VOLUME 114 MONTHLY WEATHER REVIEW MAY 1986Multigrid Solution of an Elliptic Boundary Value Problem from Tropical Cyclone TheoryPAUL E. CIESIELSKI, SCOTT R. FULTON AND WAYNE H. SCHUBERTDepartment of Atmospheric Science, Colorado State University, Fort Collins, CO 80523(Manuscript received 11 JuDy 1985, in final form 22 August 1985)ABSTRACT We consider the multigrid solution of the transverse circulation equation for a tropical cyclone. First wedevelop a standard multigrid scheme (SMG) which cycles between different levels of discretization (grids) toefficiently reduce the error in the solution on all scales. Whereas relaxation is inefficient as a solution method,it is used within the multigrid approach as a smoother to reduce the high-wavenumber errors on each grid. Theadded cost of the coarse grids is small because they contain relatively few points. The efficiency of the SMGscheme is compared to more conventional methods: Gauss-Seidel and successive-over-relaxation (SOR). Resultsshow that the SMG scheme solves to the level of truncation error 26 times faster than an optimal SOR method. In the SMG scheme an unbounded domain is approximated with a wall at a finite radius which leads tosignificant errors in the numerical solution. To better simulate an unbounded domain, we develop a secondscheme (LRMG) which naturally combines local mesh refinement with multigrid processing. In this scheme,the lateral boundary is moved far enough out that the wall boundary condition is realistic, and the grid iscoarsened in the outer region so that little additional work is required. Since finer grids are introduced onlywhere needed, the LRMG scheme maintains the usual multigfid efficiency.1. Introduction In a review (this issue), Fulton et al. (1986) havediscussed the basic concepts and techniques of multigrid methods, concentrating on their role as fast ellipticsolvers. In the present paper we apply the multigridmethod to a particular problem in tropical cyclonetheory. Our example originates from the balanced vortex equations of Eliassen (1952). Eliassen's equationshave played an important role in attempts (e.g., Ooy~ama, 1969; Sundqvist, 1970) to numerically simulatethe life cycle of tropical cyclones. The form of the balanced vortex equations which we shall use is that ofSchubert and Hack (1983). This form is based in parton a transformation of the radial coordinate from actual radius r to potential radius R, where - fR2 = rv+ - fr2, v is the tangential wind and fis the coriolisparameter. The transformed model can be used eitherprognostically to study the evolution of a vortex, ordiagnostically, as in this study, to compute the responseof a vortex to a specified forcing. In the former approach the transverse circulation equation O (OR~b*~ 0 ( &p~_~)= g OQ OoORR~b* = 0 at R -- 0, R~b* --~ 0 as R --, ooR~k* = 0 at Z = 2, R~* specified on Z = 0 (1.1)is solved for the streamfunction -* at every time step,where the potential vorticity q and the inertial stabilityc 1986 American Meteorological Societys are functions of the vortex, and the diabatic heatingQ is either specified or parameterized in terms of othermodel variables. In the diagnostic approach q, s andQ are specified functions of R and Z. The effects offrictional stresses, which we assume are confined to athin boundary layer, are incorporated in this modelthrough the bottom boundary condition; in the inviscidcase this condition becomes R~b* = 0. A variety of methods have been used in the past forsolving finite difference analogues to elliptic equationssuch as (1.1). Perhaps the most straightforward solutionmethod for such problems is Gaussian elimination.Although this direct method produces the exact solution in a finite number of computations, it becomesprohibitively expensive in terms of operations andstorage as resolution increases. Ooyama's (1969) useof Gaussian elimination to solve an analogue of (1.1)at each time step in the evolution of a tropical cyclonewas computationally feasible because his model conrained only two vertical levels. In solving for the forcedradial and vertical motions in a hurricane model, Willoughby (1979) likewise chose to use a direct methodsince his system was solved only once in a diagnosticsense. On the other hand, Sundqvist (1970) was forcedto use iterative methods to make his model simulationof tropical cyclone development with high vertical resolution computationally efficient. Employing both thesuccessive-over-relaxation (SOR) and the alternatingdirection-implicit (ADD procedures, Sundqvist foundthe latter indirect method increasingly more efficientthan SOR as the resolution of the model increased.797798 MONTHLY WEATHER REVIEW VOLUME 114 Fulton et al. (1986) describe a still faster procedurefor solving elliptic partial differential equations such'as (I. 1). This procedure, referred to as the multigrid- method, cycles between different levels of discretization(grids) to efficiently reduce the error in the solution onall scales. Relaxation is employed as a smoother toreduce the high-wavenumber errors on each grid, andthe added cost of the additional grids is small becausethey contain relatively few points. The purpose