[7] R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Clis, New Jersey, (1962).

Transcription

1 [7] R.S. Varga, Matrx Iteratve Analyss, Prentce-Hall, Englewood ls, New Jersey, (962). [8] J. Zhang, Multgrd soluton of the convecton-duson equaton wth large Reynolds number, n Prelmnary Proceedngs of 996 opper Mountan onference on Iteratve Methods, opper Mountan, olorado, (996), Vol. 2, pp. {9. 8

2 That s, M ~, and therefore M, are smlar to a symmetrc matrx, and ther egenvalues p are p real. The M-matrx D ~ has block dagonal form dag( D; : : :; D), where each D = tr(4 2; 36; 4 2). Hence, mn ( D) ~ = mn ( D): y Lemma 2 of Elman and Golub [2], the egenvalues of the trdagonal matrx D of order n are f j = 4(9 2 p 2 cos jh); j = ; 2; : : :; ng. Hence, mn ( ~ D) = mn ( D) = 4(9 2 p 2 cos h): (29) The spectral radus of ~ s bounded by Gerschgorn's theorem [7]: Hence, from (27), (29) and (3) we have the followng theorem: %( ~ ) 4( + 2 p 2): (3) Theorem 4.6 If jj = jj = p 2, the spectral radus of the lne Jacob teraton matrx for the lnear system (2) s bounded by 5 onclusons %(M) + 2 p p 2 cos h! 7 + 2p 2 :623; as h! : 73 (3) We proved that the pont Jacob and pont Gauss-Sedel methods converge for solvng the lnear system resulted from a fourth-order nte derence dscretzaton of the convecton-duson equaton when the equaton s duson-domnated. We also proved that the lne Jacob method converges when the coecent matrx s symmetrzable. References [] S..R. Denns and J.D. Hudson, ompact h 4 nte-derence approxmatons to operators of Naver-Stokes type, J. omput. Phys. 85, 39{46, (989). [2] H.. Elman and G.H. Golub, Iteratve methods for cyclcally reduced non-self-adjont lnear systems, Math. omp. 54, 67{7, (99). [3] M.M. Gupta, R. Manohar and J.W. Stephenson, A fourth order, cost eectve and stable nte derence scheme for the convecton-duson equaton, n Numercal Propertes & Methodologes n Heat Transfer. Hemsphere Publshng orp., Washngton, D, (983), pp. 2{29. [4] M. M. Gupta, R.P. Manohar, and J.W. Stephenson, A sngle cell hgh order scheme for the convecton-duson equaton wth varable coecents, Int. J. Numer. Methods Flud. 4, 64{ 65, (984). [5] M. L, T. Tang and. Fornberg, A compact fourth-order nte derence scheme for the steady ncompressble Naver-Stokes equatons, Int. J. Numer. Methods Fluds, 2, 37{5, (995). [6] W.F. Spotz and G.F. arey, Hgh-order compact scheme for the steady stream-functon vortcty equatons, Int. J. Numer. Methods Eng. 38, 3497{352, (995). 7

