Outline. Review Two Space Dimensions. Review Properties of Solutions. Review Algorithms. Finite-difference Grid

Transcription

1 Nmeric soltios o elliptic PDEs March 5, 009 Nmerical Soltios o Elliptic Eqatios Larr Caretto Mechaical Egieerig 50B Semiar i Egieerig alsis March 5, 009 Otlie Review last class Nmerical soltio o elliptic eqatios Laplace s eqatio as a eample Fiite dierece orms Iterative soltio o algebraic eqatios Sample reslts Review Properties o Soltios Cosistec trcatio error becomes zero as step sizes approach zero Stabilit errors remai boded Coverget teds to the eact soltio o the dieretial eqatio as the grid size teds towards zero Phsical realit Soltios prodce phsicall realistic reslts ccrac Ma sorces o error i Review Two Space Dimesios T T T α t ( time step, /, ) Etesio o oe space dimesio Have grid i as well as i ad time Eplicit method has almost o dierece. bt has more striget stabilit limit Caot appl Thomas algorithm directl to implicit method Deie αδt/(δ) ad αδt/(δ) merical soltios. 3 4 Old time i- Δ Fiite-dierece Grid Δ Δ - T Δ New time Node related to or earest eighbors at both time steps i CN ses all T vales at old time Δ step Δ Δ i- Δ T - i 5 Review lgorithms Eplicit T method Crak-Nicholso ( Ti Ti ) ( T T ) ( ) T T Ti ( ) T Ti T ( T T ) ( T T ) ( ) T R i i Fll implicit T T T T Ti ( ) T i i 6 ME 50B Egieerig alsis

2 Nmeric soltios o elliptic PDEs March 5, 009 Review Eplicit Stabilit mst be positive or stabilit so mst be less tha / ( T T ) ( T T ) ( ) T T i i αδt αδt ( Δ) To ct Δ ad Δ b a actor o we mst ct Δt b a actor o 4 Work icreases b a actor o 6 7 i lteratig Directio Implicit DI is techiqe or allowig se o Thomas algorithm or simple soltio t each time step, treat oe directio as eplicit ad oe as implicit Evember o time steps reqired * * * * i i α Δt ( Δ) * * * * i i α Δt ( Δ) 8 Elliptic Eqatios Soltios at a poit i the regio deped o coditios at all bodaries No ope bodaries like parabolic case Mai eamples are Laplace ad Poisso Eqatios 0 F( space) Laplace eqatio i two-dimesioal Cartesia coordiates 0 9 D Cartesia Laplace Use iite-dierece grid where i 0 iδ ad 0 Δ 0 i N ad 0 M 0 i N ad 0 M Δ ( N 0 )/N ad Δ ( M 0 )/M Mst speci bodar coditios o at all bodaries Soth 0, 0 i N Depedet North M, 0 i N variable, West i 0, 0 M ( i, ) East i N, 0 M 0 Fiite Diereces Use secod-order epressios or secod derivatives i i i ( Δ) O[( Δ) ] O[( Δ) ] β Δ Δ i O[( Δ),( Δ) ] 0 ( Δ) Fiite-dierece Eqatio Liks cetral ode to or earest eighbors Ol Δ parameter: Δ Δ i- β Δ/Δ i Δ For β, ( β ) 4 - ( ) ( β ) 0 i i β Mltipl b (Δ) ME 50B Egieerig alsis

