Numerical Methods! for! Elliptic Equations-II! Multigrid! Methods! f Analytic Solution of! g = bk. da k dt. = α. ( t)

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1 umerical Methods! or! Elliptic Equatios-II! Grétar Tryggvaso! Sprig 13! Examples o elliptic equatios! Direct Methods or 1D problems! Elemetary Iterative Methods! ote o Boudary Coditios! SOR o vector computers! Iteratio as Time Itegratio! Covergece o Iterative Methods elemetary cosideratios! Multigrid methods! Fast Direct Method! Krylov Methods! Resources! Multigrid! Methods! So ar we have covered elemetary iterative methods to solve elliptic equatios. O those the SOR method is the most useul, particularly or code developmet ad debuggig.! However, cosiderable eort has bee devoted to the solutio o elliptic equatios ad curretly there exist a umber o much more eiciet methods. For codes iteded or large problems where rutime is importat you should use such a method! Multigrid methods are amog the more popular oes! Basic Idea! Suppose we wat to solve a 1-D elliptic equatio! g x A iterative process is aalogous to solvig a usteady! problem! α g t x Ad tae it to the limit! t Aalytic Solutio o! α g t x The equatio ca be solved by Fourier series! ix a ( t) e ad assume that g ca be expaded as! ix g b e Substitutig ito the equatio,! da dt e ix α ( a (t) b ) e ix

2 Solvig or each! da α a ( t) b dt Thereore,! a ( t ) b a (t) b ( a () b )e α t Rate o covergece! ~ α τ c α 1 High wave umber modes damps out aster.! e α t For iterative method to solve elliptic equatios,! there are high wave umber ad low wave umber! compoets o errors that eed to be damped out.! Cosider a domai L 1 discretized by poits! L π 1 H π This ca also be visualized by ollowig the decay o a iitial coditios composed o two waves! (x) si(πx) +si(1πx) For explicit time step, stability coditio yields! αδt 1 ; h h Δt α I the error at various wave umber decays at! α t h e where! t Δt α è! / exp( h / ) H exp L exp π h ( π h ) or π or π L H Short-wave errors! decay much aster! t 1 t 5Δt 1 t 5Δt 1 Multigrid Method the Idea! - A low wave umber compoet o a ie grid becomes! a high wave umber compoet o a coarse grid! ( π ) L / exp h - Use a coarse grid system to coverge low wave umber! compoet o the solutio rapidly! - Map it oto the ie grid system to coverge high wave! umber compoet.!

3 1D Example! Example! d dx g i +1 + Approximate by iite diereces! Solve or i! i 1 i h g i R i h g i 1 i +1 + i 1 + R i i 1 i +1 + i 1 + h g i Iterate util covergece is slow o a ie grid! 1 i +1 + i 1 + R i i 1 i +1 + i 1 + h g i givig a approximate solutio (but ot ully coverged). Write the ully coverged solutio as the approximate solutio plus a correctio:! Substitute ito! d dx + Δ g The decompositio! gives! d Δ dx + Δ + d dx g The correctio is uow, but the secod derivative o the approximate solutio is ow, alog with the source.! Solve or the correctio o a coarser grid!! Δ i O the coarser grid the correctio is the uow, but the secod derivative o the approximate solutio oud o the ier grid is ow!! The discretizatio is thereore:! Δ i Itroduce! This approach ca be geeralized so that we solve or the correctio to the correctio o eve coarser grids.! x i i, x Δ, x ΔΔ, Where the x reers to or the various correctios, depedig o which grid we are worig o! Δ +1 + Δ 1 Δ + i +1 i 1 i g 4h h Which ca be solved iteratively as beore!!! ΔΔ Δ i