3 orollary 4.4 If = p 2, = p 2 or = p 2; = p 2, the coecent matrx A wth the followng computatonal ( p 2 7 ) 2 4(2 7 p 2) 4(2 7 p 2) 36 4(2 6 p 2) 4(2 6 p 2) ( p 2 6 ) 2 s symmetrzable by the real dagonal smlarty transformaton Q = dag(q ; Q 2 ; : : :; Q n ), where Q = dag[; p 2 7 ; ( p 2 7 ) 2 ; ( p 2 7 ) 3 ; : : :; ( p 2 7 ) n ]; Q j = ( p 2 6 )Q j; j = 2; 3; : : :; n: The symmetrzed coecent matrx ~ A has the computatonal 4 p 2 4 p p 2 4 p 2 Remark 4.5 Although the orgnal coecent matrx A s not dagonally domnant when jj = jj = p 2, the symmetrzed coecent matrx ~ A s strctly dagonally domnant. 4. A ound for Lne Jacob Method Let A be splt by the lne Jacob teraton,.e. A A A = D ; (25) where D s the dagonal block and contans the upper and lower dagonal parts of A. Suppose that A can be symmetrzed by a real dagonal smlarty transformaton Q and the symmetrzed matrx s ~ A = Q AQ. orrespondng to the lne Jacob splttng (25), ~ A s splt as ~A = ~ D ~ : (26) We now derve a bound for the spectral radus of the teraton matrx M = D based on the lne Jacob splttng of the coecent matrx A, n the case where A s symmetrzable,.e., when jj = jj = p 2. Note that M = Q ~ D ~ Q ;.e., M s smlar to ~ M = ~ D ~ and they have the same egenvalues. Hence, we can restrct our attenton to ~ M. The analyss s based on the result %( ~ D ~ ) k ~ D ~ k2 k ~ D k 2 k ~ k2 = %( ~ ) mn ( ~ D) ; (27) where the equalty follows from the symmetry of ~ D and ~. %( ~ ) s the spectral radus of ~ and mn ( ~ D) s the smallest egenvalue n absolute value of ~ D. For jj = jj = p 2, ~ D s symmetrc postve dente and can be factored symmetrcally as ~ D = LLT. Hence L T ~ D ~ L T = L ~ L T : (28) 6

4 However, Q AQ s symmetrc f and only f condtons (5), (7), (8) and (9) hold smultaneously. From (5) and (7), we have r = ; (2) 2 whch s well dened snce > by Lemma 3.. Substtutng (2) nto (9) and equatng the coecents of, we have Substtutng (5) nto (8), we obtan = : (2) + = r : (22) Substtutng (7) wth the subscrpt beng replaced by + nto (22) and comparng the coecents of +, we have = : (23) Substtutng the coecent values of (4) nto (22) and (23), after smplcaton, we have the system of two equatons whch must hold smultaneously The solutons to system (24) are ( + )( ) = ; ( + )( ) = : = = ; jj = jj = ; jj = jj = p 2: Snce we must also have 6 8 > and 5 7 > for (8) and (9) beng well dened, soluton jj = jj = s excluded by Lemma Remark 4.2 If (2) holds, () reduces to the Posson equaton and the matrx A s symmetrc and postve dente by tself. The nterestng case s when jj = jj = p 2, equaton () s convectondomnated and the lnear system (2) s nonsymmetrc and non-dagonally domnant. The followng corollares can be vered drectly: orollary 4.3 If = = 6 p 2, the coecent matrx A wth the followng computatonal 4(2 6 p 2) ( p 2 6 ) 2 4(2 7 p 2) 36 4(2 6 p 2) ( p 2 7 ) 2 4(2 7 p 2) s symmetrzable by the real dagonal smlarty transformaton Q = dag(q ; Q 2 ; : : :; Q n ), where A Q = dag[; p 2 7 ; ( p 2 7 ) 2 ; : : :; ( p 2 7 ) n ]; Q j = ( p 2 7 )Q j; j = 2; 3; : : :; n: The symmetrzed coecent matrx ~ A has the computatonal 4 p 2 4 p p 2 4 p 2 A (24) 5