3 Nmeric soltios o elliptic PDEs March 5, 009 Fiite-dierece Eqatio For D Cartesia Laplace eqatio with β Δ/Δ, is the average o its or earest eighbors i i 4 Cosider Dirichlet bodar coditios kow at all bodar odes Need to id (N )(M ) kow vales o at cetral odes Geeral Eqatio Most geeral case has ive diagoals, bt ca have dieret terms T T k k Occrs with variable properties Q& 0 Notatio poit is geeral coeiciet reers to a particlar ode Poit N(orth), S(oth), E(ast), W(est) reers to eighborig odes b directio Geeral eqatio show below S W P E N i i b 3 4 Solvig the Eqatios Tpicall have large mber o eqatios ormig sparse matri For Δ Δ.0 have 99 eqatios so matri has potetial coeiciets Ol (0.05%) are ozero Wat data strctre ad algorithm or hadlig sparse matrices Gass elimiatio ses storage or baded matrices Iterative methods sed or soltios 5 Iterative Soltios Simplest eamples are Jacobi, Gass- Seidel, ad Sccessive Over Relaatio Move rom iteratio to iteratio Iteratio 0 is iitial gess (ote all zero) Solve eqatio or ad se this as basis or iteratio S W E N b i i P P ' S' W ' E' N ' b i i 6 Iterative Soltios II Use sperscript () or iteratiomber Jacobi iteratio ses all old vales ) ' S ' W ' E' N ' b i i Gass Seidel ses most-recet vales ) ' S ' ) W ' ) E' b N ' i i Relaatio basis: Gass Seidel provides a correctio that ca be adsted ) ( ), GS ω ) [ ] Relaatio Methods Relaatio actor, ω, greater tha or less tha is over- or derrelaatio Uderrelaatio procres stabilit i problems that will ot coverge Overrelaatio procres speed i wellbehaved problems ), GS ) ω, GS ) W ' ) i [ ] ( ω) ( ω) ω[ E' i N ' S ' ) Relaatio Factor 7 8 ω b ' ] ME 50B Egieerig alsis 3

4 Nmeric soltios o elliptic PDEs March 5, 009 Ecel Relaatio Code I Gass-Seidel soltio to Poisso eqatio with costat sorce term ( ) ( ) β Q& 0 k Q i i β i i β ( ) ( β ) & k Q& k 0 Iterative Ecel Formla or Cell F9 (E9G9betaSqd*(F8F0)Sorce)/Deomiator 0 9 Ecel Relaatio Code II Modi previos soltio to iclde relaatio actor, ω ) ( ) ( β ) ) i i β ) ( ω) ω k Iterative Ecel Formla or Cell F9 oemisw*f9omega*(e9g9betasqd *(F8F0)Sorce)/Deomiator 0 Q& Visal Basic Relaatio Code or iter to maiter maresid 0 or i to N or to N old (i,) (i,) ( omega) * (i,) Oe set o iteratios (omitted oet page) - omega * ( N(i,) * (i,) E(i,) * (i,) S(i,) * (i,-) W(i,) * (i-,) - b(i,) ) resid abs( ( (i,) old ) / (i,) ) i resid > maresid the maresid resid ed i Visal Basic Relaatio Code II or iter to maiter maresid 0; or i to N or to N Code i red rom previos slide et et i i maresid < errtol eit or et iter i maresid > errtol the prit # Not coverged else call dootpt(, N, M ) ed i Covergig Iteratios Have three dieret reslts Correct soltio to dieretial eqatio Correct soltio iite-dierece eqatios Crret ad previos iteratio vales Iteratios shold approach correct soltio to iite-dierece eqatios Sice either correct soltio is kow, we se orm o error estimates Residal i iite-dierece eqatios Chage i iteratio vale 3 Covergig Iteratios II Take iiit orm (maimm absolte vale) or L orm (RMS error) Former easiest to code Relative Chage [ Residal] W ' ) i ) E' ) i ( ) ) ) S' ) N ' ) ' b 4 ME 50B Egieerig alsis 4

