A ThreePoint Combined Compact Difference Scheme


 Ezra Bryan
 3 years ago
 Views:
Transcription
1 JOURNAL OF COMPUTATIONAL PHYSICS 140, (1998) ARTICLE NO. CP A ThreePont Combned Compact Derence Scheme Peter C. Chu and Chenwu Fan Department o Oceanography, Naval Postgraduate School, Monterey, Calorna Emal: Receved February 12, 1997; revsed December 3, 1997 A new threepont combned compact derence (CCD) scheme s developed or numercal models. The major eatures o the CCD scheme are: three pont, mplct, sxthorder accuracy, and ncluson o boundary values. Due to ts combnaton o the rst and second dervatves, the CCD scheme becomes more compact and more accurate than normal compact derence schemes. The ecent twntrdagonal (or calculatng dervatves) and trpletrdagonal (or solvng partal derence equaton wth the CCD scheme) methods are also presented. Besdes, the CCD scheme has sxthorder accuracy at perodc boundares and thorder accuracy at nonperodc boundares. The possblty o extendng to a threepont eghthorder scheme s also ncluded. c 1998 Academc Press 1. INTRODUCTION The grd spacngs ( x, y) n most ocean numercal models are not small. For example, a global ocean model s consdered havng hgh resoluton when a horzontal grd s (1/8), approxmately 14.5 km. For such large grd spacng, use o hghly accurate derence scheme becomes urgent. For example, McCalpn [1] used ourthorder derencng to reduce pressure gradent error n σ coordnate ocean models. The trend toward hghly accurate numercal schemes o partal derental equatons (PDE) has recently led to a renewed nterest n compact derence schemes. Concurrently, Adam [2], Hrsh [3], and Kress [4] have proposed Hermtan compact technques usng less nodes (three nstead o ve) at each grd pont to solve PDE. Later on, as ponted out by Adam [5], the truncaton errors are usually our to sx tmes smaller than the same order noncompact schemes. Snce then, much work has been done n developng compact schemes or varous applcatons, such as: an mplct compact ourthorder algorthm [6]; a ourthorder compact derence scheme or nonunorm grds [7]; ourthorder and sxthorder compact derence schemes or the staggered grd [8]; an early orm o the sxthorder /98 $25.00 Copyrght c 1998 by Academc Press All rghts o reproducton n any orm reserved. 370
2 3POINT CCD SCHEME 371 combned compact derence scheme [9]; compact nte derence schemes wth a range o spatal scales [10]; and an upwnd thorder compact scheme [11]. These schemes are characterzed by (a) 5pont sxthorder, (b) much lower accuracy at nodes adjacent to boundares, and (c) no requrement on PDE to be satsed at boundares. Several recent work emphaszes on the mprovement o boundary accuracy. For hyperbolc system, Carpenter et al. [12, 13] ntroduced a smultaneous approxmaton term (SAT) method that solves a lnear combnaton o the boundary condtons and the hyperbolc equatons near the boundary. Ths method provdes ourthorder accuracy at both nteror and boundary. Under the assumpton that the dervatve operator admts a summatonbyparts ormula then the SAT method s stable n the classcal sense and s also tmestable. For 2D vortctystream uncton ormulaton, E and Lu [14, 15] proposed a nte derence scheme wth ourthorder accuracy at both nteror and boundary. Queston arses: can we construct a scheme (1) workng or any derental equaton and (2) wth hghorder accuracy at both nteror and boundary? A new threepont sxthorder combned compact (CCD) scheme s such a scheme wth the ollowng eatures: (a) 3pont sxthorder, (b) comparable accuracy at nodes adjacent to boundares, and (c) requrement on PDE to be satsed at boundares. Fourer analyss o errors s used to prove the CCD scheme as havng better resoluton characterstcs than any current (uncompact and compact) scheme. Two mplct solvers or the CCD scheme are also proposed or calculatng varous derences (twntrdagonal solver) and or solvng PDEs (trpletrdagonal solver). Furthermore, we use the onedmensonal convectonduson equaton and twodmensonal Stommel ocean model to llustrate the applcaton o the CCD solvers and to demonstrate the benet o usng CCD scheme. 2. CCD SCHEME 2.1. General CCD Algorthm Let the dependent varable (x) be dened on the nterval, 0 x L. Use a unorm grd, 0 = x 1 < x 2 < x 3 < < x N < x N+1 =L wth a spacng h = x +1 x = L/N. Let the dependent varable (x) at any grd pont x and two neghborng ponts x 1 and x +1 be gven by, 1, and +1 and let ts dervatves at the two neghborng ponts x 1 and x +1 be gven by 1, 1,..., (k) 1 and +1, +1 s to relate,,,..., (k) to the two neghborng ponts: 1, 1, 1 and +1, +1, +1,..., (k) +1,,..., (k) +1. The essence o the CCD scheme,..., (k) ( ) (( ) ( ) ) (( δ δ δ δ 2 ) ( δ 2 ) ) +α β 1 h + δx δx +1 δx = a 1 2h ( +1 1 ) ( δ 2 ) +α 2 (( δ 2 ) + +1 = a 2 h 2 ( ) ( δ 2 ) 1 )... 1 (( ) ( ) ) 1 δ δ + β 2 + 2h δx +1 δx 1 (2.1)
3 372 CHU AND FAN and to compute,,..., (k) by means o the values and dervatves at the two neghborng ponts. Movng rom the one boundary to the other, CCD orms a global algorthm to compute varous dervatves at all grd ponts. In ths paper we only dscuss the sxthorder CCD scheme Local Hermtan Polynomal Let H (x) be a local Hermtan polynomal dened on the closed nterval [x 1, x +1 ], representng the varable at x and and ts dervatves, at the two neghborng ponts x 1, and x +1, H (x 1 ) = 1, H (x ) =, H (x +1 ) = +1, H (x 1) = 1, H (x +1) = +1, H (x 1) = 1, H (x +1) = +1. (2.2) Expand H (x) nto Taylor seres n the neghborhood o x wth sxthorder accuracy H (x) = H (x ) + H (x )x + H (x ) x 2 + H (3) (x ) x 3 + H (4) (x ) 2! 3! 4! + H (5) (x ) x 5 + H (6) (x ) x 6. (2.3) 5! 6! The seven coecents n (2.3) are determned by the seven equatons n (2.2), x 4 H (x ) = 15 16h ( +1 1 ) 7 16 ( ) + h 16 ( +1 1 ) H (x ) = 3 h ( ) 9 8h ( +1 1 ) ( ) H (3) (x ) = 15 4h ( ) h ( ) 3 4h ( +1 1 ) H (4) (x ) = 36 h ( ) + 21 h ( ) 3 h ( ) H (5) (x ) = 45 2h ( ) 45 2h ( ) h ( ) H (6) (x ) = 360 h ( ) 225 h ( ) + 45 h ( ). (2.4) The kth dervatve at the grd pont x s approxmately gven by Substtuton o (2.5) nto (2.4) leads to (k) (x ) H (k) (x ). (2.5) 7 16 ( ) + h 16 ( +1 1 ) = h ( +1 1 ) (7) h 6 9 8h ( +1 1 ) 1 8 ( ) + = 3 1 h ( ) (8) h 6 (2.6)