5 Theorem 4. The coecent matrx A can be symmetrzed wth a real dagonal smlarty transformaton f and only f one of the followng condtons hold: = = ; (2) or jj = jj = p 2: (3) Proof. The unknowns can be ordered so that the matrx A has the block trdagonal form A = tr[a j;j; A j;j ; A j;j+]; where A j;j = tr[ 7 ; 4 ; 8 ]; A j;j = tr[ 3 ; ; ]; A j;j+ = tr[ 6 ; 2 ; 5 ]: We look for a matrx Q = dag(q ; Q 2 ; : : :; Q n ), where Q j s a real dagonal matrx of the same order as A j;j, such that Q AQ s symmetrc. Let Q j = dag( ; q(j) 2 ; : : :; q(j) ); j = ; 2; : : :; n: n We rst consder the dagonal block: Q j A j;j Q j s symmetrc f and only f = q(j) ; n ; j n; (4) where may be arbtrary. Thus, the dagonal blocks can be symmetrzed provded + = r 3 ; (5) and ths recurrence s well dened f and only f 3 = s postve, whch s true by Lemma 3.. The (equal) quanttes (4) are the (; ) and ( ; ) entres of the jth dagonal block of the symmetrzed matrx. For the odagonal blocks, we requre Q j A j;jq j = (Q j A j;jq j ) T : (6) Relaton (6) holds f and only the followng three scalar relatons hold: 4 = + 8 = + q(j) q(j) + 7 = q(j) + r 2 ; or 4 = ; (7) 2 r 6 ; or 8 = + ; (8) 6 5 ; or + = r 7 5 : (9) (7) s well dened f and only f 2 4 > whch s true for any and by Lemma 3.. (8) s well dened f and only f 6 8 >, whch s true f ether jj < and jj < both hold, or jj > and jj > both hold by Lemma 3.2. The same condtons are requred for (9) beng well dened. 4

6 Lemma 3.2 The coecents of the nne-pont stencl (4) satsfy 5 7 > and 6 8 > f one of the followng condtons hold: jj < ; jj < ; (6) or jj > ; jj > : (7) Proof. 5 7 > f ( 2 )( 2 ) > : (8) It s easy to see that (8) holds f ether (6) holds or (7) holds. The condtons for 6 8 > can be vered smlarly. 2 Lemma 3.3 The matrx A s rreducble. Proof. It s readly vered that the drected graph of A s strongly connected. 2 Lemma 3.4 The matrx A s rreducbly dagonally domnant f jj and jj. Proof. A s dagonally domnant f (3) and (9) mply ja ; j n X 2 j=;j6= ja ;j j; for = ; : : :; n 2 : (9) j j 8X j= Substtutng (4) n () after smplcaton, we have j j j: () (j + j + j j)(j + j + j j) 4: () Snce (j + j + j j) = 2 f jj, and (j + j + j j) = 2 f jj, t follows that () holds f jj and jj both hold. A s rreducble by Lemma 3.3. Snce the strct nequalty n (9) holds for at least the rst row of A for jj and jj, A s rreducbly dagonally domnant (see pp. 23 of Varga [7]). 2 From Lemmas 3.3 and 3.4, we have the followng theorem: Theorem 3.5 The pont Jacob and the pont Gauss-Sedel methods assocated wth A for jj and jj are convergent for any ntal guess. Proof. A s rreducbly dagonally domnant by Lemma 3.4. The result follows from Theorem 3.4, pp. 73 of Varga [7]. 2 4 Symmetrzaton of the oecent Matrx Theorem 3.5 establshes the convergence property of the pont Jacob and pont Gauss-Sedel methods wth the fourth-order compact scheme when the problem s duson-domnated. To analytcally show the convergence of classcal teratve methods wth ths scheme for larger cell Reynolds number, followng Elman and Golub [2], we rst show that, under certan crcumstances, A s symmetrzable by a real dagonal smlarty transformaton. 3

