JOURNAL OF COMPUTATIONAL PHYSICS 140, 370 399 (1998) ARTICLE NO. CP985899 A Three-Pont Combned Compact Derence Scheme Peter C. Chu and Chenwu Fan Department o Oceanography, Naval Postgraduate School, Monterey, Calorna 93943 E-mal: chu@nps.navy.ml Receved February 12, 1997; revsed December 3, 1997 A new three-pont combned compact derence (CCD) scheme s developed or numercal models. The major eatures o the CCD scheme are: three pont, mplct, sxth-order 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 twn-trdagonal (or calculatng dervatves) and trple-trdagonal (or solvng partal derence equaton wth the CCD scheme) methods are also presented. Besdes, the CCD scheme has sxth-order accuracy at perodc boundares and th-order accuracy at nonperodc boundares. The possblty o extendng to a three-pont eghth-order 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 ourth-order 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 ourth-order algorthm [6]; a ourth-order compact derence scheme or nonunorm grds [7]; ourth-order and sxthorder compact derence schemes or the staggered grd [8]; an early orm o the sxth-order 0021-9991/98 $25.00 Copyrght c 1998 by Academc Press All rghts o reproducton n any orm reserved. 370
3-POINT CCD SCHEME 371 combned compact derence scheme [9]; compact nte derence schemes wth a range o spatal scales [10]; and an upwnd th-order compact scheme [11]. These schemes are characterzed by (a) 5-pont sxth-order, (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 ourth-order accuracy at both nteror and boundary. Under the assumpton that the dervatve operator admts a summaton-by-parts ormula then the SAT method s stable n the classcal sense and s also tme-stable. For 2D vortcty-stream uncton ormulaton, E and Lu [14, 15] proposed a nte derence scheme wth ourth-order accuracy at both nteror and boundary. Queston arses: can we construct a scheme (1) workng or any derental equaton and (2) wth hgh-order accuracy at both nteror and boundary? A new three-pont sxth-order combned compact (CCD) scheme s such a scheme wth the ollowng eatures: (a) 3-pont sxth-order, (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 (twn-trdagonal solver) and or solvng PDEs (trple-trdagonal solver). Furthermore, we use the one-dmensonal convecton-duson equaton and two-dmensonal 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 1 + + β 1 h + δx δx +1 δx 1 +1 1 = a 1 2h ( +1 1 ) ( δ 2 ) +α 2 (( δ 2 ) + +1 = a 2 h 2 ( +1 2 + 1 ) ( δ 2 ) 1 )... 1 (( ) ( ) ) 1 δ δ + β 2 + 2h δx +1 δx 1 (2.1)
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 sxth-order CCD scheme. 2.2. 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 sxth-order 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 ( +1 + 1 ) + h 16 ( +1 1 ) H (x ) = 3 h ( 2 +1 2 + 1 ) 9 8h ( +1 1 ) + 1 8 ( + 1 + 1 ) H (3) (x ) = 15 4h ( 3 +1 1 ) + 15 4h ( 2 +1 + 1 ) 3 4h ( +1 1 ) H (4) (x ) = 36 h ( 4 +1 2 + 1 ) + 21 h ( 3 +1 1 ) 3 h ( 2 +1 + 1 ) H (5) (x ) = 45 2h ( 5 +1 1 ) 45 2h ( 4 +1 + 1 ) + 15 2h ( 3 +1 1 ) H (6) (x ) = 360 h ( 6 +1 2 + 1 ) 225 h ( 5 +1 1 ) + 45 h ( 4 +1 + 1 ). (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 ( +1 + 1 ) + h 16 ( +1 1 ) = 15 1 8 2h ( +1 1 ) 1349 7781760 (7) h 6 9 8h ( +1 1 ) 1 8 ( +1 + 1 ) + = 3 1 h ( 2 +1 2 + 1 ) 1 20160 (8) h 6 (2.6)
