Appendix A. Use of Operators in ARPS

Transcription

1 A Appedix A. Use of Operators i ARPS The methodology for solvig the equatios of hydrodyamics i either differetial or itegral form usig grid-poit techiques (fiite differece, fiite volume, fiite elemet) ca be described i four basic steps. I the first step, the appropriate cotiuous equatio or set of equatios is prescribed, as show i Eq. (A.) for a simple scalar coservatio law T t =- (ut) x - (vt) y - (wt) z. (A.) Here, T = T(x,y,z,t) is a scalar, t is time, ad u, v, ad w are the velocity compoets i the x,y, ad z directios, respectively. This equatio possesses cosiderable parallel structure i that all terms o the right had side (RHS) are similar i form but differ i directio. This is more evidet if the RHS is writte usig tesor otatio T t =- (u it) x i, (A.) where a summatio i =,,3 is implied such that x = x, x = y, x 3 = z ad u = u, u = v, u 3 = w. I a explicit umerical scheme, ad i some implicit schemes, the three (elemetwise matrix) multiplicatios of u i ad T, as well as the subsequet differetiatios, ca be performed idepedetly. I the secod step of the solutio process, the appropriate cotiuous equatios are recast ito discrete form, ofte through the use of differecig ad averagig operators as a otatioal coveiece, i a maer appropriate for a give computatioal mesh ad solutio methodology. Cosider, for example, the cetered operator otatio used by Lilly (965) A(x) x = A x+ x/ + A x- x/ (A.3) CAPS - ARPS Versio

2 δ x A(x) = A x+ x/ - A x- x/ x (A.4) where A is the depedet variable at a arbitrary locatio x, which is the idepedet variable, is a positive iteger, ad x is the grid spacig. Eq. (A.4) is the discrete represetatio of a cotiuous first-order derivative A/ x, while (A.3) represets a averagig operator that has o cotiuous couterpart. Other forms of these operators with varyig defiitios are widely used (e.g., va Leer, 977). As described i the ext sectio, Eqs. (A.3) ad (A.4), alog with other fudametal operators (e.g., oe-sided spatial differece), may be used i combiatio to create a variety of higher-order expressios provided that certai commutative ad associative properties, similar but ot idetical to those i the calculus, are obeyed. Applyig (A.3) ad (A.4) to (A.) usig secod-order quadratically coservative spatial differeces o the Arakawa C-grid (e.g., Arakawa ad Lamb, 977), alog with a secod-order leapfrog time discretizatio, yields δ t T =-δ x ut x -δ y vt y -δ z wt z (A.5) where the subscripts i, j, ad k correspod to the x, y, ad z directios, respectively (e.g., x i = i x), the superscript idicates the time level, ad t = t where t is the time step. At this poit, the full structure of the goverig equatio (A.) remais itact. I the third step, Eq. (A.5) is expaded usig the rules give i (A.3) ad (A.4) to yield a set of liear algebraic equatios: T T t = - x u (T i+,j,k + T ) - u i-,j,k (T + T i-,j,k ) - y v (T i,j+,k + T ) - v i,j-,k (T + T i,j-,k ) - z w (T + + T ) - w - (T + T - ). (A.6) Fially, these algebraic equatios are cast i a appropriate computer laguage usig the subscripted arrays. For the example show here, the code might be writte i Fortra as CAPS - ARPS Versio

3 dimesio u(x,y,z),v(x,y,z),w(x,y,z),t(x,y,z) do k =,z do j =,y do i =,x T(,future) = T(,past) : -0.5*rdx*dt*(u(,ow)*(T(i+,j,k,ow)+T(i,j,k,ow))- : u(i-,j,k,ow)*(t(,ow)+t(i-,j,k,ow))) : -0.5*rdy*dt*(v(,ow)*(T(i,j+,k,ow)+T(i,j,k,ow))- : v(i,j-,k,ow)*(t(,ow)+t(i,j-,k,ow))) : -0.5*rdz*dt*(w(,ow)*(T(+,ow)+T(,ow))- : w(-,ow)*(t(,ow)+t(-,ow))) ed do ed do ed do (A.7) where future, ow, ad past are the time idices for levels +,, ad -; rdx = / x, rdy = / y, rdz = / z; dt = t; ad x, y, ad z are the umber of grid-poits i each coordiate directio. I movig from Eq. (A.5) to the FORTRAN code, much of the fuctioal (i.e., mathematical operatio) parallel structure has bee lost. For example, it is o loger possible to perform idepedetly or to separate the three products of the scalar T with the velocities u, v, ad w. The same is true for the three differetiatios. Furthermore, resemblace to the goverig equatio (A.) has bee greatly dimiished, ad the code is iheretly proe to error due to the umber of array idex maipulatios ivolved. Extesio of this code to differet or higher order schemes, though perhaps straightforward i theory, is cumbersome ad proe to error, presetig formidable challeges to users ad greatly reducig the utility of the code. The obvious solutio to the problems illustrated above is to elimiate the operator expasio step (step 3) ad make use of the operator costructs i the computer code itself. Cosider, therefore, a set of subrouties, the details of which are described i the ext sectio, that perform the discrete operatios show i Eqs. (A.3) ad (A.4). Each routie receives o iput a depedet variable ad perhaps iformatio cocerig its dimesioality ad locatio withi the grid, ad returs o output a trasformed variable accordig to the operatio performed. For example, let AVGX(iput_var, output_var) ad DIFFX(iput_var, output_var) be the Fortra couterparts of Eqs. (A.3) ad (A.4), respectively, for the x directio (similar routies exist for the y ad z directios), ad let routie AAMULT(iput, iput, output) retur as output the elemet-wise CAPS - ARPS Versio