4 3POINT CCD SCHEME 373 whch are the schemes or computng the rstorder and secondorder dervatves at the grd pont x, respectvely. Thus, the CCD scheme wth sxthorder accuracy can be wrtten by (( ) ( ) ) ( ) 7 δ δ δ + + h (( δ 2 ) ( δ 2 ) ) 16 δx +1 δx 1 δx = 15 16h ( +1 1 ) whch s or the rst dervatve calculaton, and (( ) ( ) ) 9 δ δ 1 (( δ 2 ) ( δ 2 ) ) ( δ 2 ) + + 8h δx +1 δx (2.7) = 3 h 2 ( ) (2.8) whch s or the second dervatve calculaton. Comparng (2.7) wth (2.1), we nd that the parameters n (2.1) or the sxthorder scheme should be α 1 = 7 16, β 1= 1 16, a 1 = 15 8, α 2= 1 8, β 2= 9 4, a 2=3. For the sxthorder CCD scheme, the truncaton errors n (2.6) (7) h (7) h 6, (8) h (8) h 6 are qute small. Another benet o usng CCD scheme s the exstence o a global Hermtan polynomal wth contnuous rst and secondorder dervatves at each grd pont. We wll descrbe t n Appendx Error Estmaton We compare the truncaton errors between the CCD scheme wth current generalzed schemes [10] or rstorder dervatves, + α( ) + β( )=a b c h 4h 6h (2.9) and the secondorder dervatves, + α( ) + β( ) = a h 2 + b h h 2, (2.10) where the parameters α, β, a, b, c take derent values or varous schemes (Table 1). The comparson o truncaton errors s lsted n the last column n Table 1. We nd that the CCD scheme has the smallest truncaton error among varous sxthorder schemes. For
5 374 CHU AND FAN TABLE 1 Truncaton Errors n Varous Derence Schemes or the Frst and Second Dervatve Calculatons Parameter Dervatve approxmaton Eq. Scheme a b c Truncaton error Frst (2.12) 2ndorder central (2.12) Standard Padé scheme 1 4 (2.12) 6thorder central 0 0 (2.12) 6thorder trdagonal (2.12) 6thorder pentadagonal (2.7) 6thorder CCD / / / / / 1 3! (3) h 2 1 5! (5) h ! (7) h ! (7) h ! (7) h ! (7) h 6 Second (2.13) 2ndorder central ! (4) h 2 (2.13) Standard Padé scheme 1 10 (2.13) 6thorder central 0 0 (2.13) 6thorder trdagonal (2.13) 6thorder pentadagonal ! (6) h ! (8) h ! (8) h ! (8) h 6 (2.8) 6thorder CCD / / / / / 2 1 8! (8) h 6 example, the truncaton error o the rst dervatve usng the CCD scheme s about 41.2 tmes smaller than usng the sxthorder central scheme, 4.6 tmes smaller than usng the sxthorder trdagonal (compact) schemes, and 6.0 tmes smaller than usng the sxthorder pentadagonal (compact) scheme. The truncaton error o the second dervatve usng the CCD scheme s about 36 tmes smaller than usng the sxthorder central scheme, 8.4 tmes smaller than usng the sxthorder trdagonal scheme (compact), and 13.8 tmes smaller than usng the sxthorder pentadagonal scheme (compact). Comparng the CCD scheme wth the secondorder central derence (SCD) scheme (most commonly used n ocean models), truncaton errors or both rst and second dervatves are more than our orders o magntude smaller. Another good eature o the CCD scheme s that the CCD scheme uses the same ormulaton at all grd ponts except at the boundares, where some addtonal boundary treatment s ormulated. These addtonal schemes at the boundares are thorder accurate or the PDE wth the CCD scheme (see Secton 5). A CCD scheme wth eghthorder accuracy wll be presented n Appendx FOURIER ANALYSIS OF ERRORS Fourer analyss o errors s commonly used to evaluate varous derence schemes, descrbed extensvely n Swartz and Wendro [16], Olger and Kress [17], Vchnevetsky
6 3POINT CCD SCHEME 375 and Bowles [18], Roberts and Wess [19], Fromm [20], Orszag [21, 22], and Lele [10]. As ponted out by Lele [10], Fourer analyss provdes an eectve way to quanty the resoluton characterstcs o derencng approxmatons. For the purpose o Fourer analyss the dependent varable (x) s assumed to be perodc over the doman [0, L] o the ndependent varable,.e., 1 = N+1 and h = L/N. The dependent varable may decomposed nto Fourer seres, (x) = k=n/2 k= N/2 ˆ k e (2πkx/L), (3.1) where = 1. It s convenent to ntroduce a scaled wavenumber w = 2πkh/L = 2πk/N, and a scaled coordnate s = x/h. The Fourer modes n terms o these are smply exp(ws). The exact rstorder and secondorder dervatves o (3.1) generate a uncton wth exact Fourer coecents ˆ k = w ( h ˆ k, ˆ w 2 k h) = ˆ k. However, the Fourer coecents o the dervatves obtaned rom the derencng scheme mght not be the same as the exact Fourer coecents,.e., ( ( ˆ k ) d= w h ˆ k, ( ˆ w ) 2 k ) d= ˆ h k, where w = w (w) and w = w (w) are the moded wavenumber (both real numbers) or the rstorder and secondorder derencng. The smaller the derence between the exact and moded wavenumbers, the better the derence scheme. Accordng to Lele [10], the moded wavenumbers o the current generalzed derence schemes (2.9) and (2.10) are w (w) = a sn w + b 2 sn 2w + c sn 3w α cos w + 2β cos 2w (3.2) and w 2a(1 cos w) + b 2c (1 cos 2w) + (1 cos 3w) 2 9 (w) =, (3.3) 1 + 2α cos w + 2β cos 2w respectvely. For the CCD schemes (2.7) and (2.8), the moded wavenumbers w and w can be calculated jontly as ollows: (x) = k ˆ k e (w(x/h)) (3.4) (x) = k ˆ k e(w(x/h)) (3.5) (x) = k ˆ k e(w(x/h)) (3.6)
7 376 CHU AND FAN and [ (x)] d = k [ (x)] d = k (ˆ k ) de (w(x/h)) (3.7) ( ˆ k ) de (w(x/h)) (3.8) (x + h) = k ˆ k e (w(x/h)) e w (3.9) (x h) = k ˆ k e (w(x/h)) e w (3.10) [ (x + h)] d = k [ (x h)] d = k [ (x + h)] d = k [ (x h)] d = k (ˆ k ) de (w(x/h)) e w (3.11) (ˆ k ) de (w(x/h)) e w (3.12) (ˆ k ) de (w(x/h)) e w (3.13) (ˆ k ) de (w(x/h)) e w. (3.14) Substtuton o (3.4) (3.14) nto (2.7) (2.8), we have 7 8 [cos w + 1]w sn w(w ) 2 = 15 sn w 8 (3.15) 9 4 (sn w)w [1 14 ] cos w (w ) 2 = 6[cos w 1]. (3.16) Solvng (3.15) (3.16), we have w (w) = 9 sn w[4 + cos w] cos w + cos 2w (3.17) w (w) = cos w 33 cos 2w cos w + 2 cos 2w. (3.18) Among varous derence schemes, the moded wavenumbers o the rstorder derencng w (Fg. 1a) and o the secondorder derencng w (Fg. 1b) o the CCD scheme are closest to the exact wavenumber w. In multdmensonal problems the phase error o rstorder derencng scheme appear n the orm o ansotropy [10, 18], (C p ) d(w, θ) w (w, θ)/w = (cos θ)w (w cos θ)+(sn θ)w (w sn θ). (3.19) w Fgure 1c shows polar plots o phase speed ansotropy o varous schemes or rst dervatve approxmatons. The phase speed or wavenumber (magntude) w/π = 1 50, 5 45,..., 50 50, are plotted. Here, we also see that the CCD scheme shows mprovement.