3-POINT CCD SCHEME 373 whch are the schemes or computng the rst-order and second-order dervatves at the grd pont x, respectvely. Thus, the CCD scheme wth sxth-order accuracy can be wrtten by (( ) ( ) ) ( ) 7 δ δ δ + + h (( δ 2 ) ( δ 2 ) ) 16 δx +1 δx 1 δx 16 +1 1 = 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 1 8 +1 1 (2.7) = 3 h 2 ( +1 2 + 1 ) (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 sxth-order 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 sxth-order CCD scheme, the truncaton errors n (2.6) 1349 7781760 (7) h 6 1.73 10 4 (7) h 6, 1 20160 (8) h 6 4.9 10 5 (8) h 6 are qute small. Another benet o usng CCD scheme s the exstence o a global Hermtan polynomal wth contnuous rst- and second-order dervatves at each grd pont. We wll descrbe t n Appendx 1. 2.3. Error Estmaton We compare the truncaton errors between the CCD scheme wth current generalzed schemes [10] or rst-order dervatves, + α( +1 + 1 ) + β( +2 + 2 )=a +1 1 +b +2 2 +c +3 3 2h 4h 6h (2.9) and the second-order dervatves, + α( +1 + 1 ) + β( +2 + 2 ) = a +1 2 + 1 h 2 + b +2 2 + 2 4h 2 + +3 2 + 3 9h 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 sxth-order schemes. For
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) 2nd-order central 0 0 1 0 0 (2.12) Standard Padé scheme 1 4 (2.12) 6th-order central 0 0 (2.12) 6th-order trdagonal (2.12) 6th-order pentadagonal 1 3 17 57 0 0 1 144 3 2 3 2 14 9 90 57 0 0 3 5 1 9 1 10 0 0 (2.7) 6th-order CCD / / / / / 1 3! (3) h 2 1 5! (5) h 4 36 1 7! (7) h 6 0 4 1 7! (7) h 6 100 1 19 7! (7) h 6 1349 1544 1 7! (7) h 6 Second (2.13) 2nd-order central 0 0 1 0 0 2 1 4! (4) h 2 (2.13) Standard Padé scheme 1 10 (2.13) 6th-order central 0 0 (2.13) 6th-order trdagonal (2.13) 6th-order pentadagonal 2 11 12 97 0 0 1 194 6 5 3 2 12 11 120 97 0 0 3 5 3 11 1 10 0 0 0 18 5 1 6! (6) h 4 72 1 8! (8) h 6 184 1 11 8! (8) h 6 2672 1 97 8! (8) h 6 (2.8) 6th-order 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 sxth-order central scheme, 4.6 tmes smaller than usng the sxth-order trdagonal (compact) schemes, and 6.0 tmes smaller than usng the sxth-order pentadagonal (compact) scheme. The truncaton error o the second dervatve usng the CCD scheme s about 36 tmes smaller than usng the sxth-order central scheme, 8.4 tmes smaller than usng the sxth-order trdagonal scheme (compact), and 13.8 tmes smaller than usng the sxth-order pentadagonal scheme (compact). Comparng the CCD scheme wth the second-order 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 th-order accurate or the PDE wth the CCD scheme (see Secton 5). A CCD scheme wth eghth-order accuracy wll be presented n Appendx 2. 3. 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
3-POINT 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 rst-order and second-order 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 rst-order and second-order 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 3 1 + 2α 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)
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 + 1 8 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] 24 + 20 cos w + cos 2w (3.17) w (w) = 81 48 cos w 33 cos 2w 48 + 40 cos w + 2 cos 2w. (3.18) Among varous derence schemes, the moded wavenumbers o the rst-order derencng w (Fg. 1a) and o the second-order derencng w (Fg. 1b) o the CCD scheme are closest to the exact wavenumber w. In multdmensonal problems the phase error o rst-order 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, 50 50 are plotted. Here, we also see that the CCD scheme shows mprovement.