of this paper is to show how the multigrid approach can be used to solve (1.1) efficiently.In section 2 we present the discretized form of(1.1) onwhich our subsequent numerical methods are basedand formulate two standard methods for its solution.In section 3 the basic multigrid concepts are appliedto construct a standard multigrid scheme for solving(1.1). In section 4 we compare the efficiencies of threenumerical schemes (Gauss-Seidel, successive-over-relaxation and multigrid) for a particular test case. Forthis comparison, the outer boundary condition is approximated by a wall at finite radius which leads tosignificant differences between the numerical and analytical solutions. In section 5 we show how this problem can be overcome in an efficient and natural way'using the multigrid method in conjunction with localmesh refinement. Our conclusions are summarized insection 6.2. Standard solution methods To compare the efficiency of a multigrid methodagainst more conventional methods, wc now formulatetwo standard solution methods for (1.1) on the domainfl = [0,/~] x [0, 2]. Introducing the grid~a,,az -- (R/, Zk) = (jAR, kAZ) O ~< j ~< J, O ~ k ~< K ,where /xR = ~/J and AZ = ~/K, we let ~j,~ denotethe dimrete approximation m R~* at the ~dpoint (R~,Z~). The numerical methods in this paper are all basedon the usual second-order centered &fference approximationai-~/2,k~j-~,k -- (aj-l/2,k ~ ai+~/2,k)~j,k + ai+~/2,~j+~,k ~ bj,k--l/2~j,k--I -- (bj,k_l/2 + *j,k+ll2)~j,k + bj, k+l/2~j,k+l =~,k (2.1a)to (I. 1), where a~+m,~ - R/+m(AR)~ b~,~+l/2= s/,~+~/2 . (2.1b) OoAREquation (2. la) holds on the interior gridpoints of fi,that is, (0 < j < J, 0 < k < K). The correspondingboundary conditions are- j,k = 0, j=Oandj=J,O~<k~<K ~,k = 0, O~<j~<J,k=K -xI,/,k = specified, 0 ~< j ~< J, k = 0 (2.2)The form (2.1) preserves the energy constraint of(1.1),that is, the net outward flux of energy at the boundaryof the domain equals the net source of energy (integralof the right-hand side over fi). In addition, the matrixassociated with (2.1)-(2.2) is symmetric. The exact solution - of the discrete equations (2.1)(2.2) may be obtained in a finite number of operationsby Gaussian elimination. However, the operation countO(JK3) for factorization and O(JK2) for solution andstorage required [O(jK2)] may be prohibitively expensive. Thus, one usually turns to indirect (iterative)methods, which generate a sequence of approximations~, that converge to ~. These methods generally requireonly O(JK) storage locations and have the added adovantage of being self-correcting, so that any round-offerrors generated are reduced automatically in the iteration process. A simple iterative method is Gauss-Seidel (G~S) orsuccessive relaxation. In this method.the values ~),k ofan approximation ~ are modified point by point inlexicographic order to obtain a new approximation~,n-,~. The modification introduced at a point is suchthat the discrete equation is satisfied exactly at thatpoint; thus ~,.ew is defined byaj- l /2,k XIrj-- l,k -- ( aj- l/2,k + aj+ l/2,k)XI~,k + aj+l/2,k~tj+l,k q'- bj,k_l/2~,,~W_l - (bj, k-m + b/,k+,/2)~,[w + bj, k+,/2~j,~+, = Jj.k, (2.3)where the new approximation ~t"-w is used at the points(j - 1, k) and (j, k - 1) due to the lexicographic or-_dering. It is convenient to write (2.3) as ~ .e,~ = ~j,k rj,k (2.4) ~'j,kwhere, from this point on, r denotes the residualand d/,k = a/+~/2,k + a/-~/2,k + bj, k+m + bj, k-m. In thisform the convergence rate may be improved by scalingthe correction r/,k/d~,k in (2.4) by a relaxation parameter(I + o;). This leads to the successive-over-relaxation(SOR) method defined by (2.5) andMAY I986 CIESIELSKI, FULTON AND SCHUBERT 799 'new ~ -- --, (2.6) 'I'j,k = 'I'zk (1 +which reduces to the Gauss-Seidel method when co=0.3. A standard multigrid scheme In this section we present the details of a standardmultigrid scheme, subsequently referred to as SMG,for solving (2.1)-(2.2). We define the coextensive gridse, = (R'), ZP): = ~aR,, j: 0, ..., ZP =/CA k = 0, l= 1, ...,M (3.1)where ARt = l~/Jt and/~Zt = Z~/Kt. It is convenient touse the standard mesh ratio 1/2, i.e., ARt = I/2ARt_i andAZ~ = I/2AZM; then fi~ C fi2 C... fi~. We will oftenrefer to these grids as levels, with the highest level denoting the finest grid ~2M and the lowest level denotingthe coarsest grid ~21. Associated with each grid is a discrete problem L~~ = f~, (3.2)where Lt is the difference operator representing the lefthand side of (2. l a) for mesh spacing A Ri and A Zt. Thedifference operators on the coarse grids are computedusing (2. l b) directly on each level. (In problLms wherethe solution exhibits strong discontinuities, the coarsegrid operators should be constructed from the fine-gridoperator as discussed by Alcouffe et al. (1981)). In themultigrid approach, (3.2) is relaxed on a fine grid untilthe error in the approximate solution ~ is smooth; theproblem is then transferred to a coarser grid where theresidual problem (Brandt, 1984) is solved to correctthe fine-grid approximation. In the remainder of this section we describe the following key elements of the multigrid method on thegrid structure described above: the relaxation