8 3POINT CCD SCHEME 377 FIG. 1. Fourer analyss o error or dervatve approxmaton: (a) secondorder central scheme; (b) standard Padé scheme; (c) sxthorder central scheme; (d) sxthorder trdagonal scheme; (e) sxthorder pentadagonal scheme; () combned compact scheme; (g) exact derentaton. 4. CCD FOR DERIVATIVE CALCULATIONS The prevous secton shows that the sxthorder 3pont CCD scheme s more accurate than any other sxthorder scheme ncludng ordnary compact schemes. Nevertheless, snce the CCD scheme s mplct and combnes computaton between the rstorder and secondorder derences, we should compute and jontly and globally. An ecent and mplct CCD solver s desgned to calculate the rstorder and secondorder derences. Snce CCD s a 3pont scheme, the derence calculaton at x needs to use,, and at the two neghborng ponts x 1 and x +1. At the two boundares x 1 and x N+1, some specc treatment should be ncluded n the CCD scheme NonPerodc Boundares At both boundares, x = x 1 and x = x N+1, we propose a ourthorder onesded CCD scheme nstead o the twosded scheme to keep 3pont structure, ( ) ( ) ( δ δ δ 2 ) + α 1 + β 1 h δx 1 2 = 1 h (a b c 1 3 ) (4.1)
9 378 CHU AND FAN ( δ 2 ) ( δ 2 ) ( ) δ h + α 2 h + β ( ) ( ) ( δ δ δ 2 ) + α 1 β 1 h δx N+1 δx N N ( δ 2 h where ) N+1 ( δ 2 + α 2 h ) N β 2 ( δ δx ) N = 1 h (a b c 2 3 ) (4.2) = 1 h (a 1 N+1 +b 1 N +c 1 N 1 ) (4.3) = 1 h (a 2 N+1 + b 2 N + c 2 N 1 ), (4.4) α 1 = 2, β 1 = 1, a 1 = 7/2, b 1 =4, c 1 = 1/2, α 2 =5, β 2 = 6, a 2 =9, b 2 = 12, c 2 = 3. At the boundares, the rstorder derence, represented by (4.1) and (4.3), has a truncaton error o 22 5! (5) h 4. The secondorder derence, represented by (4.2) and (4.4), has a truncaton error o 14 5! (5) h 4. The accuracy at both boundares can be urther mproved to th or sxth order. The global CCD system, consstng o (4.1) and (4.2) or = 1, (2.7) and (2.8) or = 2, 3, 4,...,N, and (4.3) and (4.4) or = N + 1, s a wellposed system snce t has 2(N + 1) equatons wth 2(N + 1) unknowns: (δ /δx),(δ 2 / ), = 1, 2, 3,...,N, N+1. We may wrte the 2(N + 1) equatons (4.1) (4.4), (2.7), and (2.8) nto a more general orm (global CCD system), ( ) ( ) ( ) ( a j δ (1) + a j δ δx (2) + a j δ 1 δx (3) + b j δ 2 ) δx (1) +1 1 ( + b j δ 2 ) ( (2) + b j δ 2 ) (3) = s j, j = 1, 2, (4.5) wth +1 a j 1 (1) = b j 1 (1) = a j N+1 (3) = b j N+1 (3) = 0, j = 1, 2, (4.6) representng the our boundary equatons (4.1) (4.4). Here, j = 1 corresponds to the rstorder dervatve computaton (2.7), and j = 2 corresponds to the secondorder dervatve computaton (2.8). The two varables s 1 and s 2 are source terms. The 2(N + 1) 2(N + 1) coecent matrx o (4.5) has a twntrdagonal structure and can be drectly solved by two steps: twnorward elmnaton and twnbackward substtuton (see Appendx 3). For perodc boundares, we have 4.2. Perodc Boundares 0 = N, 1 = N+1, 0 = N, 1 = N+1, 0 = N, 1 = N+1. (4.7) Thus, the global CCD system, consstng o (2.7) and (2.8) or = 1, 2, 3,...,N, s wellposed snce t has 2N equatons wth 2N unknowns: (δ /δx),(δ 2 / ), = 1, 2, 3,...,N. The coecent matrx and related algorthm are lsted n Appendx 4.
