithm methods Page 7 when division makes sense. In ë27ë it was found that for è1.17è one value of æ i is Oè10,15 è and this yields a spurious eigenvalue Oè10 17 è: By changing this value of ç is seen to oscillate from a very large negative value to a very large positive one and vice versa. In the æuid dynamics literature such ëeigenvalues" are referred to as spurious eigenvalues and several of the references quoted deal with this topic. Indeed, McFadden et al. ë19ë write that the occurrence of spurious eigenvalues is due to rows of zero's in B; in this case the + 2 and 2è +2è rows of B as in è1.20è. Before proceeding we should mention that the technique of writing the diæerential equations as systems of ærst order ones and discretizing is advocated by Lindsay & Ogden ë18ë who dealt with the Orr-Sommerfeld equation and some other equations from hydrodynamics, although they did not speciæcally mention the tau coeæcients. lso, the idea of using è1.11è to remove boundary condition rows in the D 2 matrix was advanced by Haidvogel & Zang ë15ë who dealt with the solution of Poisson's equation in a two-dimensional rectangle. To begin our discussion of hydrodynamic stability eigenvalue problems we shall consider the Orr-Sommerfeld equation èd 2, a 2 è 2 ç = iareèu, cèèd 2, a 2 èç, iareu 00 ç; z 2 è,1; 1è; è1:22è see Drazin & Reid ë11ë, equation è25.12è, where D = d=dz; Re; a and c are Reynolds number, wavenumber, and eigenvalue ègrowth rateè, respectively, and ç is the amplitude of the stream function. For Poiseuille æow U =1,z 2 ;whereas for Couette æow U = z: Equation è1.22è is to be solved subject to the boundary conditions ç = Dç =0; z=æ1: è1:23è In Poiseuille æow the basic æow is driven by a pressure gradient in the x-direction whereas Couette æow is driven by the upper boundary being sheared relative to the lower one. The latter is known as shear æow but the whole class of such æows is known as parallel æow. Equation è1.22è governs the two-dimensional stability problem for parallel æow where Squire's theorem is employed to reduce the three-dimensional problem to a two-dimensional one. This is standard knowledge in the æuid dynamics literature, cf. Drazin & Reid ë11ë. The function ç is related to the stream function è by è = çèzèe iaèx,ctè : è1:24è System è1.22è, è1.23è has an inænite number of eigenvalues and associated eigenfunctions. Since the real part of the temporal growth rate in è1.24è is e c it ;c=c r +ic i ; the eigenvalue which has largest imaginary part is the most dangerous in a linear instability analysis.

9 Chebyshev tau - QZ algorithm methods Page 8 The component in è1.24è of the solution associated with an eigenvalue is referred to as a mode and the one with largest imaginary part is known as the dominant, or leading, mode èeigenvalueè. For Poiseuille æow the ærst occurrence of c i é 0 is when the Reynolds number is approximately 5772, see Drazin & Reid ë11ë. Thus, according to linearised instability theory the æow becomes unstable when Re ç 5772: However, it has long been known from experimental evidence, that instabilities are seen at much lower Reynolds numbers, even around Re = 1100: This has led to very recent analyses which investigate the possible interaction of more than one mode, and guided by experiments, interactions of modes which pertain to a three-dimensional structure solution. Butler & Farrell ë3ë and Reddy & Henningson ë23ë are particularly interesting studies which investigate the kinetic energy associated with a ænite number of modes arising in the linearised theory. For example, Butler & Farrell ë3ë show that modes associated with eigenvalues which havec i é0 can lead to energy growth, over a æxed time interval, of many orders of magnitude èperhaps 1000 timesè greater than that associated with the leading eigenvalue. They then argue that when this happens very rapid growth is present and three-dimensional instabilities can possibly give rise to growing nonlinear terms which lead to instability at Reynolds numbers well below those of classical linear theory. The analyses of ë3ë and ë23ë are very interesting and show that one ought to consider several eigenvalues in the spectrum, not just the one with greatest imaginary part. For this reason we believe it is important to have a method which yields many eigenvaluesèeigenfunctions very accurately and also in an eæcient manner. The purpose of this paper is to describe such a technique, one which we refer to as the D 2 Chebyshev tau - QZ algorithm method. When we refer to the top end of the spectrum we mean those eigenvalues with c i largest. Usually we here restrict attention to those c i with c i é,1: In analyses such as those of ë3ë, ë23ë this ought tobe suæcient, although lower values of c i are easily found. 2. The Chebyshev tau methods for solving the Orr-Sommerfeld problem. The D 4 Chebyshev tau method. Here we write, cf. Gardner et al. ë14ë, ç = +4 i=0 in equation è1.22è. Then we use the fact that D 4 ç = +4 i=0 ç i T i èzè; è2:1è ç è4è i T i èzè; è2:2è