7 However, we are not aware of any analytcal result to prove that any of the classcal teratve methods converge wth these fourth-order compact schemes. A rgorous justcaton s always desrable n spte of the fact that numercal experments have been successfully conducted, In ths paper, we gve some condtonal convergence results for some classcal teratve methods usng the fourth-order compact scheme developed by Gupta et al [4]. Although our results are lmted, they are a rst step towards the convergence analyss of such teratve methods for the fourth-order approxmaton schemes. Ths paper s organzed as follows: In Secton 2, we gve the stencl of the fourth-order compact scheme. In Secton 3 we prove the convergence of the pont Jacob and pont Gauss-Sedel methods wth the fourth-order scheme for the duson-domnated case (dened below). In Secton 4 we symmetrze the coecent matrx and gve a bound for the lne Jacob teraton matrx when the coecent matrx s symmetrzable. onclusons are gven n Secton 5. 2 The Fourth-Order ompact Scheme Let h = =(n+) be the unform meshsze. The nte derence formula for a grd pont (x; y) whch s denoted by `u ' nvolves the eght neghborng mesh ponts, whch are denoted by u ; = ; 2 : : :; 8. The dscrete values f ; = ; ; : : :; 4; are labeled smlarly. The dscretzaton results n a lnear system (for detals, see [3]): Au = f; (2) where A = (a ;j ) n 22n 2 s a square matrx, whch s usually nonsymmetrc and non-postve dente. Each equaton of (2) s of the form: 8X = u = h2 2 [(f 4 + f 3 + f 2 + f + 8f ) + p(f f 3 ) + q(f 2 f 4 )]; (3) where the coecents ; = ; : : :; 8, are descrbed by the computatonal A ( )( + ) 2( + )2 2 ( + )( + ) 2( ) ( + ) 2 2 ( )( ) 2( ) 2 2 ( + )( ) A : (4) Here = ph=2 and = qh=2 are referred to as the cell Reynolds numbers [2]. When maxfjj; jjg, we say that the lnear system (2) (and the dscretzed boundary value problem ()) s dusondomnated, otherwse t s convecton-domnated. The numercal experments conducted n [3] showed that classcal teratve methods wth ths scheme converge for any values of p and q. We also showed numercally n [8] that the multgrd method wth ths scheme converges for all values of p and q even when they are functons of x and y. 3 onvergence for the Duson-Domnated ase Lemma 3. The coecents of the nne-pont stencl (4) satsfy for all values of and. 2; = ; 2; 3; 4; (5) Proof. Drect vercaton. 2 2

8 On onvergence of Iteratve Methods for a Fourth-Order Dscretzaton Scheme 3 Jun Zhang Department of Mathematcs, The George Washngton Unversty, Washngton, D 252 May 29, 996 Abstract We prove that, under certan condtons, some classcal teratve methods converge for the lnear system resultng from a fourth-order compact dscretzaton of the convecton-duson equaton. Key words: Lnear systems, teratve methods, fourth-order compact dscretzaton scheme, convectonduson equaton. Introducton We consder the two-dmensonal constant coecent convecton-duson equaton u xx + u yy + pu x + qu y = f(x; y); (x; y) 2 ; u(x; y) = g(x; y); (x; y) () where s a smooth convex doman n R 2. Ths equaton often appears n the descrpton of transport phenomena. The magntudes of p and q determne the rato of convecton to duson. When equaton () s dscretzed usng central derences, the resultng scheme yelds a vepont formula and has a truncaton error of order h 2. lasscal teratve methods, e.g., Jacob and Gauss-Sedel methods, for solvng the resultng system of lnear equatons do not converge when the convectve terms domnate and the cell Reynolds number (dened below) s greater than a certan constant. Although the upwnd scheme s stable for all cell Reynolds number, t s only of rst-order accuracy. Recently, there has been growng nterest n developng fourth-order nte derence schemes for the convecton-duson equaton (and the Naver-Stokes equatons) whch gve hgh accuracy approxmatons, see [, 3, 4, 5, 6] and the references theren. In partcular, Gupta et al [3] proposed a fourth-order compact nte derence scheme for solvng () and showed numercally that the scheme s both hghly accurate and computatonally ecent. lasscal teratve methods wth ths scheme have been shown numercally to converge for all values of p and q [3]. In [4], ths compact scheme was extended to solve the convecton-duson equaton wth varable coecents. The new scheme has also been shown numercally to have a truncaton error of order h 4 and good numercal stablty for large values of p(x; y) and q(x; y). 3 Ths paper has been accepted for publcaton n Appled Mathematcs Letters.