10 3POINT CCD SCHEME CCD FOR SOLVING FINITE DIFFERENCE EQUATIONS (FDE) Any PDE dscretzed by the CCD scheme (called here the CCD FDE) can only be solved globally snce the CCD scheme s mplct. Unlke any other schemes, the CCD FDE solver requres the satsacton o the FDE not only on the nteror ponts, but also on the boundary nodes. Benets o such a treatment are to decrease the truncaton errors near the boundares as well as to ncrease the global accuracy. Here, we propose a trpletrdagonal solver or solvng CCD FDE Nonperodc Boundares Consder a onedmensonal derental equaton, a 1 (x) d dx +a 2(x) d2 dx 2 +a 0(x)() = s(x), 0 x L, (5.1) wth general boundary condtons d 1 (x) (x) + d 0 (x) (x) = c(x) at x = 0; x = L, (5.2) whch s the Drchlet boundary condton when d 0 = 1, d 1 = 0 and the Neumann boundary condton when d 0 = 0, d 1 = 1. The correspondng FDE can be wrtten as ( ) ( δ δ 2 ) a 1 () + a 2 () + a δx 0 () = s, = 1, 2,...,N +1, (5.3) and the boundary condtons become d l 1 ( ) ( ) δ δ + d0 l δx 1 = c l, d1 r + d0 r 1 δx N+1 = c r. (5.4) N+1 Notce that we appled the FDE (5.3) not only to the nteror ponts but also to the two boundary ponts (x 1 and x N+1 ). At each nteror grd node (2 N) we have three equatons [(5.3), (2.7), and (2.8)] wth three unknown varables,(δ/δx),(δ 2 / ). However, we have only two equatons [(5.3) and (5.4)] at both boundares but three unknowns: 1,(δ/δx) 1,(δ 2 / ) 1 or the let boundary, and N+1,(δ/δx) N+1, (δ 2 / ) N+1 or the rght boundary. To close the system we need an extra condton or both the let and rght boundares. The addtonal boundary condtons are obtaned by constructng a new thorder polynomal, P(x) = P 0 + P 1 x + P 2 x 2 + P 3 x 3 + P 4 x 4 + P 5 x 5. (5.5) For the let boundary, the sx coecents o P(x) can be obtaned by P(x 1 ) = 1, P(x 2 ) = 2, P(x 3 ) = 3, P (x 1 ) = 1, P (x 2 ) = 2, P (x 2 ) = 2 (5.6)
11 380 CHU AND FAN The addtonal let boundary condton wth thorder accuracy s then (Appendx 5) ( ) ( ) ( δ δ δ 2 ) ( δ 2 ) h 4h + 1 δx h ( ) = 0 (5.7) and the addtonal rght boundary condton wth thorder accuracy s wrtten as ( ) ( ) ( δ δ δ 2 ) ( δ 2 ) h + 4h δx N+1 δx N N+1 N 1 h (31 N+1 32 N + N 1 ) = 0. (5.8) Thus, we establsh three equatons or all grd ponts (nteror and boundary) wth three unknowns,(δ/δx),(δ 2 / ), = 1, 2,...,N+1. We may wrte the 3(N + 1) equatons (2.7), (2.8), (5.3), (5.4), (5.7), (5.8) nto a more general orm (global CCD FDE system), ( ) ( ) ( ) ( a j δ (1) + a j δ δx (2) + a j δ 1 δx (3) + b j δ 2 ) ( δx (1) + b j δ 2 ) +1 (2) 1 + b j (3) ( δ 2 ) +1 + c j (1) 1 + c j (2) + c j (3) +1 = s j, (5.9) where = 1, 2, 3,...,N+1 and j = 1, 2, 3. The superscrpt j ndcates derent equatons used at each grd pont: j = 1 corresponds to FDE (5.3), j = 2 corresponds to the rstorder dervatve calculaton (2.7), and j = 3 corresponds to the secondorder dervatve calculaton (2.8). For all the nteror and boundary ponts, the coecents o (5.9) satsy a 1 (1) = a1 (3) = b1 (1) = b1 (3) = c1 (1) = c1 (3) = 0. (5.10) For the two boundares, the coecents o (5.9) satsy a j 1 (1) = b j 1 (1) = c j 1 (1) = 0, a j N+1 (3) = b j N+1 (3) = c j N+1 (3) = 0, j = 1, 2, 3. (5.11) Thus, the coecent matrx o (5.9) ndcates a trpletrdagonal structure and can be solved n two steps: trpleorward elmnaton and trplebackward substtuton (Appendx 6) Perodc Boundares For perodc boundares (4.9), the global CCD system (5.9) s wellposed snce t has 3N equatons wth 3N unknowns:,(δ/δx),(δ 2 / ), = 1, 2, 3,...,N. The coecent matrx and the related algorthm are lsted n Appendx EXAMPLES The CCD scheme proposed here s a threepont scheme wth sxthorder accuracy. Usually a threepont scheme (e.g., central derence scheme) has only secondorder accuracy.
12 3POINT CCD SCHEME 381 Two examples are used n ths secton to show the advantage o usng ths new threepont scheme. Comparson s made between the CCD scheme and the secondorder central derence (SCD) scheme on: (a) truncaton error, (b) horzontal resoluton, and (c) CPU tme OneDmensonal Convecton Duson Equaton Consder a onedmensonal convecton duson equaton, wth the boundary condtons a(x)ψ + b(x) dψ ψ dx c(x)d2 =d(x), 0 x π, (6.1) dx2 I the coecent unctons n (6.1) are taken as ψ(0)=0, ψ(π) = 0. (6.2) a(x) = 1, b(x) = 1, c(x) = 1, d(x) = cos x + 2 sn x, 0 x π, (6.3) Eq. (6.1) has an analytcal soluton, ψ (an) (x) = sn(x). (6.4) We solved (6.1) numercally wth both CCD and SCD schemes under varous horzontal resolutons, and we recorded the CPU tme (a SUN Sparc20 was used) or each run. Comparng the numercal results wth the analytc soluton (6.4), we obtan the truncaton errors o the two schemes or the gven resoluton (represented by number o cells). We dene an averaged relatve error (err av )by err av =, j, j (an) j x y, j. (6.5), j x y Thus, we have a data set consstng o truncaton error, CPU tme, and cell number or the two schemes. The relatonshp between the cell number (N) and err av (Fg. 2a) or the CCD scheme (sold curve) and the SCD scheme (dashed curve) shows that or the same err av the cell number would be much smaller n the CCD scheme than n the SCD scheme. In other words, we may use a much coarser resoluton or the CCD scheme than or the SCD scheme the same accuracy s requred. For example, the CCD scheme needs only 18 cells when err av s around However, or the same accuracy, the SCD scheme requres 9400 cells (see Table 2). The relatonshp between the CPU tme and the averaged relatve error (Fg. 2b) or the CCD scheme (sold curve) and the SCD scheme (dashed curve) shows that or the same err av the CPU tme s much shorter n the CCD scheme than n the SCD scheme. Such strkng eatures can also be observed n Table 2. When the relatve truncaton errors are on the order o , the SCD scheme needs 3600 grd cells; however, the CCD
13 382 CHU AND FAN FIG. 2. Comparson between the CCD and SCD schemes n onedmensonal convecton duson equaton: (a) cell number versus average error; (b) CPU tme versus average error. Here sold curves denote the CCD scheme and the dashed curves represent the SCD scheme. scheme requres only 14 grd cells. The CPU tme s also more than an order o magntude smaller usng the CCD scheme ( s) than usng the SCD scheme ( s). The rato o CPU between usng SCD and CCD schemes (Ra), called the CPU rato here, s around 24.2 when the truncaton errors are on the order o Stommel Ocean Model Stommel [23] desgned an ocean model to explan the westward ntenscaton o wnddrven ocean currents. Consder a rectangular ocean wth the orgn o a Cartesan coordnate system at the southwest corner (Fg. 3). The x and y axes pont eastward and northward, respectvely. The boundares o the ocean are at x = 0,λand y = 0, b. The ocean s consdered as a homogeneous and ncompressble layer o constant depth D when at rest. When currents occur as n the real ocean, the depth ders rom D everywhere by a small
14 3POINT CCD SCHEME 383 TABLE 2 Comparson between the CCD and SCD Schemes n OneDmensonal Convecton Duson Equaton Error range Features CCD SCD Ra Cell number Average error CPU tme (s) Cell number Average error CPU tme (s) Cell number Average error CPU tme (s) Cell number Average error CPU tme (s) perturbaton. Due to the ncompressblty, a streamuncton ψ s dened by u = ψ y, ψ v= x, where u and v are the x and y components o the velocty vector. The surace wnd stress s taken as F cos(πy/b). The component rctonal orces are taken as Ru and Rv, where R s the rctonal coecent. The Corols parameter s also ntroduced. In general t s a uncton o y. The lattudnal varaton o,β=d/dy, s called the βeect n the ocean dynamcs. Under these condtons Stommel derved an equaton or the streamuncton ψ, ( 2 ) x α ( ) π = γ sn 2 y 2 x b y, (6.6) FIG. 3. Ocean basn dmensons and the coordnate system.