10 Chebyshev tau - QZ algorithm methods Page 9 where ç è4è i = 1 24c i p= +4 p=i+4 p+i even p æ p 2 èp 2, 4è 2, 3p 4 i 2 +3p 2 i 4,i 2 èi 2,4è 2æ ç p : è2:3è This expression, together with expressions like è1.8è allow us to reduce è1.22è to a system of +1 equations for ç 0 ;:::;ç +4 ; by taking the inner product with T i : To see this deæne the diæerential operator Lç ç D 4 ç, 2a 2 D 2 ç, iareèu, cèèd 2, a 2 èç +èa 4,iaReU 00 èç; è2:4è and then we solve exactly the equation Lç = ç 1 T +1 + ç 2 T +2 + ç 3 T +3 + ç 4 T +4 ; è2:5è where ç i denote tau coeæcients. There should not be any confusion with our earlier use of L as it is clear from the context which operator we are referring to. We take the inner product of è2.5è with T i for i =0;:::;:The inner product with T i for i = +1;:::;+4 leads to four equations for the tau coeæcients. The four remaining conditions are obtained from the boundary conditions è1.23è, and since T 0 nèæ1è=èæ1è n+1 n 2 ; these are +4 i=0 è,1è i ç i = +4 i=0 ç i = +4 i=0 è,1è i+1 i 2 ç i = +4 i=0 i 2 ç i =0: è2:6è èdue to the way the terms split in the discretization of è1.22è when U =1,z 2 it is then better to write è2.6è as i= +3 ç i =0; i=+4 ç i =0; i=+4 i 2 ç i =0; i=+3 i 2 ç i =0: è2:7è i=0 i even i=1 i odd i=1 i odd i=2 i even In this way, one may see that the matrix problem which arises can be split into two problems, one involving ç i, i odd, the other ç i ;ieven. Then one has to solve much smaller generalised eigenvalue problems. In the general case, however, one cannot reduce the diæerential equation to separate even and odd mode calculations.è Equations è2.7è can be solved to write ç +j j =1;2;3;4;as a linear combination of ç 0 ;:::;ç :The terms ç +j ;j=1;2;3;4;can then be removed in a manner not dissimilar to that for the simple example in section 1, and what results is a generalised eigenvalue problem like è1.21è with x =èç 0 ;:::;ç è; although the matrix B which results is nonsingular.

11 Chebyshev tau - QZ algorithm methods Page 10 Even though B is non-singular it is important to realise that the discretization involving the fourth order derivative D 4 leads to a matrix whose terms grow like OèM 7 è; where M = +1 is the number of polynomials. This is evident from è2.3è. In actual calculations we ænd that a large number of polynomials are required, perhaps M = 500; and thus this growth rate can lead to serious round oæ error problems. It is worth noting that in speciæc calculations we use the form for D 4 given by Canuto et al. ë4ë, p. 196, which rearranges è2.3è to give smaller round oæ error. evertheless, even with the Canuto et al. rearrangement the growth is still OèM 7 è: Thus, we believe it is desirable to reduce the order of the diæerential equations whenever possible. The D 2 Chebyshev tau method. D 2 method writes è1.22è as two equations L 1 èç; çè ç èd 2, a 2 èç, ç =0; L 2 èç; çè ç èd 2, a 2 èç, iareèu, cèç + iareu 00 ç =0: è2:8è We solve exactly the equations L 1 èç; çè =ç 1 T +1 + ç 2 T +2 ; L 2 èç; çè =ç 3 T +1 + ç 4 T +2 ; è2:9è by writing ç = +2 i=0 ç i T i èzè; ç = +2 i=0 ç i T i èzè; and then by multiplying each of è2.9è in turn by T i ;i=0;:::;: This yields 2è +1è equations for the coeæcients ç i ;ç i :The equations obtained by taking the inner product of è2.9è with T +1 ;T +2 yield equations for the tau coeæcients. The diæculty with the above approach, as pointed out by McFadden et al. ë19ë, p. 232, is that the boundary conditions are all on ç i and none are on ç i : Thus, we cannot remove boundary condition rows by removing the ç +1 ;ç +2 ;ç +1 ;ç +2 terms which arise in the resulting D 2 matrices. èif the boundary conditions are those appropriate to surfaces free of tangential stress then there are two boundary conditions on ç and two onçand one can remove the oæending boundary condition rows. This we have done for Orr-Sommerfeld problems and we obtain highly accurate results and no spurious eigenvalues. lso, in a practical multi-component diæusion problem involving penetrative convection Straughan & Walker ë28ë arrived at similar conclusions. For porous convection problems the natural boundary conditions allow boundary condition removal in the matrix and very satisfactory results are yielded, Straughan & Walker ë26,27ë.è Wemay instead write in the boundary conditions as rows of the matrix. This is also done by Lindsay & Ogden ë18ë who generalized the Gardner et al. ë14ë method and solved è1.22è,

