A ThreePoint Combined Compact Difference Scheme


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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
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