4 product of two iput matrices iput ad iput. Assumig that the appropriate ested DO-loops are cotaied withi each operator routie, ad for clarity eglectig other argumets passed to this routie, the operator-based code for solvig Eq. (A.) ca be writte as dimesio u(x,y,z), v(x,y,z), w(x,y,z), T(x,y,z) dimesio temp(x,y,z), temp(x,y,z),temp3(x,y,z) call avgx(t, temp)! temp cotais T x call avgy(t, temp)! temp cotais T y call avgz(t, temp3)! temp3 cotais T z call aamult(u, temp, temp) call aamult(v, temp, temp) call aamult(w, temp3, temp3)! secod temp cotais U T x! secod temp cotais V T y! secod temp3 cotais W T z call diffx(temp, temp)! secod temp cotais δ x (U T x ) call diffy(temp, temp)! secod temp cotais δ y (V T y ) call diffz(temp3, temp3)! secod temp3 cotais δ z (W T z ) do k =,z do j =,y do i =,x T(,future) = T(,past) : -.*dt*(temp(,ow)+temp(,ow)+temp3(,ow)) ed do ed do ed do (A.8) A umber of importat poits are worth otig about this code relative to that foud i (A.7). First, the fudametal mathematical structure of the three types of operatios (averagig, matrix multiplyig, ad differecig) is clearly evidet. Secod, it is clear that, although the types of operatios must be performed sequetially for each pairig of variables (e.g., the matrix multiply of u ad the average of T caot occur util the average of T is available), computatios withi each type (e.g., all averagig operatios) ca be performed idepedetly ad simultaeously. I distributig this parallelism or graularity, oe might choose a fuctioal decompositio for a shared memory computer or workstatio cluster i which avgx is executed o oe processor, avgy o aother ad avgz o yet aother. No commuicatio would ever be required amog the processors sice each has a copy of the e- CAPS - ARPS Versio

5 tire array T. I a data decompositio mode for a distributed memory parallel computer, oe could load the T array across all processig odes ad the simply perform each averagig operatio idepedetly, first i x, the i y, the i z. Agai, o commuicatio is required after the computatios begi. Third, the code i Eq. (A.8) bears more of a resemblace to the goverig equatio (A.) or its discrete operator represetatio (A.3) tha does (A.7) because (A.8) is writte i a maer aalogous to the cotiuous problem usig fuctioally similar costructs. As a result, oce the operators are kow to be correct ad properly applied, the code ca be verified largely by ispectio, thereby facilitatig its debuggig, maiteace, ad correct usage by those ufamiliar with it. Implemetig higher order or more complex umerical schemes, virtually all of which ca be expressed usig some forms of operator otatio, simply ivolves usig other operators or differet combiatios thereof. Fourth, the complexity of the array idex maipulatios, all of which are extremely basic ad occur i the operator routies (all of which cotai less tha 0 lies of executable code), is hidde from the user. By redefiig a particular operator, oe ca chage literally hudreds of lies of code i a matter of miutes. Fially, the storage costs associated with the operator methodology rage from oe to two times that of a more stadard code, depedig upo the use of temporary arrays ad overlaid storage. It is our experiece that the beefits associated with the operators ad some use of temporary storage far outweigh the associated overhead, although this issue is somewhat problematic. The obvious simplicity ad apparet uiversality of the discrete operator methodology suggests its applicatio to broad classes of problems ivolvig differetial or itegral equatios. I much the same way that graphics primitives are used to build images of complex objects, the discrete operators ca, we believe, serve as the buildig blocks of complex umerical models. CAPS - ARPS Versio