4 O the -grid, write:! [ ] Δ 1 Δ + Δ + 4 h +1 1 ( g + i+1 + i 1 i ) or! where! Δ 1 [ Δ + Δ + R +1 1 ] R 4( h g + i +1 + i 1 i ) This ca be geeralized to the coarser grids! Δx 1 [ Δx +1 + Δx 1 + 4( m h g + x i +1 + x i 1 x i )] or! where! Δx 1 [ Δx +1 + Δx 1 + R ] R 4( m h g + x i +1 + x i 1 x i )! Δ! Δx i x i Oce the correctio has bee oud, the solutio o the ier grid ca be corrected:! x i x i + Δx whe grid poits overlap! x i +1 x i Δx + Δx +1 whe grid poits! do ot overlap!! Trasormig the source term to the coarser grid! c 1D multigrid example rom c MUDPACK c Solve ''(t)-1, ()(1) real r(518),(518),d,d1,s,t,u Iteger it,,,l,ll,m,,l 6 ** it4 t.1 u.7 c iput right side do 1 i1, r(i)1. 1 c iitialize s(1./)** do i1, r(i)s*r(i) m l1 ll l++- do 3 i1,l 3 (i). d1. 4 c Gauss-Seidel Iteratio 5 dd1 d1. +1 il 6 ii+1 s.5*((i-1)+(i+1)+r(i)) d1d1+abs(s-(i)) (i)s i(i.lt. ll) goto 6 write(*,*)' DIFF: ',d1 i(.lt. it) goto 5 i(d1.lt. t) goto 1 i((d1/d).lt. u) goto 5 i(.eq. ) goto 5 c coarse mesh---slow covergece ill+ 8 ll+ ii+1 i(l.gt. ll)goto 9 (i). r(i)4.*(r(l)+(l-1)-*(l)+(l+1)) go to 8 9 / write(*,*)' UMBER OF MESH & ITERVALS: ', llll++1 ll+1 go to 4 c ier mesh---aster covergece 1 i(.eq. m)goto 1 il-3 ll 11 (i)(i)+() (i+1)(i+1)+.5*(()+(+1)) ii- -1 i(.gt. l)go to 11 + write(*,*)' UMBER OF MESH & ITERVALS: ', lll- li (i+1)(i+1)+.5*(+1) go to 4 c prit aswer 1 m+1 do 13 i1,m i-1 s/loat() 13 write(*,*),s,(i) ed DIFF: DIFF: DIFF: DIFF: UMBER OF MESH ITERVALS: 3 DIFF: DIFF: DIFF: DIFF: UMBER OF MESH ITERVALS: 16 DIFF: DIFF: DIFF: DIFF: UMBER OF MESH ITERVALS: 8 DIFF: DIFF: DIFF: DIFF: UMBER OF MESH ITERVALS: 4 DIFF: DIFF: DIFF: DIFF: DIFF:.5377 DIFF: DIFF: DIFF: DIFF: DIFF: DIFF: E-5! UMBER OF MESH ITERVALS: 8 DIFF: DIFF: DIFF: DIFF: DIFF: DIFF: DIFF: E-5 UMBER OF MESH ITERVALS: 16 DIFF: DIFF: DIFF: DIFF: DIFF: DIFF: DIFF: E-5 UMBER OF MESH ITERVALS: 3 DIFF: DIFF: DIFF: DIFF: DIFF: DIFF: E-5 UMBER OF MESH ITERVALS: 64 DIFF: DIFF: DIFF: DIFF: DIFF: E ! ! D Example!

5 Suppose we are solvig a Laplace equatio:! i+ 1, i 1, L + R Δx + Δy + ad iterate util residual becomes smaller tha tolerace! Deie! + Δ Δ R i, Coverged solutio! Solutio ater -th iteratio! Correctio! Sice! ad! we have! L L + LΔ L, L i,, Δ Coarse grid correctio! Steps:!!Fie grid solutio!!è Iterpolatio oto coarse grid (restrictio)!!è Coarse grid correctio!!è Iterpolated oto ie grid (prologatio)!!è Coverge o ie grid! Two-Level Multigrid Method - (x+1, y+1); (x/+1, y/+1)! 1. Do iteratios o the ie grid ( 3~4) usig G-S! L,. I R i, < the stop.! 3. Iterpolate the residual R i, oto the coarse grid!!(restrictio, iectio)! 4.!Iteratios o the coarse grid! L Δ i,, 5.!Iterpolate the correctio Δ i, oto the ie grid!!(prologatio)! + Δ 6.!Go to 1.! Details o Prologatio! Δ i, Δ Δ Δ i, + i, i, Δ i+, Coarse Grid! Fie grid! Δ i+1, 1 ( Δ + Δ i+, ) Δ +1 1 ( Δ + Δ + ) Δ i+1, +1 1 ( 4 Δ + Δ + Δ + Δ i+1, +1 i+1, + i+, +1) 1 ( 4 Δ + Δ + Δ + Δ i+, + i+, + ) The process ca be exteded to multi-level grids! (x+1,y+1), (x/+1,y/+1), (x/4+1,y/4+1),, (3,3)! Various Strategies! Fie! Coarse! V-Cycle! exact solutio! W-Cycle! Example: 4-Level V-Cycle! Fie è Coarse! 1. Iterate o Grid 1 or times! L R. Restrictio by iectio to Grid! 3. Iterate o Grid! LΔ 4. Restrictio by iectio to Grid 3! I 3 5. Iterate o Grid 3 LΔ I R! 1 (Save Δ ) I R )! 3 3 3!!! I! R + L! Δ(! (Save Δ ) I R )! 6.!Restrictio by iectio to Grid 4! 7.!Iterate o Grid 4! LΔ 1 I1, 1 R I1, + LΔ( ) I1 R i, 3 R i, R 3) 4 3 I3 R i, I3, (, 1 ( 3,