12 Chebyshev tau - QZ algorithm methods Page 11 è1.23è as a system of four ærst order equations. We refer to their technique as a D-method and outline this below. s we have pointed out in section 1 when we use the Lindsay & Ogden ë18ë technique on the simple harmonic motion equation with homogeneous boundary conditions we detect a spurious eigenvalue. D 2 -method for è1.22è, è1.23è appropriate to Poiseuille æow eventually solves an equation like è1.21è where with and r = 0 D 2, a 2 I,I BC1 0 :::0 BC2 0 :::0 x =èç 0 ;:::;ç +2 ;ç 0 ;:::;ç +2 è T ; 0 D 2,a 2 I BC3 0 :::0 BC4 0 :::0 1 C B r = 0; B i = ; i = ,aReI 0 :::0 0::: :::0 0:::0 0:::0 0:::0,2aReI areèp, Iè 0 :::0 0:::0 0:::0 0:::0 where P is the Chebyshev matrix representing z 2 ;= r +i i ; and B = B r + ib i : èp is the matrix obtained by writing z 2 = 1 2 è1 + T 2èzèè; and then taking the inner product èt i ;z 2 çè:è The rows BC1;:::;BC4 refer to the boundary conditions on ç n 1 C 1 C and for the Orr- Sommerfeld problem we ænd it preferable to use the form è2:7è 1 as BC1; BC2 and è2:7è 2 as BC3;BC4: The D Chebyshev tau method. In this case we write è1.22è as four equations L 1 Y ç Dç, ç =0; L 2 Y ç Dç, ç =0; L 3 Y ç Dç, æ =0; L 4 Y ç Dæ, æ 2a 2 + iareèu, cè æ ç + æ a 4 + ia 3 ReèU, cè+iareu 00æ ç =0; where L i denote the operators indicated and Y =èç; æ; æ; æè: Then write ç = +1 i=0 ç i T i èzè; ç = +1 i=0 ç i T i èzè; ç = +1 i=0 ç i T i èzè; æ = +1 i=0 æ i T i èzè: è2:10è è2:11è

13 Chebyshev tau - QZ algorithm methods Page 12 ow solve exactly the tau system L m Y = ç m T +1 ; m =1;:::;4; è2:12è by multiplying each equation by T i : This yields èl m Y;T i è=0; m=1;:::;4; i=0;:::;; è2:13è and the tau coeæcients may be determined from èl m Y;T +1 è=ç m kt +1 k 2 : è2:14è Equations è2.13è give 4è+1è equations for the coeæcients èç 0 ;:::;ç +1 è; èç 0 ;:::;ç +1 è; èç 0 ;:::;ç +1 è; èæ 0 ;:::;æ +1 è: Here, a similar problem arises with the boundary conditions as with the D 2 method. The boundary conditions è2.7è involve two conditions on each of ç i ;ç i ; but none on ç i ;æ i ; although they are somewhat simpler being i= i=0 i even ç i =0; i=+1 i=1 i odd ç i =0; The matrix problem thus becomes è1.21è with i = r = 0 0 i= i=0 i even ç i =0; i=+1 i=1 i odd D,I 0 0 BC1 0 :::0 0:::0 0:::0 0 D,I 0 BC2 0 :::0 0:::0 0:::0 0 0 D,I 0:::0 BC3 0 :::0 0:::0 a 4 I 0,2a 2 I D 0:::0 BC4 0 :::0 0:::0 1 C ç i =0: :::0 0:::0 0:::0 0::: :::0 0:::0 0:::0 0::: :::0 0:::0 0:::0 0:::0 a 3 ReU + areu 00 0 areu 0 0 :::0 0:::0 0:::0 0:::0 1 C è2:15è