15 384 CHU AND FAN wth the boundary condtons (0, y)= (λ, y)= (x,0)= (x,b)=0. (6.7) Here, the two parameters α and γ are dened by α = Dβ R, γ = Fπ Rb. The analytcal soluton o (6.6) wth the boundary condtons (6.7) s gven by ( ) b 2 ( ) π (pe = γ sn π b y Ax +qe Bx 1 ), (6.8) where A = α 2 + α The physcal parameters are selected as [23] ( ) π 2, B= α b 2 α p = ( 1 e Bλ)/( e Aλ e Bλ), q = 1 p. ( ) π 2 b (6.9) λ = 10 7 m, b = 2π 10 6 m, D = 200 m, F = m 2 s 2, R = ms 1. The parameter β s taken as 0 or the case wthout the βeect case, and t s taken as m 1 s 1 or the case wth the βeect case Computatonal Algorthm Use a unorm grd, 0 = x 1 < x 2 < <x Nx <x Nx +1=λ, and 0 = y 1 < y 2 < < y Ny <y Ny +1=b wth grd spacng x = x +1 x = λ/n x and y = y j+1 y j = b/n y. For smplcty and no loss o generalty, we assume that the cell number n both the x and y drectons are the same, N x = N y = N. The alternatng drecton mplct (ADI) method s used or solvng FDE. The teraton k to k + 1 can be separated nto two parts: (a) teraton along the xaxs to obtan ntermedate varables, j,(δ /δx),j, and (δ2 / ),j, ( δ ),j +α ( δ 2 δy 2 ) k (( ) 7 δ 16 δx +1,j 15 8 ( ) δ δx,j,j y 6 y 2,j = s,j 3 ( k y 2,j+1 +,j 1 k ) ( (δ δy ( ) δ ) + δx 1,j ) k,j 1 ( ) δ δx,j ( ) ) δ k δy,j+1 x 16 (( δ 2 ) +1,j ( δ 2 δy 2 ( δ 2 ) k ),j+1 (6.10) 1,j 1 2 x ( +1, j 1, j ) = 0 (6.11) ) 9 8 x (( ) δ δx +1,j ( ) δ ) 1 (( δ 2 δx 1,j 8 ) +1,j ( δ 2 + ) 1,j ) ( δ 2 + ),j 3 1 x 2( +1, j 2, j + 1, j ) = 0 (6.12)
16 3POINT CCD SCHEME 385 and (b) teraton along the yaxs to obtan varables at the next teraton k + 1,,j k+1, (δ /δx), k+1 j, and (δ 2 / ),j k+1, ( δ 2 ) k+1 6 ( ) δ δy 2,j x 2 k+1,j =s,j α 3 δx,j x 2 ( +1, j + 1, j )+ 1 ( δ 2 ) 8 +1, j + 1 ( δ 2 ) + 9 (( ) δ ( ) δ ) (6.13) 8 1, j 8 x δx 1,j δx +1,j 7 16 ( (δ 15 8 δy ) k+1 +,j+1 ( ) ) δ k+1 + δy,j 1 ( ) δ k+1 y δy,j 16 ( (δ 2 δy 2 ) k+1, j+1 ( δ 2 δy 2 ) k+1, j 1 1 ( k+1, j+1 2 y k+1, j 1) = 0 (6.14) 9 8 y ( (δ δy ( δ 2 + δy 2 ) k+1,j+1 ) k+1,j ( ) ) δ k+1 1 δy,j 1 8 ( (δ 2 δy 2 ) k+1,j+1 ( δ 2 + δy 2 ) k+1,j ( k+1 y 2,j+1 2 k+1,j +,j 1) k+1 = 0. (6.15) Such an teratve process stops when the correcton at the teraton k + 1, s smaller than corr (k+1) = Case 1: Wthout the βeect, j, k j x y, (6.16) x y, j k+1, j, k j The condton β = 0 leads to α = 0 n (6.6). The analytcal soluton o (6.6) becomes = γ ( ) b 2 ( π sn )(1 π b y 1 e π b λ e π b λ b x e π b λeπ e π b λ 1 e π b λ e π b λe ) π b x ) ) (6.17) whch s depcted n Fg. 4. We solved (6.6) numercally wth both CCD and SCD schemes under varous horzontal resolutons, and we recorded the CPU tme (a SUN Sparc20 was used) or each run. Comparng the numercal results wth the analytc soluton (6.17), we obtan the truncaton errors o the two schemes or varous resolutons (represented by the number o cells). The relatonshp between N and err av (Fg. 5a) or the CCD scheme (sold curve) and the SCD scheme (dashed curve) shows that or the same err av the cell number (N) would be much smaller or the CCD scheme than or the SCD scheme. Ths s to say that we may use a much coarser resoluton or the CCD scheme than or the SCD scheme or the same accuracy. The relatonshp between the CPU tme and the averaged relatve error (Fg. 5b) or the CCD scheme (sold curve) and the SCD scheme (dashed curve) shows that or the same err av the CPU tme s much shorter n the CCD scheme than n the SCD scheme. Table 3 lsts err av, cell number, CPU tme or the two schemes, and CPU rato (Ra). When the relatve truncaton errors are on the order o , the SCD scheme needs
17 FIG. 4. Streamuncton (m 2 /s) obtaned rom Stommel ocean model wth beta = 0. FIG. 5. Perormance o the CCD and SCD schemes n Stommel ocean model (beta = 0): (a) average error versus cell number n the SCD scheme; (b) average error versus cell number n the CCD scheme; (c) CPU tme versus cell number n the SCD scheme; (d) CPU tme versus cell number n the CCD scheme.
18 3POINT CCD SCHEME 387 TABLE 3 Comparson between the CCD and SCD Schemes n Stommel Ocean Model (beta = 0) Error range Features CCD SCD Ra Cell number Average error CPU tme (s) Cell number Average error CPU tme (s) Cell number Average error CPU tme (s) ,500 grd cells; however, the CCD scheme requres only 196 grd cells. The CPU rato between usng SCD and CCD schemes (Ra) s Case 2: Wth the βeect For ths case, β = m 1 s 1 s used. The analytcal streamuncton, ψ an, s plotted n Fg. 6. We solved (6.6) numercally wth both CCD and SCD schemes under varous horzontal resolutons, and we recorded the CPU tme (a SUN Sparc20 was used) or each run. Comparng the numercal results wth the analytc soluton (6.8), we obtan the truncaton errors o the two schemes or varous gven resolutons (represented by the number o cells). The relatonshp between N and err av (Fg. 7a) or the CCD scheme (sold curve) and the SCD scheme (dashed curve) shows that or the same err av the cell number (N) would be FIG. 6. Streamuncton (m 3 /s) obtaned rom Stommel ocean model wth beta = m 1 s 1.