schemeused to smooth the error on a given level, the gridtransfers used to carry information from one level toanother, and the control algorithm which determineswhen to switch from one level to another.a. Relaxation scheme for interior points In a multigrid scheme the role of relaxation is tosmooth the error, i.e., to reduce its high-wavenumbercomponents. The proper choice of a smoother is criticalto the success of a multigrid scheme. Fortunately, theperformance of a smoother can be predicted by localmode analysis (Brandt, 1982) before it is incorporatedin a multigrid context, enabling us to optimize ourprocedures and debug our code. In this analysis theerror in the solution is expanded in a Fourier seriesallowing us to compute the factor ~ by which the errorin each Fourier component is reduced. Since lowwavenumbers on a given grid are represented as higherwavenumbers on a coarser grid, ~ need only be smallfor high wavenumbers. Thus we are interested in obtaining a suitable smoothing factor g, where this factoris the largest ~ over the high wavenumbers (i.e., themodes not visible on coarser grids). When a = b in (2. la), local mode analysis gives thesmoothing factor ~ = 1/~ for Gauss-Seidel relaxationwith lexicographic ordering, which is quite satisfactory(Brandt, 1977). However in the anisotropic case (i.e.,where a and b differ), the smoothing rate for GaussSeidel degrades significantly. For example, in the caseof a strong vortex analyzed by Schubert and Hack(1983), the ratio ofb/a in (2.1a) varied by a factor of~ 100, resulting in ~ = 0.98 for Gauss-Seidel relaxation.In such cases, that is where b/a is quite large or quitesmall, line relaxation provides efficient smoothing. Forexample, when b/a >~ 1 in (2. la) Z-line relaxation isappropriate; in Z-line relaxation the values along a lineof points in the Z-direction (constant R~) are replacedby new values which simultaneously satisfy all theequations on that line. Thus, Z-line relaxation is defined by - w + = Jj, t< -- [aj-i/2,~?ff?~: + aj+,/2,~:'~j+,,a] (3.3)where we assume the lines to be taken in lexicographicorder and the notation is the same as in (2.3). Eachline to be solved for results in a tridiagonal and diagonally dominant system of equations, which is easyand efficient to solve. Z-line relaxation results in ~= 0.45 for any b/a >~ 1. However, in regions whereb/a ---, 0 the smoothing rate degrades significantly, andR-line relaxation, defined analogously but for constantZ~, is more suitable. Following one sweep of R-linerelaxation with a sweep of Z-line relaxation results inalternating direction line relaxation (ADLR), which hasthe smoothing rate fi = 0.45 for any b/a. Such a schemeis needed for our purposes due to the spatial variationof b/a in a vortex of hurricane intensity as mentionedearlier. Although line relaxation as presented above is notdirectly veetorizable, by changing the order in whichlines are relaxed one obtains a vectorizable algorithmknown as Zebra relaxation. One sweep of Zebra relaxation consists of the following two stages: In the firststage the tridiagonal systems for all white (even) linesare solved simultaneously with new values of the unknown replacing the old ones; then using these updat~lvalues, the same procedure is repeated for all the black(odd) lines. Each stage involves solving a set of tridiagonal systems; since they are decoupled they can besolved in parallel. For problems, such as ours, whereb/a is large in some regions and small in others, alternating direction Zebra (ADZ) is appropriate. In theSMG scheme, one sweep of ADZ relaxation is implemented as the following four steps:800 MONTHLY WEATHER REVIEW VOLUME 114 1) R-line relaxation for all even k simultaneously, 2) R-line relaxation for all odd k simultaneously, 3) Z-line relaxation for all even j simultaneously, 4) Z-line relaxation for all odd j simultaneously. In contrast to schemes with standard ordering, thelocal mode analysis for ADZ relaxation is more complicated in that several modes are coupled by thisscheme (Stiiben and Trottenberg, 1982). We considerhere two theoretical estimates of the performance ofADZ relaxation for the discrete problem (2.1a) withconstant (frozen) coefficients. These estimates wereobtained as described in Brandt (1982). The first andsimplest measure of multigrid efficiency is the smoothing factor ~, which is the error reduction due solely tothe relaxation scheme; the second predictor, whichconsiders the effects of grid transfers as well, is the twolevel convergence factor. In Fig. 1 these predictors ofefficiency show that ADZ relaxation is robust (relativelyinsensitive to changes in the coefficients a and b) andefficient (~ < 0.23 for any ratio b/a). The third curvein this figure represents asymptotic convergence factorsobtained numerically by solving (2.1)-(2.2) with theSMG scheme. The details of the numerical results arediscussed in section 4.b. Grid transfers Multigrid methods require frequent switching between grid levels to maintain the efficiency of relaxa0.5o 0.1~-0.05tO -'~5-4 -3 -z -~ o 2 ~ 4 5 log (b/o) FIG. 1. Theoretical smoothing factor (dashed), and two-level convergence factor (solid) plotted as a