14 Chebyshev tau - QZ algorithm methods Page 13 with B i = :::0 0:::0 0:::0 0::: :::0 0:::0 0:::0 0::: :::0 0:::0 0:::0 0:::0 a 3 ReI 0,aReI 0 0 :::0 0:::0 0:::0 0:::0 B r = 0 1 C where in this case BC1, BC4 are è2:15è 1, è2:15è 4 : The vector x is now x =èç 0 ;:::;ç +1 ;ç 0 ;:::;ç +1 ;ç 0 ;:::;ç +1 ;æ 0 ;:::;æ +1 è T : It should be observed that in each of the D 2 and D methods the B matrices involverows of zero's in addition to zero blocks. This is due to the way the boundary condition rows are added to : Further, from è2.3è, è1.8è, and è1.19è, the growth of the matrix coeæcients for each ofthed 4,D 2 ;and D methods is OèM 7 è;oèm 3 è and OèMè: Finally, the D 4, D 2 ; and D methods essentially give generalised eigenvalue problems which involve ; B of order M æ M, 2Mæ2M; and 4M æ 4M: The recent papers of Gardner et al. ë14ë and McFadden et al. ë19ë are very relevant to the present contribution and a brief discussion is in order. Gardner et al. ë14ë reduce a fourth order system to two second order ones, or, in general, reduce systems of fourth order equations to systems of second order ones. While in æuid dynamics one is often faced with an even order system, the system is not necessarily one composed of a system of fourth order equations. Gardner et al. ë14ë illustrate their method by application to the equation d 4 u dx + R d3 u 4 dx, s d2 u 3 dx =0; 2 x2è,1;1è; in which R is a constant and s is the eigenvalue. The boundary conditions they take are u =0; x=æ1; u 0 =0; x=æ1: To solve this they write as a second order system v 00 + Rv 0, sv =0; u 00 = v; è2:16è

15 Chebyshev tau - QZ algorithm methods Page 14 and use a tau method with uèxè = +2 i=0 u i T i èxè; vèxè = +2 i=0 v i T i èxè: By multiplying è2.16è by T i they derive 2è+1è equations for the u i ;v i and four equations for the tau coeæcients. The boundary conditions yield a further four equations for the coeæcients u i ;v i : The procedure of ë14ë is to eliminate the v i coeæcients since the boundary conditions are all on u i : To do this they partition the resulting ænite dimensional system as B 1 b + B 4 y, sb 1 = 0; B 2 b + B 5 y, sb 2 = 0; T = B 3 b + B 6 y, sy; b = Qa; è2:17è where the matrices B 1 ;:::;B 6 and Q are deæned in ë14ë and where the vectors are deæned by b =èb 0 ;:::;b è T ; b 1 =èb 0 ;:::;b,2 è T = Q 1 a; b 2 =èb,1 ;b è T = Q 2 a; y =èb +1 ;b +2 è T ; T =èç 1 ;ç 2 è T ; a=èa 0 ;:::;a +2 è T : In ë14ë they ærst solve for y and then reduce è2.17è to a single equation in a, namely èb 1 Q, B 4 B,1 5 B 2Qèa = sèq 1, B 4 B,1 5 Q 2èa: è2:18è This is of form è1.21è and may be solved by the QZ algorithm. This method is extended in ë14ë to the Orr-Sommerfeld problem. lthough the boundary condition rows may effectively be removed in this manner the problem of growth of matrix coeæcients is still present. This is due to the matrices in è2.18è èequation è3.9bè of ë14ëè. The matrices B 1 and Q each involve ç 2 F 2 èin the notation of ë14ëè and this term is like OèM 3 è; thus the product grows at least as OèM 6 è and hence the modiæed tau method of ë14ë still does not remove the growth problem. lso, it is not so evident how one would extend the Gardner et al. ë14ë method to more complicated systems such as the Butler & Farrell ë3ë one in three-dimensions, or the two æuid system studied in section 6 of this article. McFadden et al. ë19ë discuss the fourth order and second order Chebyshev tau methods, paying particular attention to the diæerential equation system d 4 u dx = ç d2 u ; x 4 2 è,1; 1è; dx2 uèæ1è = u x èæ1è = 0:

16 Chebyshev tau - QZ algorithm methods Page 15 They suggest a modiæed tau method by discretizing è2.19è as it stands to obtain an equation of type u è4è i = çu è2è i è2:20è together with boundary conditions of type è2.6è. They write equation è2.20è plus boundary conditions in the form è1.21è and then suggest setting the last two columns of the B matrix equal to zero in order to remove spurious eigenvalues. They then include an elegant proof to show that this approach is in some ways equivalent toad 2 method which they call a stream - function vorticity method. Of course, the formulations cannot be entirely equivalent due to the growth of coeæcients in the respective matrices. McFadden et al. ë19ë also discuss the Orr-Sommerfeld problem for Poiseuille æow. In the two-dimensional case for the Orr-Sommerfeld equation the D 2 method is essentially the stream function - vorticity technique discussed by McFadden et al. ë19ë. For, in that case, the vorticity has only one component, in the y-direction,! =! 2 with! =u ;z, w ;x =è ;zz + è ;xx =e iaèx,ctè èd 2, a 2 èç: Thus the functions ç and ç are essentially è and!: We believe that the D 2 method is more general than the stream function - vorticity method. For example, the Butler & Farrell ë3ë problem for Poiseuille and Couette æow in three dimensions results in a coupled system involving a fourth order equation for w and a second order equation for the normal vorticity! 3 = v ;x, u ;y : In this three-dimensional situation one may still reduce things to three second order equations and use a D 2 method which is then not equivalent to the usual stream function - vorticity method. Other areas where the D 2 method diæers are in three-dimensional convection studies in anisotropic porous media where the principal axes of the permeability tensor are not orthogonal to the layer, or the Hadley æow problem, see Straughan & Walker ë26,28ë, respectively. lthough we do not include explicit details of analysis for problems such as that of Butler & Farrell ë3ë we stress that the extension to such three-dimensional studies is very important. Details of such studies together with the consequences for the relevant æuid mechanics may be found in a porous media setting in ë26,28ë, although the treatment of boundary conditions is diæerent for porous media æow. In the remainder of the paper we make a systematic study of Chebyshev tau methods applied to Poiseuille æow, Couette æow, Hagen-Poiseuille æow, and ænally we show how to apply the method to the situation where one æuid is overlying another. We stress

17 Chebyshev tau - QZ algorithm methods Page 16 that we concentrate on ænding many eigenvalues, including eigenvalues which are diæcult to obtain, and we investigate parameter ranges which have previously proved diæcult. Consideration is given to loss of accuracy due to the various D 4 ;D 2 or D methods, and to loss of accuracy due to insuæcient polynomials, or insuæcient resolution due to lack of precision in the representation of real numbers. 3. The eigenvalue problem for plane Poiseuille æow In this section we study è1.22è, è1.23è with U =1,z 2 :There have been many calculations of the spectral behaviour for Re not too large, say Re ç 10 4 ; see Butler & Farrell ë3ë, Drazin & Reid ë11ë, Orszag ë22ë, Reddy & Henningson ë23ë, and the references therein. bdullah & Lindsay ë1ë, Davey ë7ë and Fearn ë12ë report studies on particular eigenvalues for Re extending to 10 9 : We use the D 2 èstream function - vorticityè method and show what can go wrong in calculating the spectrum for large Re and how one can put things right. We have written our own codes for the D 4, D 2 and D methods, and a comparison of results for these methods is now given. Undoubtedly the advantage of the D 2 method is the growth rate removal, and in this respect the D method is even better, with terms growing only like OèMè; this important feature does not appear to have been realised in Lindsay & Ogden ë18ë. gainst this, the time taken by the QZ algorithm appears to scale like OèMsize 3 è where M size is the matrix width, i.e. M size = M;2M;4M; with the D 4 ;D 2 ;D methods, respectively. The problems associated with doubling the matrix size have been commented on by McFadden et al. ë19ë. For example, on a SU sparc station èipcè, with 50 polynomials, the D 4 ;D 2 ;D methods take, respectively, 4.1 seconds, 17.1 seconds, and seconds. One test of the D method with M = 150 took seconds, i.e. approximately 49 minutes. Clearly, when many computations are required this is an important factor. dditionally, the memory requirements of the D method are substantial, requiring approximately 16MB for the 150 polynomial case èusing full precisionè. The D 2 method we have found to yield high accuracy, although as reported below, for Re high enough extended precision arithmetic is required. Unless explicitly stated, our calculations are based on full precision, i.e. 64 bit, arithmetic. The D 2 and D methods necessarily produce a B matrix in è1.21è which has one or more rows of blocks of 0's and so is singular. One approach to solving this problem is the QZ algorithm of Moler & Stewart ë20ë. This algorithm relies on the fact that there