3-POINT CCD SCHEME 377 FIG. 1. Fourer analyss o error or dervatve approxmaton: (a) second-order central scheme; (b) standard Padé scheme; (c) sxth-order central scheme; (d) sxth-order trdagonal scheme; (e) sxth-order pentadagonal scheme; () combned compact scheme; (g) exact derentaton. 4. CCD FOR DERIVATIVE CALCULATIONS The prevous secton shows that the sxth-order 3-pont CCD scheme s more accurate than any other sxth-order scheme ncludng ordnary compact schemes. Nevertheless, snce the CCD scheme s mplct and combnes computaton between the rst-order and secondorder derences, we should compute and jontly and globally. An ecent and mplct CCD solver s desgned to calculate the rst-order and secondorder derences. Snce CCD s a 3-pont 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. 4.1. Non-Perodc Boundares At both boundares, x = x 1 and x = x N+1, we propose a ourth-order one-sded CCD scheme nstead o the two-sded scheme to keep 3-pont structure, ( ) ( ) ( δ δ δ 2 ) + α 1 + β 1 h δx 1 2 = 1 h (a 1 1 + b 1 2 + c 1 3 ) (4.1)
378 CHU AND FAN ( δ 2 ) ( δ 2 ) ( ) δ h + α 2 h + β 1 2 2 ( ) ( ) ( 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 2 1 + b 2 2 + 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 rst-order derence, represented by (4.1) and (4.3), has a truncaton error o 22 5! (5) h 4. The second-order 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 well-posed 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 second-order 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 twn-trdagonal structure and can be drectly solved by two steps: twn-orward elmnaton and twn-backward 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.
3-POINT CCD SCHEME 379 5. 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 trple-trdagonal solver or solvng CCD FDE. 5.1. Nonperodc Boundares Consder a one-dmensonal 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 th-order 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)
380 CHU AND FAN The addtonal let boundary condton wth th-order accuracy s then (Appendx 5) ( ) ( ) ( δ δ δ 2 ) ( δ 2 ) 14 + 16 + 2h 4h + 1 δx 1 1 2 h (31 1 32 2 + 3 ) = 0 (5.7) and the addtonal rght boundary condton wth th-order accuracy s wrtten as ( ) ( ) ( δ δ δ 2 ) ( δ 2 ) 14 + 16 2h + 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 rst-order dervatve calculaton (2.7), and j = 3 corresponds to the second-order 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 trple-trdagonal structure and can be solved n two steps: trple-orward elmnaton and trple-backward substtuton (Appendx 6). 5.2. Perodc Boundares For perodc boundares (4.9), the global CCD system (5.9) s well-posed 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 7. 6. EXAMPLES The CCD scheme proposed here s a three-pont scheme wth sxth-order accuracy. Usually a three-pont scheme (e.g., central derence scheme) has only second-order accuracy.
3-POINT 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 second-order central derence (SCD) scheme on: (a) truncaton error, (b) horzontal resoluton, and (c) CPU tme. 6.1. One-Dmensonal Convecton Duson Equaton Consder a one-dmensonal 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 Sparc-20 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 0.38 10 7. 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 0.2 10 6, the SCD scheme needs 3600 grd cells; however, the CCD
382 CHU AND FAN FIG. 2. Comparson between the CCD and SCD schemes n one-dmensonal 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 (0.28 10 2 s) than usng the SCD scheme (0.32 10 1 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 4.37 10 7. 6.2. 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
3-POINT CCD SCHEME 383 TABLE 2 Comparson between the CCD and SCD Schemes n One-Dmensonal Convecton Duson Equaton Error range Features CCD SCD Ra 0.36 0.83 10 4 Cell number 7 200 Average error 0.3649 10 4 0.8292 10 4 1.22 CPU tme (s) 0.0015 0.001833 0.27 0.35 10 5 Cell number 10 1000 Average error 0.2734 10 5 0.343 10 5 4.42 CPU tme (s) 0.002 0.008833 0.23 0.26 10 6 Cell number 14 3600 Average error 0.2395 10 6 0.2577 10 6 11.3 CPU tme (s) 0.002833 0.032 0.37 0.38 10 7 Cell number 18 9400 Average error 0.3747 10 7 0.3779 10 7 24.2 CPU tme (s) 0.0035 0.08483 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 β-eect n the ocean dynamcs. Under these condtons Stommel derved an equaton or the streamuncton ψ, ( 2 ) x + 2 + α ( ) π = γ sn 2 y 2 x b y, (6.6) FIG. 3. Ocean basn dmensons and the coordnate system.