function of the base 10 logarithmof b[a. The dotted curve is the observed asymptotic convergencefactor of the SMG scheme using V-cycles.tion. This switching is accomplished by the fine-tocoarse transfer of the residual r/ = f/- Lt~~ and thecoarse-to-fine interpolation of the coarse-grid correction. For the SMG scheme we use bilinear interpolationfor the coarse4o-fine transfer. After one sweep of ADZ relaxation the resulting residual field contains a large high-frequency component,since the residual is zero on every other line due fo theorder of relaxation. Therefore, to transfer a representative value of the residual to the coarse grid, one mustselect a transfer operator which uses all available information in the fine grid residual field. Full residualweighting (Brandt, 1984) accomplishes this task. Sincewe know a priori that residuals on lines for j odd arezero after an ADZ relaxation sweep, full residualweighting in this case at the point (j, k) reduces to I ~1+I 1 t'~l+l ~1+1 xr~,k = ~ -2j,2k + ~ x-2;,2~-~ + '2j,2k+l). (3.4)c. Control algorithm The final element needed to carry out multigrid processing is a cycling algorithm which controls when toswitch from one level to another. For the SMG schemewe have chosen to use an accommodative algorithmreferred to as CYCLE C (Brandt, 1977). In contrast tofixed algorithms, which switch levels after a preassignednumber of relaxation sweeps (e.g., V-cycle or W-cycle),the accommodative CYCLE C algorithm switches gridsusing internal checks based on the relative magnitudeof the residuals. The CYCLE C algorithm was used ina correction scheme mode, in which a coarse grid variable stores a correction to the finer grid approximation.With Dirichlet boundary conditions, such as (2.2), relaxation is performed only on the interior points; inthe correction scheme this means using specified valueson the boundaries of the finest grid and zeros alongthe boundaries of the coarser grids where no correctionis needed.4. Test case and numerical results In this section we apply three numerical methods(Gauss-Seidel, successive-over-relaxation, and themultigrid scheme SMG) to obtain the solution of(2.1)(2.2) for a particular test case, and examine the resultsand efficiencies in detail.a. Test case We consider a test case On the domain fl = [0, J~]x [0~ ~r], where/~ = 960 km and 2 - 16 kin. In orderto have an analytic solution to (1.1) against which wecan compare our numerical results, we consider aphysical situation in which an inviscid, resting atmosphere is heated. Under such conditions the potentialradius (R) equals the actual radius and q and s in (1.1)are constants given by 2.3 x 10-9 m3 s-2 kg-~ and 1.31986 CIESIELSKI, FULTON AND SCHUBERT 801x 10-4 m3 s-2 kg-~, respectively. For this case the analytical solution to (1.1) is ~b*(R, Z) = -'~aRe-'~ sin ~- , (4.1)where ~ = gQ2 0o = 300-K and g = 9.81 m s-2. aqOo 'Here Qo (1.5 x 10-m K s-~) and a (1.6 X 10-6 m-~)are paramete~ which specify the magnitude and efolding ~smn~ of the solution (-*), res~ctively. Fromthe field Re* (Fig. 2) one can define the componen~of the transver~ circulation as follows w*) (4.2) p OZ'p RORJ'where for simplicity we a~ume the pseudo-density isa constant ~ven by ~ = 1.1 kg m-3. The anal~ical u*and w* fields associated with (4.1) are shown in Figs.3 and 4, respectively. Using (4.1) in (1.1) we obtain theforcing~R, Z) = Qoe-a~[a-~(X2 - a2)(a-~ + R) + 3] sin ~ , (4.3)where, x = (S~'/2 ~is the inveme of the generalized Rossby radius. Knowing Q and w*, we can compute the local warning (~0/OT) from the themodynamic equation 00 0o . O~ = o - ~ pqw . (4.4)This equation shows that the local warning is the difference between the specified heating and the a~abaticcooling associated with the fomed secondau cimulation; the loc~ warning field is of pfimau interest tous since its spatial distribution is critical to intensitychanges in the vo~ex. The warning field derived anv '"' ,oo 12 ~00 :o ~ 3oo E 8 ~ 4 600 7 o "; ~, , ~'~ , '~ ,~o~S- 0 ~00 400 800 B00 R (kin)RG. 3, Anal~c~ ~ (ms-~) Mth contour inte~ 0.03 m s-~ and ~hed lines for ne~tive v~ues.alytically by using (4.2) and (4.3) in (4.4) is shown inFig. 5. From thermal wind balance the distribution inthis warming field results in cyclonic wind tendenciesat lower levels and anticyclonic tendencies above thelevel of maximum warming. Further details on howthe warming fields and corresponding wind tendencieschange as the vortex evolves are given in Schubert andHack (1982).b. Results and efiiciencies of numerical schemes We consider the numerical solution of (2.1)-(2.2)over the discretized version of f~, where AR = 10 kmand AZ = 250 m resulting in a grid ofJ = 97 by K= 65 points. For the SOR results presented in this section we used an optimal relaxation parameter o0 = 0.93.This value was determined by trial and error; in a prognostic model optimizing o~ repeatedly as the coefficientsa and b change would be expensive. Figure 6 shows the scaled discrete 12 norm lrll -- [~ (rj,k)2ARAZ]m (4,5) j,kof the residual r as a function of execution time for thethree numerical schemes. Included in this executiontime is the computation of the residuals which were,6~ .... .... ,oo[,o 0 ' 0 g00 400 600 B00 R (kin)~G. 2. Anfl~cfl s~eamfunction field R-* (10* ~ s-t) ~ contour inte~ 40 X 10~ kg s-L2oo ~300 E400500 r'~6O0?008509,50161410 8 6200400 600 BOOR (km)FIG. 4. Analytical w* (10-~ m s-t) with contour interval 10-4 m s-~.lOO2oo ~300 ~ v400500 1:~600?0085095O802 MONTHLY WEATHER REVIEW VOLUME 114161412IO 8 6 4 0o 200 400 o00 ~00 R (km)1o0200300,tOO500600?00B50950FIG. 5. Analytical warming field O0/OT (K d-l) with contour interval 0.02 K d-~.obtained dynamically (i.e., during the course of therelaxation sweep). In this figure the values plotted alongthe SOR and GS curves represent the number of relaxation sweeps; the values along the SMG curve arework units where a work unit is defined as the computational work of one ADZ sweep on the finest grid.The truncation error ~-, defined as the residual whichresults from inserting the true continuous solution R~*in the discrete equations (2.1)-(2.2), is ~ 10-8 m s-3for the test case. To solve the discrete problem to thelevel of truncation error, SMG required '-~4 work unitsor 0.033 seconds of execution time on a CRAY1 computer. In contrast the SOR scheme took 100 relaxationsweeps (0.85 seconds) to achieve this same level of accuracy, and the GS scheme required 2200 relaxationsweeps (19 seconds). Thus, in solving the test case tothe level of truncation error the SMG scheme was 26times faster than SOR and 575 times faster than GS.The curve associated with SMG becomes horizontalafter 19.9 work units indicating that the solution hasreached machine accuracy. If we were solely interested in the solution q~, solving(2.1)-(2.2) to its truncation error r would be adequate.However the warming field is often a small differencebetween two large terms in (4.4): Q and a term involving w* 'which is a derivative of the 9 field. Errors ineither of these terms are magnified in the warming field.Thus to maintain a reasonable hope for computing thewarming field to the level of accuracy of its truncationerror, the - field, in practice, is solved to several ordersof magnitude below r and w* is computed with fourthorder finite differences. A further comparison of the efficiencies of the SMGand SOR schemes is presented in Table 1, where theexecution time per grid point is listed for several caseswith different grid resolutions. These results were computed by solving (2.1)-(2.2) for the test case to a tolerance of 10-8 in the 12 norm of the residual. The ex 10's~ 2.7 - 4.3, I0qO 5f~19 E . '~u~~'lOq2' 9.2~0.8'-'"t~ ~)4. I iWSl ~15.3 . ~17,0 ~8.7 19.9 ~ ~ ~ 0 .2 ~ ~ ~ ; ; ; I I t I I I I I I I I I I - 4 .6 .8 I.O 1.2 1.4 1.6 1.8 2.0 2.2 EXECUTION TIME (seconds)FIG. 6. Scaled discrete 12 norm of the residual r (m s-3) as a function of CRAY 1 execution time in secondsfor three different numerical schemes: Gauss-Seidel (GS), optimal successive-over-relaxation (SOR), and thestandard multigrid scheme (SMG). Values labeled along the GS and SOR curves represent the number ofrelaxation sweeps; values along the SMG curve are work units, where a work unit is defined as the computational work of one ADZ sweep on the finest grid.M^-1986 ClESIELSKI, FULTON AND SCHUBERT 803 TABLE 1. Etticiencies for the SOR and SMG schemes in terms of CRAYI execution time (in sec) per grid point for five cases with different resolutions. The last column gives the ratio of the execution times for the two schemes.Execution time per grid pointNumber of grid points Grid resolution (kin) Number of grid levels in SMG SOR KM JM x K~4 ARst AZst (M) SOR SMG SMG17 17 289 60.0 1.0 4 8.59 (-5) 1.31 (-5) 6.633 33 1089 30.0 0.5 5 1.02 (-4) 8.98 (-6) 10.265 65 4225 15.0 0.25 6 1.31 (-4) 7.23 (-6) 18.197 65 6305 10.0 0.25 6 1.67 (-4) 6.49 (-6) 25.8129 65 8450 7.5 0.25 6 3.22 (-4) 5.88 (-6) 54.8ecution time per grid point for SOR increases as thegrid resolution increases, consistent with the theoreticalasymptotic convergence rate, 1 - O(h), for SOR whereh represents the mesh spacing. On the other hand, asthe resolution increases in the SMG runs, the executiontime per grid point decreases. Since multigrid smoothing rates are independent of grid resolution (Brandt,1977), this behavior reflects the efficiency Of vectorization as a function of vector length. The overhead,that is, the time required for setting up the problem,was 2% more in the case of the SMG runs reported inTable I in comparison to the SOR runs; this additionalexpense of the SMG scheme resulted from computingthe coefficients (a and b) of (2. I a) and performipg anLU decomposition of the tridiagonal systems on eachgrid level. The asymptotic convergence rate computed (over aV-cycle) for the SMG solution of the test case was 0.28.This rate is considerably larger than predicted by thetheoretical convergence factor in Fig. 1 at the corresponding ratio of b/a for the test case, in which log(b/a) = 4.75. This discrepancy can be understood by realizing that the coefficients a and b in the test case arefunctions of R, whereas the theoretical results werecomputed by assuming constant coefficients. In caseswith slowly varying coefficients (such as the test case)- local mode analysis is still valid on the finer grids wherecoefficients are locally constant; however, on coarsergrids the changes in the coefficients are more abrupt,invalidating a basic assumption in the local modeanalysis. On the other hand, the numerical results inFig. 