18 Chebyshev tau - QZ algorithm methods Page 17 exist unitary matrices Q and Z such that QZ and QBZ are both upper triangular. The algorithm then yields sets of values æ i ;æ i which are the diagonal elements of QZ and QBZ: The eigenvalues ç i of è1.21è are then obtained from the relation ç i = æ i =æ i ; provided æ i 6=0:This is very important, since the way wehave constructed B means it contains a singular band, corresponding to inænite eigenvalues, and the æ i =0must be æltered out. Indeed, with the technique advocated here one ought always to consider the æ i and æ i ; since as Moler & Stewart ë20ë point out, the æ i and æ i contain more information than the eigenvalues themselves. The QZ algorithm is available in the routines ZGGHRD, ZHGEQZ and ZTGEVC of the LPCK Fortran Subroutine library, nderson et al. ë2ë. The coeæcients of the eigenvector x yielded by the QZ algorithm are extremely useful convergence indicators. In this regard we throughout render the eigenvector unique by normalising so that the sum of squares of the moduli of the components is equal to one and the component of largest modulus is real. In fact, the eigenvectors can be used to indicate the presence of spurious eigenvalues. We have found by computation that the ëeigenvector" for such a spurious eigenvalue typically does not demonstrate the convergence evident in the eigenvector for a real eigenvalue. n alternative way is to compute the ç - coeæcients, cf. Gardner et al. ë14ë, but as these are based on the eigenvector we ænd it simpler to just examine the eigenvectors themselves. nother possible method of assessing whether an eigenvector is spurious is to compute the residuals for è1.21è, i.e. compute r èiè = æ i x èiè, æ i Bx èiè : When an ëeigenvalue" is spurious we have found these to have all components between Oè10 22 è and Oè10 18 è: For a real eigenvalue, the residuals corresponding to those æ i =0 are Oè10 18 è as are two or three corresponding to the presence of spurious eigenvalues, whereas the rest converge from Oè10 6 èdown to Oè10,8 è; which is consistent with the fact that the discretization only allows us to see the ëtop end" of the true spectrum. Even though we are using a D 2 method we do ænd a spurious eigenvalue may be produced. When we apply the method to the problem of two æuids in section 6 then we always see spurious eigenvalues. In connection with this, we have calculated the sensitivities for the eigenvector, cf. Stewart & Sun ë25ë, and these indicate that the spurious eigenvalues are connected with the discretization procedure rather than the QZ algorithm used to ænd the matrix eigenvalues. Details appropriate to the superposed æuid problem are given in section 7. referee has kindly pointed out that relying on the eigenvector or the residuals assumes the forward stability of the QZ algorithm which cannot be assumed. However, we

19 Chebyshev tau - QZ algorithm methods Page 18 have found numerically that these are reliable guides and our computations agree with, or are better than, numerical results reported by other workers. Orszag ë22ë gives for Re =10 4 and a =1; c= 0: : i è3:1è as the exact value of the ærst eigenvalue èto 8 d.p.è. He used 56 polynomials to achieve this accuracy èalthough he only uses even ones, thereby only 28 terms are in his expansionè. The D 4 method gives c = 0: : i with M = 50:We found this to be the best approximation and thereafter on increasing M the value diverges from è3.1è. The D 2 and D methods, however, agree with è3.1è for M = 56 and beyond. We can also ænd the eigenfunctions very accurately. For example, the D 2 method with 56 polynomials yields ç r and ç i as in ægure 4.20 of Drazin & Reid ë11ë. With the boundary conditions è2.7è or è2.15è, respectively, by using the D 2 and D methods we obtained only even modes for cases when the eigenfunction is symmetric and odd modes in the skew symmetric case, and the convergence is better than that for the D 4 technique. nother feature of the D 4 method is that even though the boundary conditions are removed we still saw twovalues in the list with æ =0:The corresponding æ values were large, Oè10 15 è; and real. These we believe are spurious but are easily æltered out by examining the æ i given by the QZ algorithm. This does raise an important point. It questions whether the only way spurious eigenvalues can arise is through rows of zero's in the B matrix due to inserting boundary condition rows in the equivalent rowèsè of : Orszag ë22ë table 5 gives a list of the 32 least stable modes for Re =10 4 ;a=1:with the D 2 method we obtained complete agreement with this list in the sense: for the ærst 12 eigenvalues with 70 polynomials, for the ærst 14 eigenvalues with 80 polynomials, and complete agreement with all 32 eigenvalues by using 96 polynomials. We did, however, ænd an extra eigenvalue; between positions 17 and 18 of Orszag ë22ë table 5 we obtain the value c = 0: , 0: i: è3:2è The eigenvector coeæcients from the D 2 method ènormalised as indicated earlierè with 96 polynomials indicate the value è3.2è corresponds to a skew-symmetric solution and with this number of polynomials the eigenvector had converged to Oè10,13 è;oè10,14 è; for the ç and è= D 2 ç, a 2 çè terms. Interestingly, we ænd both symmetric and skew symmetric modes with the D èand D 2 è methodèsè. Lindsay & Ogden ë18ë appear to report only symmetric ones. For Re =10 4 ;a =1;the spectrum, in the range c i 2 è,1; 0è; c r 2è0; 1è;

20 Chebyshev tau - QZ algorithm methods Page 19 is given in ægure 1, indicating which are even and which are odd modes. When we refer to the spectrum in a ægure here and throughout the paper we mean that part displayed in the relevant ægure. We have produced an mpeg movie which may be accessed with a web browser, such as Mosaic or etscape, at walkerèeigenproblems.html This movie contains the parametric evolution of the spectrum of the plane Poiseuille æow problem for a =1;with Re ranging from 100 to 10 4 in steps of 10. This may yield useful insight into resonance mechanisms. The evolution of the upper branches is clearly visible and the emanence of the eigenvalues from c r =2=3 is evident.