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 2 4 + The physcal parameters are selected as [23] ( ) π 2, B= α b 2 α 2 4 + 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 = 0.3 10 7 m 2 s 2, R = 0.6 10 3 ms 1. The parameter β s taken as 0 or the case wthout the β-eect case, and t s taken as 10 11 m 1 s 1 or the case wth the β-eect case. 6.2.1. 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 x-axs to obtan ntermedate varables, j,(δ /δx),j, and (δ2 / ),j, ( δ 2 + 1 8 ),j +α ( δ 2 δy 2 ) k (( ) 7 δ 16 δx +1,j 15 8 ( ) δ δx,j,j 1 + + 9 8 y 6 y 2,j = s,j 3 ( k y 2,j+1 +,j 1 k ) 1 + 8 ( (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)
3-POINT CCD SCHEME 385 and (b) teraton along the y-axs 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 1 3 1 ( 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 10 6. corr (k+1) = 6.2.2. Case 1: Wthout the β-eect, 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 Sparc-20 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 0.68 10 4, the SCD scheme needs
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.
3-POINT CCD SCHEME 387 TABLE 3 Comparson between the CCD and SCD Schemes n Stommel Ocean Model (beta = 0) Error range Features CCD SCD Ra 0.86 0.9 10 4 Cell number 9 9 50 50 Average error 0.866 10 4 0.894 10 4 27.0 CPU tme (s) 3.10 83.8 0.76 0.77 10 4 Cell number 10 10 100 100 Average error 0.766 10 4 0.761 10 4 271.7 CPU tme (s) 4.6 1250 0.68 0.69 10 4 Cell number 14 14 150 150 Average error 0.685 10 4 0.68 10 4 356.8 CPU tme (s) 16.2 5780 22,500 grd cells; however, the CCD scheme requres only 196 grd cells. The CPU rato between usng SCD and CCD schemes (Ra) s 356.8. 6.2.3. Case 2: Wth the β-eect For ths case, β = 10 11 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 Sparc-20 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 = 10 11 m 1 s 1.
388 CHU AND FAN FIG. 7. Perormance o the CCD and SCD schemes n Stommel ocean model (beta = 10 11 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 0.73 10 4, 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 254.87. 7. CONCLUSIONS (1) From ths study, t can be stated that the three-pont sxth-order 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 sxth-order polynomal whch not only satses the FDE at all the grd nodes ncludng boundary ponts but also the boundary condtons.
3-POINT CCD SCHEME 389 TABLE 4 Comparson between the CCD and SCD Schemes n Stommel Ocean Model (beta = 10 11 m 1 s 1 ) Error range Features CCD SCD Ra 0.20 0.24 10 2 Cell number 14 14 50 50 Average error 0.236 10 2 0.204 10 2 1.98 CPU tme (s) 8.12 16.1 0.22 0.24 10 3 Cell number 19 19 150 150 Average error 0.238 10 3 0.225 10 3 78.79 CPU tme (s) 14.9 1174 0.73 0.74 10 4 Cell number 27 27 250 250 Average error 0.73 10 4 0.735 10 4 254.87 CPU tme (s) 33.9 8640 (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 sxth-order schemes. The truncaton error o the rst dervatve usng the CCD scheme s about 41.2 tmes smaller than usng the sxth-order central scheme, 4.6 tmes smaller than usng the sxth-order trdagonal (compact) scheme, and 6.0 tmes smaller than usng the sxth-order pentadagonal (compact) scheme. The truncaton error o the second dervatve usng the CCD scheme s about 36 tmes smaller than usng the sxth-order central scheme, 8.4 tmes smaller than usng the sxth-order trdagonal scheme (compact), and 13.8 tmes smaller than usng the sxth-order pentadagonal scheme (compact). Comparng the CCD scheme wth the second-order 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 sxth-order accuracy at all grd ponts ncludng boundary nodes. For nonperodc boundares, the CCD scheme has sxth-order accuracy at all nteror grd ponts, ourth-order accuracy n the dervatve computaton, and th-order accuracy n the FDE solutons at the boundary nodes. (4) Both twn-trdagonal and trple-trdagonal 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 eghth-order CCD scheme. APPENDICES Appendx 1: Global Hermtan Polynomal The rst-order and second-order CCD derences are obtaned mplctly and globally by the two jont equatons (2.7) and (2.8). A twn-trdagonal technque was developed to