1, which were computed with a and b constant,closely match the theoretical convergence rates. At ratios of b/a where the numerical results differ from thetheoretical estimates, the two curves can be broughtinto agreement by using W-cycles, which solve theproblem more accurately on the coarser grids; thetheoretical estimates assume the problem was solvedexactly on a coarse grid. The asymmetry in the numerical results in Fig. I reflects the order of the ADZrelaxation; first Zebra relaxation is done along Z-lines,then along R-lines. Switching this order of relaxationcauses the asymmetry to reverse itself about log(b/a)=0. Figures 7-10 present the numerically computed solution fields for comparison with their analytical counterparts in Figs. 2-5. Inspection of these fields revealsa poor numerical simulation which one would expectfrom using a wall (i.e., - = 0) at a finite radius as anouter boundary condition. This condition acts to suppress the forced secondary circulation and to concentrate the warming within the finite radius specified bythe wall; this effect is seen in the numerical warmingfield (Fig. 10) which shows significantly more warmingthan in the analytical case (Fig. 5).5. Multigrid with local refinement To improve the numerical, solution shown in Figs.7-10, let us consider the following changes in how theouter boundary condition is imposed. First, we couldapply a more realistic boundary condition such as thatdiscussed by Schubert and Hack (1983), who simulatean infinite domain by applying (at R '-- 1000 km) acondition derived from the analytical far-field solutionof (1.1). This method appears quite acceptable, although it is complicated in the sense of being nonlocal,i.e., it must be applied mode-by-mode in the vertical.Thus, it requires the solution of an associated (discretized) vertical structure problem and repeated projections onto the resulting basis functions. A second al ~4 ~ I ~oo ~2 : ~o ~ 8 ~ 6 . 4 .- ~,~ "~ 2 :\~' ~~ 0 0 200 400 600 800 R (kin)~G. 7. Streamfunction R~ (106 ~ s-~) comput~ numefic~ly~th the SMG scheme Mth contour inte~ 40 X l0s kg s-L200300 ~40O500 ~:~~00?00850950804 MONTHLY WEATHER REVIEW VOLUME 11414 14 300 300 ~ 8 ~ ~ 8 "-" 400 4006 500 ~ ~ 6 50O ~4 600 4 600~ 700 2 700 850 8500 950 0 950 o ~oo 400 ~oo soo o ~oo ~oo Boo ~oo R (km) R (km) DG. 8. Plot of u* (m s-~) computed numerically from streamfunction field in Fig. 7 with contour interval 0.03 m s-~ and dashedlines for negative values. FIG. 10. Warming field 80/3T (K d-t), with contour interval 0.05K d-~, computed numerically using (4.3) and the streamfunctionfield in Fig. 7.ternative is based on the fact that solutions of (1.1)become smoother as R increases. Thus, the lateralboundary can be moved far enough out such that- = 0 is a realistic assumption, while the grid is coarsened in the outer regions so that not much additionalcomputational work is required. In this approach westill make use of the analytical far-field solution, butonly as a rough guide to the rate at which the grid iscoarsened as R increases.a. Far field solution and radial extent of various grids' To examine the far-field solution of(1.1 ), let us consider R large enough (R >/~, say) such that g 00 'Pq --'* 0-~ O-~ ~- N2, pS ._, f 2,and OQ/OR --* O. In addition, if we make the Boussinesqapproximation and assume N is a constant, the farfield equation becomes f2 021~ vs OlORC~*l-~N2--=O for R>/~. (5.1) OR \ ROR ] OZ2~e4 86!~"' ~ 400 ~ 500 ~ 4 ~ 600 ~ ~ 700 850 0 ~'~ ..... ' ' ~ ' ~ 950 0 ~00 400 600 ~00 R (km) ~G. 9. Plot of w* (10-n m s-~) computed numerically fromstrmmfunction field in Fig. 7 with contour int~al 10-~ m s-~ anddashed lines for negative values.For the frictionless case the solution of (5.1) is~b*(R, Z) = ~ AnK~(X,R)sin mrZ (5.2)where X, = f/c,, c, = N2/mr and K~ is a modifiedBessel function of the first order. The coefficients A,are determined by the details of the solution in theinner region (R </~). For the present purpose the crucial feature of (5.2) is that the various modes decaywith radius at different rates. This is illustrated by thecharacteristic decay scales, X~I, given in the third column of Table 2, where the higher vertical modes decayfaster with R. Suppose that a single mode n is excited and that thelateral boundary/~ of our finite difference grid is placedfar enough away so that the solution decays to about10% or less of its value at R =/~. Then/~ must satisfygl(~knt~)'= ~ 0.1. (5.3)Ki(X,R)Convenient values of/~ satisfying (5.3) for different nwith ~ = 800 km are given in the last column of Table2. If, in addition to the internal modes (n > 0), an TABLE 2. Characteristics of the vertical modes in (5.2): c~ is thephase speed, 3,,-~ the characteristic decay scale, and/~ a convenientradius at which the solution has decayed to less than 10% of its valueat R =/q.n c, = --nr (m s-') h~-~ = y (kin) ~ (km)I 61.12 1220 28802 30.56 611 19204 15.28 306 14408 7.64 153 116016 3.82 77 960MAY 1986 CIESIELSKI, FULTON AND SCHUBERTTABLE 