21 Chebyshev tau - QZ algorithm methods Page P c i S c r Figure 1 The spectrum for plane Poiseuille æow. Re =10 4 ;a =1;open circle èæè = even eigenfunction, cross èæè = odd eigenfunction. The upper right branch consists of ëdegenerate" pairs of even and odd eigenvalues. When Re is increased eventually mode crossing is seen, i.e. eigenvalues exchange positions in the sense that the imaginary part of one eigenvalue decreases relative to that of another eigenvalue whose imaginary part eventually becomes larger than that of the former. bdullah & Lindsay ë1ë are critical of the papers of Davey ë7ë and Fearn ë12ë in their analysis of higher Re values. ccording to linear theory for è1.22è and è1.23è the most unstable eigenvalue has largest imaginary part, i.e. c i largest. Davey ë7ë claims the most unstable mode is symmetric, presumably on the basis that this is so for Re =10 4 ;a =1: Fearn ë12ë solves a symmetric problem and simply refers to the solution to è1.22è, conærming Davey's ë7ë results. bdullah & Lindsay ë1ë claim that these writers are using a tracking technique and miss the leading eigenvalue since mode crossing occurs. We have partially conærmed the ændings of bdullah & Lindsay ë1ë and Lindsay & Ogden ë18ë who only give the ærst æve eigenvalues, although it would appear they ænd only symmetric

22 Chebyshev tau - QZ algorithm methods Page 21 ones. This is especially important, since for Re =10 5 ;a =1;we conærm mode crossing has occurred, but we ænd the leading eigenvalue to be skew-symmetric. We present in table 1 the leading eigenvalues for Re =10 5 and a =1;our calculations being made by a D 2 method, but determining odd modes and even modes separately, with M = 200 in each case, i.e. equivalent tom= 400 for the full problem. It is seen from table 1 that the ëleading" eigenvalue is a skew - symmetric one. In table 2we include components of the ènormalisedè eigenvector for this skew mode. It is seen that machine precision is reached with 86 odd polynomials, i.e. up to T 171 : èll components are not included, just a sample to demonstrate convergence.è We have used the separate ëodd" and ëeven" codes with M = 200 to compute the behaviour of the eigenvalues in the transition region. For a =1;in table 3 it is seen that the structure at Re =10 4 is maintained at Re =80;822 but by Re =80;828 the ærst two eigenvalues exchange positions, then by Re = 80;830 the eigenvalue which is second at Re =80;828 exchanges with the one occupying third position for the same Re value. The position of these three is maintained in table 1. bdullah & Lindsay ë1ë report further mode crossings for higher Re values. However, much care must be taken when Re increases. We now report ændings for the spectrum and in particular in the region near the joining of the so called, P and S branches - the three groups of branches in ægure 1. s indicated in the ægure the branches are the upper left ones, the P branch is the upper right one composed of degenerate pairs, and the S branch is the lower one emanating from c r =2=3:This notation is standard in the æuid dynamics literature, see Drazin & Reid ë11ë. The eigenvalues near the branch point are particularly sensitive tochange, Orszag ë22ë, and we ænd great care must be taken even with Re around 2:3 æ 10 4 : We have computed many cases and ægures 2-5 below are just a sample. Even though they are only for even modes they illustrate the important points regarding round oæ error. The same ændings are true for odd mode cases, and for the full code which ænds odd and even modes together. Figures 2 to 5 are obtained with the D 2 method solving è1.22è, è1.23è for even modes only, i.e. employing only even polynomials, using full precision arithmetic è64 bitè in ægures 2 to 4, whereas extended precision è128 bitè is employed in ægure 5. Figure 2 demonstrates inaccuracy caused by having insuæcient polynomials, even though M = 85; èequivalent to 170 in the odd and even codeè. This splitting in the tail is symptomatic of insuæcient polynomials and we ænd similar behaviour in all of the problems reported here. By increasing the number of polynomials we are able to overcome the splitting of the tail problem as in ægure 3 where M = 200: evertheless, the eigenvalues at the intersection