6 Coarse è Fie! 1. Prologate rom Grid 4 to Grid 3! 3 I 4 Δ( 4) 3 3 ) Δ( 3) + I 4 ( 4) Δ( Δ 3. Iterate o Grid 3 LΔ( 3) I, (saved)! 3. Prologate rom Grid 3 to Grid! Δ( I 3 3) ) Δ( ) + I3 ( 3) Δ( Δ 1 4. Iterate o Grid LΔ I1, (saved)! 1 5. Prologate rom Grid to Grid 1! Δ( I ) 1 + I Δ( ) 6. Iterate o Grid 1! L i, 7. I < the stop. Else, repeat the etire cycle.! R i, Test Problem (Taehill, p. 17)! - Laplace equatio i a square domai! - Dirichlet coditios o our boudaries! - 5 Levels o resolutio!!9 9, 17 17, 33 33, 65 65, 19 19! - 4 Methods!!1. Covetioal Gauss-Seidel (GS)!!. Gauss-Seidel-SOR with optimal ω (GS opt!)!!3. Multigrid with -level grids (MG)!!4. Multigrid with maximum levels o grid (MGMAX)! Grid size! GS! GSω opt! MG! MGMAX! A umber o pacages exist already ad usually it is ot ecessary to write your ow multigrid solver, particularly or simple geometries.! Multigrid methods are also used to solve steady-state problems such as low over airplaes! See, or example:! MUDPACK: Multigrid Sotware or Elliptic Partial Dieretial Equatios! Fortra Code with OpeMP Directives or Shared Memory Parallelism! byjoh C. Adams! Fast Direct Methods! Although iterative methods are the domiat techique or solutios o elliptic equatios i CFD, Fast Direct Methods exists or special cases. The methods require simple domais (rectagles), simple equatios (separable), ad simple boudary coditios (periodic, or the derivative or the uctio equal to zero at each boudary)! a(x, y) x x + x + y S separable! b(x, y) y o-separable! y S

7 The ast Fourier trasorm! iverse! ˆ l l 1 l 1 ˆ l e i π l e i π l Ca be evaluated i log operatios! The Cooley-Tuey algorithm! Write! b, Δ, +1, + 1, +, +1 +, 1 4, Tae the double Fourier Trasorm! Δ,, l 1 m 1 l 1 m 1 ˆ l,m e i π ( l + m ) ˆ l,m e i π l + m e i π l + e i π l + e i π m + e i π m 4 cos π l + cos π m 4 Have! Δ, ˆl,m e i π ( l + m ) cos π l + cos π m also! but! l 1 m1 b, ˆbl,m e i π l 1 m1 ( l + m ) Δ, b, The algorithm is! 1 Fid b ˆ l,m by FFT! Fid ˆ as show beore! l,m, 3 id by FFT! Solve:! ˆ ˆb l,m l,m cos π l + cos π m As outlied the method is applicable to periodic boudaries oly. Other simple boudary coditios ca be hadled by simple chages (usig cosie or sie series).! Other ast direct methods, such as Cyclic Reductio, are based o similar ideas! See FISHPACK! subroutie hwscrt (a,b,m,mbdcd,bda,bdb,c,d,,bdcd,bdc,bdd, 1 elmbda,,idim,pertrb,ierror,w) c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c * i s h p a * c * * c * a pacage o ortra subprograms or the solutio o * c * separable elliptic partial dieretial equatios * c * (versio 3.1, october 198) * c * by * c * oh adams, paul swarztrauber ad rolad sweet * c * o * c * the atioal ceter or atmospheric research * c * boulder, colorado (837) u.s.a. * c * which is sposored by * c * the atioal sciece oudatio * c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c * * * * * * * * * purpose * * * * * * * * * * * * * * * * * * c subroutie hwscrt solves the stadard ive-poit iite c dierece approximatio to the helmholtz equatio i cartesia c coordiates: c c (d/dx)(du/dx) + (d/dy)(du/dy) + lambda*u (x,y).

8 Examples o elliptic equatios! Direct Methods or 1D problems! Elemetary Iterative Methods! ote o Boudary Coditios! SOR o vector computers! Iteratio as Time Itegratio! Covergece o Iterative Methods elemetary cosideratios! Multigrid methods! Fast Direct Method! Krylov Methods! Resources!