3. Grid domains and resolutions required to adequately resolve vertical modes in Eq. (5.2).805Number of grid points Extent ofResolution (kin) domain (kin)Grid level Vertical modeI resolved (n) J/ K/ Jt x Kt ARtI 0 16 3 48 320.0 8.0 48002 1 19 5 95 160.0 4.0 28803 2 25 9 225 80.0 2.0 19204 4 37 17 629 40.0 1.0 14405 8 51 33 1683 20.0 0.50 11606 16 97 65 6305 10.0 0.25 960external type response is possible d~ue to boundary-layerfrictional effects, an even larger R of about 4800 kmis needed. Assuming that eight grid points per verticalwave length are necessary to resolve a given mode, Table 3 summarizes the grid domains and resolutionsrequired for the various modes discussed above. Thesegrids are depicted in Fig. 11. In practice, of course, all vertical modes are excitedand the lateral boundary of the computational domainshould be placed at about 4800 km. A grid with uniform resolution to this distance with AR = 10 km andAZ -- 0,25 km would require 31 850 points, which isnot an attractive prospect. A more practical grid construction would include nonuniform resolution suchas would result from overlapping the grids in Fig. 11.Gi (AR)I =:SZOkm ~_~ (aZ)I = 8.0kin I I (,,R)2~ (~z)z04,I,I160 km ~4.0 krn 0 1(aR)$ = 80 krn (aZ)$ = 2.Okm 1920(aR)4 =4ok~.J(~Z)4 -' I,O km144o~R)e=20 ~~Z)5=aSkm ~4~ )0 Ii~6~ ~=oz5i ) I ' kmJ..~, m ~ ~[ , ,I izt ~:~.~ j I ~ ( ~ n, ~ ~ ,P i I 8~ ~ ~ ~ 44 n, / i '~ ~*0 oi . . t ..... 0 8~ I~ ' '~' ' '~' ' '~' ' ~ FIG. l I. Grid domains and resolutions r~uired to remlve the vettirol m~es n which result from ~e mlution of the f~-field equation(~. 1). The outer boun~ of a ~ven ~d Gt is plaid at the ra~usRt ~ven in Table 3 such that the mlution for the m~e n demysaway to 10% or le~ ofiB v~ue at R = ~.A grid constructed in this manner would provide adequate resolution where it is needed without an excessof additional grid points. In fact, for the case consideredhere the number of points exterior to G6 (i.e., the regionof finest resolution) is 9% of the number interior toG6; this compares to a 405% increase ira grid with theuniform resolution of G6 was used out to 4800 km. Computing with standard finite differences on a gridwith nonuniform mesh spacing is an awkward procedure. An advantage of the multigrid method is that itallows nonuniformity to be organized through the useof uniform grids on various levels of discretization, butwith each of these uniform grids extending over different domains. In the discussion that follows we demonstrate how the concepts of multigrid and local meshrefinement can be merged in an efficient and naturalway to obtain a numerical scheme which accommodates nonuniform resolution.b. Local refinement We now present an alternative to the SMG scheme,which will subsequently be referred to as LRMG (i.e.,local refinement with multigrid). The multigrid approach with local refinement involves relaxing (2.1)(2.2) on a certain grid G/which has uniform resolution.If convergence is slow on that grid, the problem isswitched to a coarser but noncoextensive gridhaving different but uniform resolution. On the otherhand, if the solution has converged on G/, the problemis transferred to a finer grid Gt+~ where the outerboundary values are obtained naturally by interpolatingfrom the coarser grid. This process is repeated untilthe solution has converged to a specified tolerance onthe finest grid level. Such a scheme is computationallyconvenient in that each grid has uniform resolution.Since finer grids are introduced only where needed,this approach solves the problem with the usual multigrid efficiency. The grid structure chosen for the LRMG scheme issummarized in Table 3. Due to the noncoextensivestructure of these grids, the parts of the domain on acoarse grid not covered by a finer grid must certainlycarry the full solution. This is accomplished by usingthe Full Approximation Scheme (FAS), in which the806 MONTHLY WEATHER REVIEW VOLUME 114full current approximation is stored on each grid level(as opposed to the Correction Scheme, where a coarsegrid stores a correction to the finer grid approximation!.Details of the FAS mode of multigrid processing aregiven in Fulton et al. (1986). For the LRMG scheme we have chosen to use FullMultigrid (FMG) as the control algorithm. The FMGalgorithm, unlike cycling algorithms, works itself fromthe coarsest level (l = 1) to the finest level (l = M). Inthis manner a good first approximation is achieved oneach grid in turn with minimal expense. Figure 12shows a tlowchart of the accommodative FAS-FMGalgorithm used in LRMG. In this flowchart m denotesthe current finest level, th~at is, the finest level for whichan approximate solution - has already been computed;l refers to the current working level. The differenceoperator L/ as well as the initial right hand side f/arecomputed using (2. lb) directly on each level. As in theSMG scheme, we again choose ADZ as the relaxationscheme for our current application. However, in contrast to the SMG scheme, the values inserted along theouter boundaries of the finer grids (1 > 1) representapproximate solutions interpolated from coarser grids.When transferring the problem to a coarser grid, theFMG