5 Nmeric soltios o elliptic PDEs March 5, 009 Eectio Times ad Errors Eamie sqare regio with zero bodar coditios at 0, ma, ad 0; two cases or ma Case : costat vale o N () Case : N () si(π) First case has discotiit or ma at 0 ad ma Use overrelaatio (SOR) with variable relaatio actors 5 Eectio Time (secods 0 Eect o Relaatio Factor o Eectio Time Sqare ( L H ) 64 b 64 Grid Zero bodar o let, right ad bottom Top bodar has (,H) si(π/l) "Other" is dieret code with (,H) grid 33 grid 6464 grid 88 grid Other code Relaatio Factor Eectio Times ad Errors II For secod order algorithm the error depeds o the bodar coditio Case : costat vale o N () Case : N () si(π) First case has discotiit or ma at 0 ad ma First case gives irst order error de to discotiit Secod case gives secod order error Eect o Iteratios o Errors Compare three error measres sig the maimm vale o the grid Tre iteratio error: dierece betwee the crret vale ad the vale od b a eact soltio o the dierece eqatios Dierece i betwee two iteratios Residal i i- - 4 Eact error is dierece betwee iteratio vale ad eact soltio 7 8 Errors Eects o Iteratios o Laplace Eqatio Errors.E00.E-0.E-0.E-03.E-04.E-05.E-06 Dierece.E-07 Residal.E-08 Iteratio error.e-09 Sqare ( L H ) Eact Error.E-0 64 b 64 Grid Zero bodar o let,.e- right ad bottom.e- Top bodar has.e-3 (,H) si(π/l).e-4.e Iteratios 9 dvaced Solvers dvaced soltio techiqes treat matri or iite-dierece eqatios Leads to dimesioal cosio Start with D grid ( ad idices) Treat as matri eqatio where kows orm a colm vector (oe-dimesioal) The coeiciets i the matri orm a twodimesioal displa Eamie small grid eample 30 ME 50B Egieerig alsis 5

6 Nmeric soltios o elliptic PDEs March 5, 009 Small Grid (N 6, M 5) i 0 i i i 3 i 4 i 5 i Bodar 3 odes 0 Comptatioal Molecle Grid Eqatios (β ) N 6 ad M 5 gives (6 )( 5 ) 0 eqatios ol eight show Diagoal strctre icorrect here N 6, M 5 Matri Strctre Real smmetric matri Zero coeiciets becase o bodar Geeral Matri Strctre Cosio abot two twodimesioal represetatios Grid has two space dimesios with (N )(M ) kowodes orms a oe dimesioal colm matri o kows (at right) Coeiciet matri has ive diagoals Right-had side has bodar vales S W P E N b i i M S Geeral Eqatio i a Matri Look at separatio betwee coeiciets N 3 coeiciets that are zero Not preset i irst N rows W Not preset i irst eqatio ad zero i eqatio N ad ever N eqatios thereater P E N 3 coeiciets that are zero Not preset i last N rows 35 N Zero i eqatio N ad ever N eqatios thereater. Not preset i last eqatio Sparse Matri Strctre 0 eqatios ca have 400 coeiciets Here each eqatio has o more tha ive coeiciets (00 possible) Bodaries give aother (N M - ) zero coeiciets (8 i this eample) Ths, we have 8 ozero coeiciets ad zeros i matri Nearl 80% o coeiciets are zero Fractio icreases or larger grids 36 ME 50B Egieerig alsis 6

7 Nmeric soltios o elliptic PDEs March 5, 009 How Sparse is the Matri? The M b N grid has (M )(N ) odes with eqatios givig (M ) (N ) possible coeiciets Withot bodaries we have ol 5 (M )(N ) ozero coeiciets Bodaries give (N M ) (M ) (N ) additioal zero coeiciets Nozero 5( N )( M ) ( N ) ( M ) Fractio ( N ) ( M ) 5 37 ( N )( M ) ( N )( M ) ( N ) ( M ) What Makes Sparseess? Each ode is coected ol to a small mber o earest eighbors Problem here has or eighbors Higher order schemes ad 3D Laplace eqatio ca have more eighbors Ca have comple coeiciets so log as mber o eighbors is limited Look at eve grid spacig as a eample o this ME 50B Egieerig alsis 7