19 388 CHU AND FAN FIG. 7. Perormance o the CCD and SCD schemes n Stommel ocean model (beta = m 1 s 1 ): (a) average error versus cell number n the SCD scheme; (b) average error versus cell number n the CCD scheme; (c) CPU tme versus cell number n the SCD scheme; (d) CPU tme versus cell number n the CCD scheme. much smaller n the CCD scheme than n the SCD scheme. The relatonshp between the CPU tme and the averaged relatve error (Fg. 7b) or the CCD scheme (sold curve) and the SCD scheme (dashed curve) shows that or the same err av the CPU tme s much shorter n the CCD scheme than n the SCD scheme. Table 4 lsts err av, cell number, CPU tme, and Ra or the two schemes. When the relatve truncaton errors are on the order o , the SCD scheme needs 22,500 grd cells; however, the CCD scheme requres only 729 grd cells. The CPU rato between usng SCD and CCD schemes (Ra) s CONCLUSIONS (1) From ths study, t can be stated that the threepont sxthorder CCD scheme s a promsng hghly accurate method or both dervatve computaton and FDE solutons. The advantage o ths scheme s the exstence o a global sxthorder polynomal whch not only satses the FDE at all the grd nodes ncludng boundary ponts but also the boundary condtons.
20 3POINT CCD SCHEME 389 TABLE 4 Comparson between the CCD and SCD Schemes n Stommel Ocean Model (beta = m 1 s 1 ) Error range Features CCD SCD Ra Cell number Average error CPU tme (s) Cell number Average error CPU tme (s) Cell number Average error CPU tme (s) (2) Fourer analyss shows that the CCD scheme has the least error among other same order schemes, ncludng the normal compact scheme. Also, the CCD scheme has the smallest truncaton error among varous sxthorder schemes. The truncaton error o the rst dervatve usng the CCD scheme s about 41.2 tmes smaller than usng the sxthorder central scheme, 4.6 tmes smaller than usng the sxthorder trdagonal (compact) scheme, and 6.0 tmes smaller than usng the sxthorder pentadagonal (compact) scheme. The truncaton error o the second dervatve usng the CCD scheme s about 36 tmes smaller than usng the sxthorder central scheme, 8.4 tmes smaller than usng the sxthorder trdagonal scheme (compact), and 13.8 tmes smaller than usng the sxthorder pentadagonal scheme (compact). Comparng the CCD scheme wth the secondorder central derence (SCD) scheme (most commonly used n ocean models), the truncaton errors or both rst and second dervatves are more than our orders o magntude smaller. (3) For perodc boundares, the CCD scheme has sxthorder accuracy at all grd ponts ncludng boundary nodes. For nonperodc boundares, the CCD scheme has sxthorder accuracy at all nteror grd ponts, ourthorder accuracy n the dervatve computaton, and thorder accuracy n the FDE solutons at the boundary nodes. (4) Both twntrdagonal and trpletrdagonal technques are proposed or the CCD scheme or calculatng dervatves and solvng FDEs. (5) Two examples (the convecton duson model and the Stommel ocean model) show strkng results (great reducton n truncaton error and CPU tme), whch may lead to a wde applcaton o the CCD scheme n computatonal geophyscs. (6) Future studes nclude applyng the CCD scheme to nonunorm and/or staggered grd systems, as well as desgnng even hgher order schemes such as an eghthorder CCD scheme. APPENDICES Appendx 1: Global Hermtan Polynomal The rstorder and secondorder CCD derences are obtaned mplctly and globally by the two jont equatons (2.7) and (2.8). A twntrdagonal technque was developed to
What is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationHomework 11. Problems: 20.37, 22.33, 22.41, 22.67
Homework 11 roblems: 0.7,.,.41,.67 roblem 0.7 1.00kg block o alumnum s heated at atmospherc pressure such that ts temperature ncreases rom.0 to 40.0. Fnd (a) the work done by the alumnum, (b) the energy
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationIMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS 1
Nov Sad J. Math. Vol. 36, No. 2, 2006, 009 IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS Modrag S. Petkovć 2, Dušan M. Mloševć 3 Abstract. A new theorem concerned
More informationRing structure of splines on triangulations
www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAMReport 201448 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon
More informationEE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN
EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson  3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson  6 Hrs.) Voltage
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationPoint cloud to point cloud rigid transformations. Minimizing Rigid Registration Errors
Pont cloud to pont cloud rgd transformatons Russell Taylor 600.445 1 600.445 Fall 000014 Copyrght R. H. Taylor Mnmzng Rgd Regstraton Errors Typcally, gven a set of ponts {a } n one coordnate system and
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationHYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION
HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION Abdul Ghapor Hussn Centre for Foundaton Studes n Scence Unversty of Malaya 563 KUALA LUMPUR Emal: ghapor@umedumy Abstract Ths paper
More information9.1 The Cumulative Sum Control Chart
Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationA Computer Technique for Solving LP Problems with Bounded Variables
Dhaka Unv. J. Sc. 60(2): 163168, 2012 (July) A Computer Technque for Solvng LP Problems wth Bounded Varables S. M. Atqur Rahman Chowdhury * and Sanwar Uddn Ahmad Department of Mathematcs; Unversty of
More informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2  Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of noncoplanar vectors Scalar product
More informationLesson 2 Chapter Two Three Phase Uncontrolled Rectifier
Lesson 2 Chapter Two Three Phase Uncontrolled Rectfer. Operatng prncple of three phase half wave uncontrolled rectfer The half wave uncontrolled converter s the smplest of all three phase rectfer topologes.
More informationTime Domain simulation of PD Propagation in XLPE Cables Considering Frequency Dependent Parameters
Internatonal Journal of Smart Grd and Clean Energy Tme Doman smulaton of PD Propagaton n XLPE Cables Consderng Frequency Dependent Parameters We Zhang a, Jan He b, Ln Tan b, Xuejun Lv b, HongJe L a *
More informationMAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPPATBDClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
More information"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *
Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationMonte Carlo Simulation
Chapter 8 Monte Carlo Smulaton Chapter 8 Monte Carlo Smulaton 8. Introducton Monte Carlo smulaton s named ater the cty o Monte Carlo n Monaco, whch s amous or gamblng such as roulette, dce, and slot machnes.