algorithm generates a new right hand side, F/, 'atcoarse grid points which are interior to the finer grid.The transfer operators are denoted in_ Fig. 12 as follows:I~+~ (full weighting of residuals), [~+~ (injection), I~_~(bilinear interpolation) and H,~_~ (bicubic interpolation). A detailed explanation of the steps in this FMGalgorithm is given in Brandt (1979). An advantage of the FMG algorithm in Fig. 12 isthat on each grid level m > 1 it generates an appropriateconvergence tolerance e,~ internally. Setting the pa- I-~=1lFIG. 12. Accommodative FAS-FMG algorithm (after Brandt, 1979).'rameter 3' = 1 makes em an approximation to the truncation error; setting 3~ < 1 allows us to solve belowtruncation error, as discussed in section 4b. For problems where the truncation error is not known in advance this feature is very useful. When (2.1)-(2.2) wassolved for the test case described in section 4 with theLRMG scheme, the computed approximate truncationerror on the finest grid (~M) was 7.4 x 10-9 m s-3,whereas the actual truncation error (r) was 2.1 x 10-8m s-3.c. Results from the LRMG scheme We now apply the LRMG scheme to solve (2.1)(2.2) for the test case described in section 4. For theresults to be presented the parameters in the FMG algorithm were set as follows: ~ = 0.3, 3~ = 1.0, and ~= -1.0. By specifying ~ to be negative we ignore thetest for fast convergence which avoids a second (andin our case, unnecessary) relaxation sweep; one ADZsweep substantially reduces the error in high-wavenumber modes, and further relaxation on the samelevel will reduce the remaining low-wavenumbermodes very slowly. Using the LRMG scheme, the test case was solvedto the tolerance eM in 3.25 work units (0.053 seconds).In spite of the added complexity of the LRMG scheme,its computational efficiency is comparable to that oftheimpler SMG scheme. Moreover, the solution fieldsof ~, u* and w* computed from the LRMG schemeare virtually indistinguishable from the correspondinganalytical solutions (Figs. 2-4) and thus are not shownhere. A comparison of the computed warming field(Fig. 13) and its analytical counterpart (Fig. 5) revealsa slight amplitude error in the numerical field inside200 km, and a notable disparity near the boundary R= 0. The former error can be reduced by solving theproblem to a smaller tolerance (i.e., by setting 3' < 1).The latter error is the result of less accurate finite differences for w* at the j = 0 and I points; these inaccuracies in w*, although slight, are then magnified inthe warming field. In the test case solved above, the coefficients q ands were constant. We have also used the LRMG schemeto solve other problems which are more realistic. Forexample, in the case mentioned in section 3a [wheres/q in (1. I) varied by a factor of 100], LRMG solvedthe problem to the tolerance tim (1.2 x 10-9 m S-3) in5 work units (0.067 seconds). From our experience,LRMG in a prognostic mode will solve (1.1) to theapproximate truncation error ~ in 6 work units or lessfor any ratio of s/q that occurs as the vortex evolves.6. Concluding remarks The multigrid schemes described in this paper weredesigned to handle the specific features of (1 ..1), namelyits anisotropy and the unbounded domain. Theseschemes proved considerably more efficient than theM^YI986 CIESIELSKI, FULTON AND SCHUBERT 8071412 40 ~00 400 600 BOOR (km)loo~oo30040050060O?0085095O lqG. 13. Warming field O0/OT(K d-~) with contour interval 0.02K d-t obtained using (4.3) and the streamfunction field computedvia the LRMG scheme.SOR method. In solving (1.1) diagnostically, efficiencyis not a practical concern; however in a prognosticmode this equation must be solved on the order of1000 times during a typical numerical simulation of atropical cyclone. In such cases efficiency is more crucial(especially with high resolution), and we recommendthat the multigrid approach be given serious consideration. In designing the multigrid scheme with local refinement, the gross features of the far-field solution wereused as a rough guide to construct the grids. Alternatelythe choice of local refinements can be made totallyadaptively, i.e.,, as required by the numerical solutionas it evolves. This method is beyond the scope of thepresent paper, and the reader is referred to Bai andBrandt (1984) for further discussion. In this paper the multigrid approach was used todesign an efficient solver for an elliptic boundary valueproblem from tropical cyclone theory. Similar problems with variable coefficients occur in other areasof meteorology such as quasi-geostrophic, semi-geostrophic and nonlinear balance theory. We believe themultigrid approach has great potential for solving theseproblems efficiently, and we encourage others to trymultigrid.methods in such cases. Acknowledgments. The authors gratefully acknowledge the help and guidance provided by Achi Brandt,especially during his sabbatical year at the Institute forComputational Studies at Colorado State University.We are also indebted to Gerald Taylor for his theoretical analysis of the relaxation scheme and to OdiliaPanella for her help in preparing the manuscript. 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