23 Chebyshev tau - QZ algorithm methods Page 22 are not accurate. Increasing the number of polynomials compounds the problem and we ænd a ëtriangle of numerical instability" begins to form, ægure 4, where M = 500: We have found that this behaviour is due to the precision to which we are working. By increasing from 64 to 128 bit arithmetic this eæect is removed èin this caseè, see ægure 5. We have not seen the latter eæect reported before. The 128 bit arithmetic calculations were performed with the LPCK routines on an IBM RS machine. referee raised the very interesting question as to whether the D 2 method operating at 128 bit arithmetic is as expensive as the D method at 64 bit arithmetic. From the computational time point of view the two methods are comparable. However, we have ran the D method in 64 bit arithmetic for the cases of ægures 2 to 4. The instability at the intersection of the branches is not removed. Thus, the D 2 method at 128 bit arithmetic is not equivalent to the D method at 64 bit arithmetic. The instability seen in ægures 2 to 4 is not due to growth of coeæcients in the matrix and we believe that it can only be removed by increasing the precision. This, of course, raises the question of when to use a certain precision, and how many polynomials we should employ a priori. This point was succinctly raised by a referee. t present we do not have an analytical answer to this. We have proceeded by numerical experiment and comparison with previous work. We are investigating further this aspect and, indeed, the question of how spurious eigenvalues form. evertheless, we believe the ændings we report here are worthy and ought to be brought to the attention of workers dealing with practical hydrodynamical stability problems.

24 Chebyshev tau - QZ algorithm methods Page 23 Symmetry S S S S S S Eigenvalue : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i Table 1. The six odd and six even eigenvalues with largest imaginary part for the Orr- Sommerfeld problem with U =1,z 2 ;Re=10 5 ;a=1:=anti-symmetric, S=symmetric. Com. o. ç r ç i r i E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-14 Table 2. selection of the components of the eigenvector corresponding to ç è1è = 0: , 0: i; Re =10 5 ;a=1:com. o. refers to the component of the eigenvector, ç = ç r + iç i ;= r +i i : The eigenvector is normalised with the sum of squares of the moduli of the components equal to one and the component of largest modulus is real.

25 Chebyshev tau - QZ algorithm methods Page 24 Symmetry Eigenvalue Re : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i E +00,: E, 01i S : E +00,: E, 01i S : E +00,: E, 01i S : E +00,: E, 01i S : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i S : E +00,: E, 01i S : E +00,: E, 01i S : E +00,: E, 01i S : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i : E +00,: E, 01i S : E +00,: E, 01i S : E +00,: E, 01i S : E +00,: E, 01i S : E +00,: E, 01i Table 3. The four leading odd and even eigenvalues in the transistion region, M = 200; a =1:

26 Chebyshev tau - QZ algorithm methods Page c i -0.5 c i c r c r Figure 2 Figure c i -0.5 c i c r c r Figure 4 Figure 5 Figure 2 The approximate eigenvalues associated with even modes for plane Poiseuille æow. Eæect of too few polynomials. Re =2:7æ10 4 ;a=1;m =85;64 bit arithmetic. Figure 3 The approximate eigenvalues associated with even modes for plane Poiseuille æow. Eæect of ænite precision. Re =2:7æ10 4 ;a =1;M = 200; 64 bit arithmetic. Figure 4 The approximate eigenvalues associated with even modes for plane Poiseuille æow. Eæect of ænite precision. Re =2:7æ10 4 ;a =1;M = 500; 64 bit arithmetic. Figure 5 The approximate eigenvalues associated with even modes for plane Poiseuille æow. Re =2:7æ10 4 ;a=1;m = 200; 128 bit arithmetic.

27 Chebyshev tau - QZ algorithm methods Page 26 Remarks. To conclude this section we note that we have drawn attention to three important types of error which are present in solving diæcult eigenvalue problems. The ærst is round oæ error due to growth of matrix coeæcients. In this respect a D 2 method is preferable to one using D 4 : Secondly, too few polynomials causes the ëtail", i.e. the S branch, to split. Thirdly, increasing the number of polynomials in a cavalier fashion to compute sensitive eigenvalues can lead to inaccuracy due to ill conditioning and insuæcient precision in real number representation. Clearly care must be taken to avoid these errors and while our work regarding the latter two is heuristic, we believe itisworth drawing attention to these points, while we continue in a search for analytical tests to determine the correct number of polynomials coupled with the best precision. 4. The eigenvalue problem for plane Couette æow The problem of this subsection is è1.22è and è1.23è when U = z: Physically it corresponds to the lower plate æxed while the upper plate is moved with constant velocity, generating a linear shear. This problem is always stable according to linear theory, cf. Drazin & Reid ë11ë. evertheless, it is not a trivial eigenvalue problem from a numerical standpoint. Indeed, we studied this problem at the suggestion of Dr. lison Hooper who informed us she had diæculty in calculating the spectrum for Reynolds numbers as low as With 150 polynomials in the D 2 method we obtain excellent accuracy for Re = 3000;a= 1; in 64 bit arithmetic. Breakdown at the intersection of the branch points is evident around Re = 3500: However, the same code operating at 128 bit arithmetic yields an accurate spectrum.