More informationFaraday's Law of Induction
Introducton Faraday's Law o Inducton In ths lab, you wll study Faraday's Law o nducton usng a wand wth col whch swngs through a magnetc eld. You wll also examne converson o mechanc energy nto electrc energy
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationForecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network
700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School
More informationLinear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
More informationNasdaq Iceland Bond Indices 01 April 2015
Nasdaq Iceland Bond Indces 01 Aprl 2015 Fxed duraton Indces Introducton Nasdaq Iceland (the Exchange) began calculatng ts current bond ndces n the begnnng of 2005. They were a response to recent changes
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationMultivariate EWMA Control Chart
Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant
More informationUsing Series to Analyze Financial Situations: Present Value
2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated
More informationAPPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT
APPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT Toshhko Oda (1), Kochro Iwaoka (2) (1), (2) Infrastructure Systems Busness Unt, Panasonc System Networks Co., Ltd. Saedocho
More informationIDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS
IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationINTRODUCTION. governed by a differential equation Need systematic approaches to generate FE equations
WEIGHTED RESIDUA METHOD INTRODUCTION Drect stffness method s lmted for smple D problems PMPE s lmted to potental problems FEM can be appled to many engneerng problems that are governed by a dfferental
More informationNONCONSTANT SUM REDANDBLACK GAMES WITH BETDEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia
To appear n Journal o Appled Probablty June 2007 OCOSTAT SUM REDADBLACK GAMES WITH BETDEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate
More informationA machine vision approach for detecting and inspecting circular parts
A machne vson approach for detectng and nspectng crcular parts DuMng Tsa Machne Vson Lab. Department of Industral Engneerng and Management YuanZe Unversty, ChungL, Tawan, R.O.C. Emal: edmtsa@saturn.yzu.edu.tw
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationHollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA )
February 17, 2011 Andrew J. Hatnay ahatnay@kmlaw.ca Dear Sr/Madam: Re: Re: Hollnger Canadan Publshng Holdngs Co. ( HCPH ) proceedng under the Companes Credtors Arrangement Act ( CCAA ) Update on CCAA Proceedngs
More informationTime Value of Money Module
Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the
More informationFinite difference method
grd ponts x = mesh sze = X NÜÆ Fnte dfference method Prncple: dervatves n the partal dfferental eqaton are approxmated by lnear combnatons of fncton vales at the grd ponts 1D: Ω = (0, X), (x ), = 0,1,...,
More informationSection 5.4 Annuities, Present Value, and Amortization
Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today
More informationCHAPTER 9 SECONDLAW ANALYSIS FOR A CONTROL VOLUME. blank
CHAPTER 9 SECONDLAW ANALYSIS FOR A CONTROL VOLUME blank SONNTAG/BORGNAKKE STUDY PROBLEM 91 9.1 An deal steam turbne A steam turbne receves 4 kg/s steam at 1 MPa 300 o C and there are two ext flows, 0.5
More informationDamage detection in composite laminates using cointap method
Damage detecton n composte lamnates usng contap method S.J. Km Korea Aerospace Research Insttute, 45 EoeunDong, YouseongGu, 35333 Daejeon, Republc of Korea yaeln@kar.re.kr 45 The contap test has the
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationChapter 7. RandomVariate Generation 7.1. Prof. Dr. Mesut Güneş Ch. 7 RandomVariate Generation
Chapter 7 RandomVarate Generaton 7. Contents Inversetransform Technque AcceptanceRejecton Technque Specal Propertes 7. Purpose & Overvew Develop understandng of generatng samples from a specfed dstrbuton
More informationImmersed interface methods for moving interface problems
Numercal Algorthms 14 (1997) 69 93 69 Immersed nterface methods for movng nterface problems Zhln L Department of Mathematcs, Unversty of Calforna at Los Angeles, Los Angeles, CA 90095, USA Emal: zhln@math.ucla.edu
More informationCHAPTER 14 MORE ABOUT REGRESSION
CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp
More informationSafety and Reliability of Distributed Embedded Systems
Saety and Relablty o Dstrbuted Embedded Systems Techncal Report ESL 0401 Smulaton o Vehcle Longtudnal Dynamcs Mchael Short Mchael J. Pont and Qang Huang Embedded Systems Laboratory Unversty o Lecester
More informationCHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES
CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable
More informationAryabhata s Root Extraction Methods. Abhishek Parakh Louisiana State University Aug 31 st 2006
Aryabhata s Root Extracton Methods Abhshek Parakh Lousana State Unversty Aug 1 st 1 Introducton Ths artcle presents an analyss of the root extracton algorthms of Aryabhata gven n hs book Āryabhatīya [1,
More informationA Binary Particle Swarm Optimization Algorithm for Lot Sizing Problem
Journal o Economc and Socal Research 5 (2), 2 A Bnary Partcle Swarm Optmzaton Algorthm or Lot Szng Problem M. Fath Taşgetren & YunCha Lang Abstract. Ths paper presents a bnary partcle swarm optmzaton
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationModelling of Hot Water Flooding
Unversty of Readng Modellng of Hot Water Floodng as an Enhanced Ol Recovery Method by Zenab Zargar August 013 Department of Mathematcs Submtted to the Department of Mathematcs, Unversty of Readng, n Partal
More informationOn fourth order simultaneously zerofinding method for multiple roots of complex polynomial equations 1
General Mathematcs Vol. 6, No. 3 (2008), 9 3 On fourth order smultaneously zerofndng method for multple roots of complex polynomal euatons Nazr Ahmad Mr and Khald Ayub Abstract In ths paper, we present
More informationOn the Optimal Control of a Cascade of HydroElectric Power Stations
On the Optmal Control of a Cascade of HydroElectrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationMulticomponent Distillation
Multcomponent Dstllaton need more than one dstllaton tower, for n components, n1 fractonators are requred Specfcaton Lmtatons The followng are establshed at the begnnng 1. Temperature, pressure, composton,
More informationNumerical Study of Wave Runup around Offshore Structure in Waves
Journal of Advanced Research n Ocean Engneerng Journal of Advanced Research n Ocean Engneerng 2(2) (2016) 061066 http://dx.do.org/10.5574/jaroe.2016.2.2.061 Numercal Study of Wave Runup around Offshore
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationComparison of Control Strategies for Shunt Active Power Filter under Different Load Conditions
Comparson of Control Strateges for Shunt Actve Power Flter under Dfferent Load Condtons Sanjay C. Patel 1, Tushar A. Patel 2 Lecturer, Electrcal Department, Government Polytechnc, alsad, Gujarat, Inda
More informationChapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT
Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the
More informationCalculating the high frequency transmission line parameters of power cables
< ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,
More informationFinancial Mathemetics
Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,
More information2. Linear Algebraic Equations
2. Lnear Algebrac Equatons Many physcal systems yeld smultaneous algebrac equatons when mathematcal functons are requred to satsfy several condtons smultaneously. Each condton results n an equaton that
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationSection 5.3 Annuities, Future Value, and Sinking Funds
Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme
More informationComputational Fluid Dynamics II
Computatonal Flud Dynamcs II Eercse 2 1. Gven s the PDE: u tt a 2 ou Formulate the CFLcondton for two possble eplct schemes. 2. The Euler equatons for 1dmensonal, unsteady flows s dscretsed n the followng
More informationChapter 4 Financial Markets
Chapter 4 Fnancal Markets ECON2123 (Sprng 2012) 14 & 15.3.2012 (Tutoral 5) The demand for money Assumptons: There are only two assets n the fnancal market: money and bonds Prce s fxed and s gven, that
More informationThe eigenvalue derivatives of linear damped systems
Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by YeongJeu Sun Department of Electrcal Engneerng IShou Unversty Kaohsung, Tawan 840, R.O.C emal: yjsun@su.edu.tw
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationA. Te densty matrx and densty operator In general, te manybody wave functon (q 1 ; :::; q 3N ; t) s far too large to calculate for a macroscopc syste
G25.2651: Statstcal Mecancs Notes for Lecture 13 I. PRINCIPLES OF QUANTUM STATISTICAL MECHANICS Te problem of quantum statstcal mecancs s te quantum mecancal treatment of an Npartcle system. Suppose te
More informationParallel Numerical Simulation of Visual Neurons for Analysis of Optical Illusion
212 Thrd Internatonal Conference on Networkng and Computng Parallel Numercal Smulaton of Vsual Neurons for Analyss of Optcal Illuson Akra Egashra, Shunj Satoh, Hdetsugu Ire and Tsutomu Yoshnaga Graduate
More informationActuator forces in CFD: RANS and LES modeling in OpenFOAM
Home Search Collectons Journals About Contact us My IOPscence Actuator forces n CFD: RANS and LES modelng n OpenFOAM Ths content has been downloaded from IOPscence. Please scroll down to see the full text.
More informationCHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol
CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL
More informationBrigid Mullany, Ph.D University of North Carolina, Charlotte
Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte
More informationState function: eigenfunctions of hermitian operators> normalization, orthogonality completeness
Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators> normalzaton, orthogonalty completeness egenvalues and
More informationFREQUENCY OF OCCURRENCE OF CERTAIN CHEMICAL CLASSES OF GSR FROM VARIOUS AMMUNITION TYPES
FREQUENCY OF OCCURRENCE OF CERTAIN CHEMICAL CLASSES OF GSR FROM VARIOUS AMMUNITION TYPES Zuzanna BRO EKMUCHA, Grzegorz ZADORA, 2 Insttute of Forensc Research, Cracow, Poland 2 Faculty of Chemstry, Jagellonan
More informationData Broadcast on a MultiSystem Heterogeneous Overlayed Wireless Network *
JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 24, 819840 (2008) Data Broadcast on a MultSystem Heterogeneous Overlayed Wreless Network * Department of Computer Scence Natonal Chao Tung Unversty Hsnchu,
More informationNonlinear data mapping by neural networks
Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal
More informationQuestions that we may have about the variables
Antono Olmos, 01 Multple Regresson Problem: we want to determne the effect of Desre for control, Famly support, Number of frends, and Score on the BDI test on Perceved Support of Latno women. Dependent
More informationThe Effect of Mean Stress on Damage Predictions for Spectral Loading of Fiberglass Composite Coupons 1
EWEA, Specal Topc Conference 24: The Scence of Makng Torque from the Wnd, Delft, Aprl 92, 24, pp. 546555. The Effect of Mean Stress on Damage Predctons for Spectral Loadng of Fberglass Composte Coupons
More informationSDN: Systemic Risks due to Dynamic Load Balancing
SDN: Systemc Rsks due to Dynamc Load Balancng Vladmr Marbukh IRTF SDN Abstract SDN acltates dynamc load balancng Systemc benets o dynamc load balancng:  economc: hgher resource utlzaton, hgher revenue,..
More informationI. SCOPE, APPLICABILITY AND PARAMETERS Scope
D Executve Board Annex 9 Page A/R ethodologcal Tool alculaton of the number of sample plots for measurements wthn A/R D project actvtes (Verson 0) I. SOPE, PIABIITY AD PARAETERS Scope. Ths tool s applcable
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationCharacterization of Assembly. Variation Analysis Methods. A Thesis. Presented to the. Department of Mechanical Engineering. Brigham Young University
Characterzaton of Assembly Varaton Analyss Methods A Thess Presented to the Department of Mechancal Engneerng Brgham Young Unversty In Partal Fulfllment of the Requrements for the Degree Master of Scence
More informationSection B9: Zener Diodes
Secton B9: Zener Dodes When we frst talked about practcal dodes, t was mentoned that a parameter assocated wth the dode n the reverse bas regon was the breakdown voltage, BR, also known as the peaknverse
More informationAnalysis of Reactivity Induced Accident for Control Rods Ejection with Loss of Cooling
Analyss of Reactvty Induced Accdent for Control Rods Ejecton wth Loss of Coolng Hend Mohammed El Sayed Saad 1, Hesham Mohammed Mohammed Mansour 2 Wahab 1 1. Nuclear and Radologcal Regulatory Authorty,
More informationConsider a 1D stationary state diffusiontype equation, which we will call the generalized diffusion equation from now on:
Chapter 1 Boundary value problems Numercal lnear algebra technques can be used for many physcal problems. In ths chapter we wll gve some examples of how these technques can be used to solve certan boundary
More informationSorting Online Reviews by Usefulness Based on the VIKOR Method
Assocaton or Inormaton Systems AIS Electronc Lbrary (AISeL) Eleventh Wuhan Internatonal Conerence on e Busness Wuhan Internatonal Conerence on ebusness 5262012 Sortng Onlne Revews by Useulness Based
More informationNumerical Methods 數 值 方 法 概 說. Daniel Lee. Nov. 1, 2006
Numercal Methods 數 值 方 法 概 說 Danel Lee Nov. 1, 2006 Outlnes Lnear system : drect, teratve Nonlnear system : Newtonlke Interpolatons : polys, splnes, trg polys Approxmatons (I) : orthogonal polys Approxmatons
More informationSliding mode Control of Chopper Connecting Wind Turbine with Grid based on synchronous generator
Proceedngs o the 4 Internatonal onerence on Power Systems, Energy, Envronment Sldng mode ontrol o hopper onnectng Wnd Turbne wth Grd based on synchronous generator Ahmed TAHOUR, Abdel Ghan AISSAOUI, Mohamed
More informationSIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA
SIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA E. LAGENDIJK Department of Appled Physcs, Delft Unversty of Technology Lorentzweg 1, 68 CJ, The Netherlands Emal: e.lagendjk@tnw.tudelft.nl
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More informationOptimal resource capacity management for stochastic networks
Submtted for publcaton. Optmal resource capacty management for stochastc networks A.B. Deker H. Mlton Stewart School of ISyE, Georga Insttute of Technology, Atlanta, GA 30332, ton.deker@sye.gatech.edu
More informationThe Current Employment Statistics (CES) survey,
Busness Brths and Deaths Impact of busness brths and deaths n the payroll survey The CES probabltybased sample redesgn accounts for most busness brth employment through the mputaton of busness deaths,
More informationAn experimental study of interface relaxation methods for composite elliptic differential equations
Avalable onlne at www.scencedrect.com Appled Mathematcal Modellng (8) 66 www.elsever.com/locate/apm An expermental study of nterface relaxaton methods for composte ellptc dfferental equatons P. Tsompanopoulou
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationAn O(N log N) fast direct solver for partial Hierarchically SemiSeparable matrices With application to radial basis function interpolation
An O() fast drect solver for partal Herarchcally SemSeparable matrces Wth applcaton to radal bass functon nterpolaton Svaram Ambkasaran Erc Darve Receved: date / Accepted: date Ths artcle descrbes a fast
More informationLecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.
Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook
More information